Class 6 Maths - Integers

Verify: $$-(-x) = x$$ for $$x = \dfrac{2}{15}$$.

Given,

$$x=\dfrac { 2 }{1 5 } $$

Then,

$$-(-x)$$

$$=-\left( -\dfrac { 2 }{1 5 } \right) =\dfrac { 2 }{ 15 } $$

As we know that by the sign rule of multiplication: (-ve) multiplied (-ve) equals (+ve)

Hence, proved $$-(-x)=x$$.

Write the following in increasing order:
$$21, -8, -26, 85, 33, -333, -210, 0, 2011$$

Writing down the numbers on real axis and move from left to right we can see that smallest number among given numbers is $$-333$$ and largest is $$2011$$

Hence in increasing order above numbers can be written as,

$$-333,-210,-26,-8,0,21,33,85,2011$$

Write the following in decreasing order
$$85, 210, -58, 2011, -1024, 528, 364, -10000, 12$$

Writing down every number on real line and moving from right to left we can see that the highest number among given numbers is $$2011$$ and smallest is $$-10000$$

So decreasing order of given numbers is, $$2011,528,364,210,85,12,-58,-1024,-10000$$

Subtract $$-6$$ from $$3$$.

Given that

Subtract

−6−6 from

33.

writing the expression

$$3-(-6)$$

we know that $$(-) \times (-) = +$$

$$ = 3+6$$

$$ = 9$$

Find the greatest value of $$c$$ in order that the equation $$7x+9y=c$$ may have exactly six solutions in positive integers.

The greatest value of $$c=(n+1)ab-a-b=7\times7\times9-7-9=425$$

Here, $$n=$$ no of solutions

$$a $$ and $$b$$ are coefficents of $$x$$ and $$y$$ respectively

Subtract $$1732765$$ from $$3673124$$ in the scale of eight.

$$3673124$$ from scale of eight to decimal is $$4\times8^0+2\times8^1+1\times8^2+3\times8^3+7\times8^4+6\times8^5+3\times8^6=1013332$$

$$1732765$$ from scale of eight to decimal is $$5\times8^0+6\times8^1+7\times8^2+2\times8^3+3\times8^4+7\times8^5+1\times8^6=505333$$

Difference between them is $$=1013332-505333=507999$$

Now we need to represent this in scale of $$8$$ is:

When $$507999$$ is divided by $$8$$ quotient is $$63499$$ and remainder is $$7$$
When $$63499$$ is divided by $$8$$ quotient is $$7937$$ and remainder is $$3$$
When $$7937$$ is divided by $$8$$ quotient is $$992$$ and remainder is $$1$$
When $$992$$ is divided by $$8$$ quotient is $$124$$ and remainder is $$0$$
When $$124$$ is divided by $$8$$ quotient is $$15$$ and remainder is $$4$$

When $$15$$ is divided by $$8$$ quotient is $$1$$ and remainder is $$7$$

When $$1$$ is divided by $$8$$ quotient is $$0$$ and remainder is $$1$$
Hence, the difference is $$1740137$$ in scale of eight.

Solve in positive integers:
$$23x+17y+11z=130$$.

$$23x+17y+11z=130$$

Here, the maximum value of $$x=[\dfrac{130}{23}]=5$$

If $$x=5,17y+11z=15$$. This will not have any positive integral solutions.

If $$x=4,17y+11z=38$$. This will not have any positive integral solutions.

If $$x=3,17y+11z=61\Rightarrow y=1$$ and $$z=4$$.

If $$x=2,17y+11z=84\Rightarrow y=3$$ and $$z=3$$.

If $$x=1,17y+11z=107\Rightarrow y=5$$ and $$z=2$$.

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.
Which city has the highest temperature?

Cities	Temperature
Banglore	20
Ooty	15
Nainital	-3
Manali	-7
Kasauli	-9

Among the $$5$$ cities, Banglore has the highest temperature ( i.e $$20$$ ).

Simplify:
$$(-6) + (-10) + 5 + 17$$.

Use BODMAS rule,

Firstly open brackets,

$$-6-10+5+17$$

=> $$-16+22$$

=> $$6$$

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.
Write the temperatures of the cities marked on it?

From the number line temperatures (in $$^\circ C$$) of the $$5$$ cities are as:

In the number line, the number increases from left to right and decreases from left.

Cities	Temperature
Banglore	$$22$$
Ooty	$$14$$
Shimla	$$5$$
Srinagar	$$-2$$
Lahulspiti	$$-8$$

Some integers are marked on the number line. Which is the biggest and which is the smallest?

Since number Increases from left to right and decreases from right to left .

Therefore, smallest number $$ = -3 $$ and largest number $$ = 2$$.

Find the following using number line.
- $$6 + 7 + 7=? $$

Draw a Number line:- $$-6+7+7$$

for $$-$$ve No. we have to go left & for the no. we have to go right from zero.

$$\cdot$$ first we go left $$-6$$ unit after that we go right $$(17)$$ unit & again $$7$$ unit.

Ans is $$+8$$.

1167386_700532_ans_ed7d17b6a6bf48328707b65dc712cf7c.jpg

Simplify:
$$(-6) - (5) - (+2)$$.

Given: $$ (−6)−(5)−(+2)$$ .

$$\Rightarrow - 6 -5 -2 = - 13$$

The temperature of 55 cities in India on a particular day are shown on the number line given. Name the cities with temperature more than 0oC?

Cities	Temperature
Banglore	$$20^0$$ C
Ooty	$$15^0$$ C
Nainital	$$-3^0$$ C
Manali	$$-7^0$$ C
Kasauli	$$-9^0$$ C

Among the $$5$$ cities Banglore and Ooty have temperature more than $$0^0$$ C.

These $$2$$ cities have less temperature than $$0^0$$ C..

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.
Which city has the lowest temperature?

Cities	Temperature
Banglore	$$ 20^0$$ C
Ooty	$$15^0$$ C
Nainital	$$-3^0$$ C
Manali	$$-7^0$$ C
Kasauli	$$ -9^0$$ C

Among the $$5$$ cities, Kasauli has the lowest temperature.

Represent $$6,-4,0$$ and $$4$$ on a number line.

This is the representation of $$ 6 , -4 , 0 $$ and $$4$$ on a number line .

Represent the following subtractions on the number line:
$$7-2$$

For the subtraction operation, we go towards left direction on number line.

Thus, for $$(7-2)$$ we go from $$7$$ towards left by $$2$$ units

This is the representation of the following subtractions on the number line .

Represent the following addition on a number line:
$$5+(-2)$$

Hence, this is the addition of the $$2$$ numbers on a number line.

Represent the following addition on a number line:
$$5+(-7)$$

Represent the following addition on a number line:
$$5+2$$

Hence, this is the representation of the following addition on a number line.

Compute the following:
$$9+(-7)$$

To compute the following

$$ 9 + (-7) $$

As we know that $$ (+) \times (-) = (-) $$

Therefore, we have

$$ = 9 -7 $$

$$ = 2 $$

Compute the following:
$$4+(-4)$$

To compute the following

$$ 4 + (-4 ) $$

As we know that $$ (+)\times(-) = (- )$$

Therefore, we have

$$ = 4 -4 $$

$$ = 0 $$

Solve the following:
$$17-(-14)$$

Let us solve

$$ 17 - (-14) $$

As we know that $$ (-) * (-) = (+) $$

Therefore, $$17 + 14 $$ $$ = 31 $$

Compute the following:
$$14+(-6)$$

To compute the following

$$ 14 + (-6) $$

As we know that $$ (+) \times (-) = - $$

Therefore, we have

$$ = 14 -6 $$

$$ = 8 $$

Compute the following:
$$8+(-3)$$

To compute the following:

$$ 8 + (-3) $$

As we know that $$ (+) \times (-) = - $$

Therefore, we have

$$ = 8 -3 $$

$$ = 5 $$

Show the following numbers on a numberline:
$$2, (-4), (-1)$$.

We have to show the given number $$(-4),(-1)$$ and $$2$$ on the number line. At first show a number line as shown in the given figure.

The given number are show in the above figure.

1087355_710297_ans_2d8ded99a4b74b3a93420a7099393e7c.PNG

Show the following numbers on a numberline:
$$(-3), 5, 8$$.

We have to show the given number $$(-3),5$$ and $$8$$ on the Number Line.

At first draw a Number Line as shown in the figure.

The given number are shown in the above figure.

1087330_710296_ans_664ba3121ab7496789ce4503e5105a5f.PNG

Show the following numbers on a numberline:
$$0, 3$$

We have to show the given number $$0$$ and $$3$$ on the Number Line.

Number Line

At first draw a Number Line as shown in the figure.

The given number are shown in the above figure.

1087321_710294_ans_0e06a9130dae43e3b6dd07415bad92e3.PNG

Show the following numbers on a numberline:
$$4, 6, (-6)$$.

Integers are $$4,6,(-6)$$ in numberline.
1161593_710295_ans_251349b8e2d1462e91b1e675b99c2a3c.png

Express '$$-6$$' as the difference between a negative integer and a whole number.

SI.No. Negative integer $$+$$ Whole No. $$=-6$$
1 $$(-6)$$ $$+$$ $$0$$ $$=-6$$
2 $$(-7)$$ $$+$$ $$1$$ $$=-6$$
3 $$(-8)$$ $$+$$ $$2$$ $$=-6$$
4 $$(-9)$$ $$+$$ $$3$$ $$=-6$$ etc

Solve the following:
$$13-(-8)$$

To Solve the following

$$ 13 - (-8) $$

As we know that $$ (-)\times(-) = (+) $$

Therefore ,

$$ = 13 + 8 $$

$$ = 21 $$

Solve the following:
$$19-(-5)$$

To Solve the following

$$ 19 - (-5) $$

As we know that $$ (-)\times(-) = (+) $$

Therefore ,

$$ = 19 + 5 $$

$$ = 24 $$

Find:
$$(-2) + 9$$

$$(-2)+9=7$$.

Arrange the following integers in descending order with the help of numberline:
$$(-18), (-3), (-2), (-7)$$.

We arrange the given interger $$\left( { - 18} \right),\left( { - 3} \right),\left( { - 2} \right),\& \left( { - 7} \right)$$ in the descending order with the help of number line .

Then,

Descending order $$=-18,-7,-3,-2$$

1087368_710321_ans_e5f23d0c2cea423299aef3904d9f8e44.PNG

Place appropriate sing $$<$$ or $$>$$ in the blank:
(1) $$(-2)$$ __ $$(-1)$$
(2) $$(-4)$$ $$(-6)$$
(3) $$(-12)$$ _ $$(-5)$$
(4) $$(-3)$$ _ $$(-6)$$
(5) $$5$$ __ $$(-3)$$
(6) $$(-6)$$ _ $$(-1)$$
(7) $$0$$ __ $$(-5)$$
(8) $$0$$ _ $$7$$
(9) $$5$$ _ $$(-5)$$
(10) $$(-8)$$ __ $$0$$.

$$(1)<(2)>(3)<(4)>5>(6)<(7)>(8)>(9)>(10)$$<

Do following addition with the help of numberline:
$$2 + (-5)$$.

$$2+(-5)= 2-5 = -3$$.

Do following addition with the help of numberline:
$$(-4) + (-5)$$.

$$(-9)$$.

Arrange the following integers in an ascending order with the help of numberline:
$$(-5), 4, (-1), 12$$.

Ascending order:-

$$-5<-1<4<12$$

1161800_710316_ans_a0db17b729d5448a94dc43db1f757ee7.JPG

Do following addition with the help of numberline:
$$0 + (-4)$$.

Simplify by adding the numbers,$$0+(-4)$$.

This time around, we are adding one negative.To start locate the first number which is $$0$$.Then add negative $$4$$ to it which means moving the existing point $$4$$ to the left of the number line.

Therefore,we have $$0+(-4)=-4$$

1070660_710329_ans_e14f46269d854bcb9cb45e076cd59899.JPG

Find the following:
$$8 + (-6)$$

$$8+(-6)= 8-6 = 2$$.

Do following addition with the help of number line:
$$(-4) + 5$$.

Referring to above number line,

$$(−4)+5=1.$$

1497361_710336_ans_5dc869b7bc3d40f1bf5d47fd01e9036f.PNG

Do following addition with the help of numberline:
$$(-5)+ (-7)$$.

Let subtract $$(-5)+(-7)$$,

Step(1) Rewrite as an addition problem $$(-5)+(-7)$$

Step(2) start at the first number in the Subtract problem, int the case$$-12$$ on numberline.

1070787_710337_ans_328934b72cd74dfdb221c93e9aa18574.JPG

Subtract the following with the help of numberline:
$$(-3) - 6$$.

Let subtract $$(-3)-6$$.

Step(1):-Rewrite as an addition problem

$$(-3)-6$$

Step(2) start at the first number in the subtract problem, in this case$$ =-9$$.

Subtract the following with the help of numberline:
$$4 - 1$$.

Let subtract $$4-1$$

Step(1) Rewrite as an addition problem $$4-1$$

Step(2) start at the first number in the subtraction problem, in this case$$3$$.

1070804_710340_ans_f6b45c414e6d424b8e0c893fa06a7d95.JPG

Subtract the following with the help of numberline:
$$0 - 4$$.

$$0 - 4 = -4$$.

What is the sum of $$4 + (-10)$$?

$$ 4+ (-10) = -6 $$.

Do following addition with the help of number line:
$$5 + (8)$$.

At first draw a number Line.

Take $$5$$ from the point $$0$$ and Again Take $$8$$ from the point $$5$$.Then we can reach at point $$13$$.

Hence,

$$5+(8)=13$$

1087383_710339_ans_a3f8621ad96142e2a374d83817dbaf1e.jpeg

Subtract the following with the help of numberline:
$$3 - (-6)$$.

$$3 - (-6) = 3 + 6 = 9$$.

Subtract the following with the help of numberline:
$$(-3) - (-2)$$.

$$(-3) - (-2) = -3 + 2 = -1$$.

Subtract the following with the help of number line:
$$(-2) - (-5)$$.

Given, $$-2 - (-5)$$

$$= -2 + 5= 3$$

1066380_710346_ans_752ae62190644f3092953e735d054778.jpeg

Arrange the following integers into ascending order with the help of numberline:
$$3, (-2), (-6), (-3)$$.

$$(-6), (-3), (-2), 3$$.

Arrange the following integers in descending order with the help of numberline:
$$13, (-3), 4, (-2)$$.

$$13, 4, (-2), (-3)$$.

Do the following addition with the help of numberline:
$$(-2) + (-5)$$.

$$(-2) + (-5) = (-7)$$.

Subtract the following with the help of numberline:
$$(-3) - (-4)$$.

$$(-3) - (-4) = -3 + 4 = 1$$.

Subtract the following with the help of numberline:
$$5 - 9$$.

$$5 - 9 = -4$$.

Arrange the following integers into ascending order with the help of numberline:
$$1, (-5), 2, (-9)$$.

$$-9 < -5 < 1 < 2 $$

$$(-9), (-5), 1, 2$$.

Arrange the following integers in descending order with the help of numberline:
$$(-6), 10, 0, (-1)$$.

$$10, 0, (-1), (-6)$$.

Arrange the following integers in descending order with the help of numberline:
$$7, (-5), (-1), 4$$.

$$7, 4, (-1), (-5)$$.

Arrange the following integers into ascending order with the help of numberline:
$$(-9), (-1), (-6), (-4)$$.

$$-9 < -6 < -4 < -1 $$

Thus, ascending order of given numbers is { -9, -6, -4, -1 }

Arrange the following integers into ascending order with the help of numberline:
$$(-5), 8, (-13), 0$$.

$$(-13), (-5), 0, 8$$.

Do the following subtraction with the help of number line:
$$8 - (-4)$$.

$$8 - (-4) = 8 + 4 = 12$$.

Do the following addition with the help of numberline:
$$(-7) + 4$$.

$$-7 + 4 = -3$$.

Do the following subtraction with the help of number line:
$$(-3) - (-9)$$.

Given, $$-3 - (-9)$$

$$= -3 + 9 = 6$$

1497362_710400_ans_44218671c1a94b7db95a59d95b02afd8.PNG

The value of $$7 + (-10)$$ is

$$\textbf{Step 1: Find the value.}$$

$$\text{To find the value of : }7 + (-10)$$

$$\Rightarrow 7 + (-10)=7-10$$

$$=-3$$

$$\textbf{Hence, the required answer is }\mathbf{-3.}$$

Do the following subtraction with the help of number line:
$$8-2$$.

Given, $$8- 2$$

$$ = 6$$

1113593_710406_ans_9f9d82dd703242589a31aad02a859f38.jpeg

The value of $$0 + (-7)$$ is

$$0 + (-7) = 0 - 7 = (-7)$$.

Do the following addition with the help of numberline:
$$(-3) + 8$$.

$$-3 + 8 = 5$$.

In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack's scores in five successive rounds were $$ 25, -5, -10, 15$$ and $$10$$, what was his total at the end?

Scores of jack $$= 25, -5, -10, 15, 10$$

$$\text{Total at end }= 25 + (-5) + (-10) + 15 + 10$$

$$=25 - 5 - 10 + 15 + 10$$

$$=35$$

Simplify :
$$21-27+3+12\times(-1)$$

$$21-27+3+12\times(-1)$$

$$=21-27+3-12$$

$$=-6+3-12$$

$$=-3-12$$

$$=-15$$.

$$-23+\Box =34$$

$$-23+\Box =34$$Let $$-23+x=34$$
On moving $$-23$$ to right side it will become $$+23$$$$x=34+23$$$$x=57$$
Substituting value of x
So, $$-23+57=34$$

Find the sum of $$-23$$ and $$3$$.

The required sum is

$$-23+3=-20$$

Add
(-7)+(-8)+(-90)

$$(-7)+(-8)+(-90)=-7-8-90=-105$$

Find: $$(-13)+32-8-1$$

$$(-13)+32-8-1=-13+32-8-1=10$$

Write the additive inverse of $$\dfrac {8}{5}$$

we know that Additive inverse of $$x$$ is $$-x$$ as $$x+(-x)=0$$

$$\therefore$$ Additive inverse of $$\dfrac{8}{5} = \dfrac{-8}{5}$$

Add without using number line: $$11 + (-7)$$

$$11+(-7)$$

$$=11-7=4$$ -------(The multiplication of signs : "$$+ \times-=-$$")

Using the numbers $$7, 9, 0, 1, 2, 5$$ the biggest number that can be formed is_______

Given numbers $$7,9,0,1,2,5$$

Arrange them in descending order $$9,7,5,2,1,0$$ the number thus formed will be the biggest number

$$\therefore$$ Biggest number formed $$=9,75,210$$

Find $$(-32)-(-50)$$

Product of two negative terms is a positive term.

$$ \displaystyle - 32 + 50 = 18 $$

Solve
$$50-(-40)-(-2)$$

$$50 + 40 + 2 = 92$$ $$[\because -(-40)=+40$$ and $$-(-2)=+2]$$

Solve
(-7)-8-(-25)

$$(-7)-8-(-25)=-7-8+25=10$$

Show that every positive integer is either even or odd..

Let $$I$$ be a positive integer

We can now write $$I$$ in the from of

$$I=pq+r$$

So, $$pq+r$$ can either be even or odd.

Since, $$pq+r$$ yields either even or odd number

$$\Rightarrow$$ if $$pq+r$$ is even

$$pq+r=2x$$

and if odd

$$pq+r=2x+1$$

So, $$I=2x$$ or $$2x+1$$

$$\Rightarrow I=$$odd or even

The sum of two integers is $$-12$$. If one of them is $$34$$, find the other.

Let $$ \displaystyle \quad a + b = -12 $$

one of them $$ = 34 $$

$$ \displaystyle a + 34 = -12 $$

$$ \displaystyle a = - 12 - 34 = -46 $$

$$\therefore$$ the number is $$-46$$

Solve: $$3 + 4\left( {7 - h} \right) - 4\left( {3h - 1} \right)$$

Given: $$3 + 4\left( {7 - h} \right) - 4\left( {3h - 1} \right)$$

$$\Rightarrow 3+28-4h-12h+4$$

$$=35-16h$$

What is additive identity?

The additive identity is the element, by adding which to any 'other element' we get that 'other element'.

For example,

$$0$$ is the additive identity for integers.

As $$a+0=0+a=a$$ for $$a$$ being integer.

Fill the appropriate sign in given box. $$('<', '=', '>')$$ .
$$\frac{1}{2} \square \frac{1}{5}$$

$$\dfrac12=0.5$$

$$\\ \dfrac15=0.2$$

So, $$\dfrac12>\dfrac15$$

Verify: $$\left( -5 \right) { - } \left( +7 \right) =\left( -12 \right) $$

$$\left( { - 5} \right) - \left( 7 \right) = \left( { - 12} \right)$$

$$L.H.S$$

$$= - 5 - 7$$

$$ = - 12$$

$$R.H.S$$

$$ - 12$$

$$L.H.S = R.H.S$$

Prove: $$\left( +3 \right) \boxed { + } \left( -7 \right) =\left( -4 \right) $$

$$\left( { + 3} \right) + \left( { - 7} \right) = \left( { - 4} \right)$$

$$L.H.S$$

$$= \left( { + 3} \right) + \left( { - 7} \right)$$

$$= + 3 - 7$$

$$R.H.S$$

$$ = - 4$$

$$L.H.S = R.H.S$$

Eighteen integers are multiplied together. What will be the sign of their product, if :
(i) 15 of them are negative and 3 are positive ?
(ii) 12 of them are negative and 6 are positive ?
(iii) 9 of them are positive and the remaining are negative ?
(iv) all are negative ?

Let the positive numbers be $$1$$ and negative numbers be $$-1$$

Now,

$$(i) (-1)^{15}(1)^3=-1\\(ii)(-1)^{12}(1)^6=1 \\(iii)(-1)^9(1)^9=-1\\(iv)(-1)^{18}=1$$

Arrange the following integers in ascending order.
$$-4,7,6,0,-8,-11,2,14,-15,9$$.

$$-15,-11,-8,-4,0,2,6,7,9,14$$

Find the value of $$703-1+1-400$$.

consider the given expression.

$$=703-1+1-400$$

$$=703-400$$

$$=303$$

Hence, the value is $$303.$$

Prove: $$\left( -29 \right) \boxed { - } \left( -7 \right) =\left( -22 \right)$$

$$\left( { - 29} \right) - \left( { - 7} \right) = \left( { - 22} \right)$$

$$L.H.S$$

$$ = - 29 - \left( { - 7} \right)$$

$$= - 29 + 7$$

$$= - 22$$

$$R.H.S$$

$$ = - 22$$

therefore ,

$$L.H.S = R.H.S$$

Use the sign of $$>, <$$ or $$=$$ in the box to make the statement true.
$$23-41+11\square 23-41-11$$

Calculating expression on either sides,
$$23-41+11\square 23-41-11$$

$$-18+11\square-18-11$$

$$-7\square-29$$

$$-7>-29$$(since $$-7$$ is a greater than $$-29$$)

$$23-41+11> 23-41-11$$

(Also, $$23-41$$ is same on either sides. Since $$11>-11$$, the box should be filled by $$>$$ symbol.)

Use the sign of $$>, <$$ or $$=$$ in the box to make the statement true.
$$39+(-24)-(15)\square 36+(-52)-(-36)$$

$$39+(−24)−(15)□36+(−52)−(−36) $$

$$39-24−15□36-52+36 $$

$$15−15□-16+36 $$

$$0□20$$

$$0<20$$

$$\therefore 39+(−24)−(15)<36+(−52)−(−36) $$

Fill in the blank.
$$12+(-12)=$$ ______.

The addition of any number with its additive inverse gives zero and the additive inverse of any number $$a$$ is $$-a$$.

Hence, value of expression $$12+ (-12)$$ is equal to $$0$$.

Fill in the blank.
$$13+$$ _______$$=0$$.

1159472_1063548_ans_c9b6bc9a7dae43378dffd7a23bb71cf2.jpg

Use the sign of $$>, <$$ or $$=$$ in the box to make the statement true.
$$-231+79+51\square -399+159+81$$

$$−231+79+51□−399+159+81 $$

$$−152+51□−240+81 $$

$$−101□−159$$

$$-101>-159$$

$$\therefore−231+79+51>−399+159+81 $$

The number which should be subtracted from $$3,8403$$ to get the difference of $$1,980$$ is

Let x should be subtracted

then,

$$38,403-x=1,980$$

$$x=38,403-1,980$$

$$x=36,423$$

Solve: $$(-8) + (-9) + (+ 17)$$.

Now,

$$(-8) + (-9) + (+ 17)$$

$$=-8-9+17$$

$$=-17+17$$

$$=0$$.

What is Additive Inverse?

Additive inverse is the element, by adding the inverse to that element we will get the additive identity.

For example,

For integers $$0$$ is the additive identity.

Now for any integer $$a$$, $$-a$$ is the additive inverse of $$a$$.

As $$a+(-a)=(-a)+a=0$$.

Solve:
$$\left( { - 4} \right) + 6 - 2 + 3 + 2 = $$

Consider the given expression.

$$=-4+6-2+3+2$$

$$=2+3$$

$$=5$$

Hence, the value is $$5$$.

Find the sum of $$1005$$ and $$-277$$.

Let $$x = 1005$$ and $$ y = -277$$

Sum of these two numbers $$(x + y)$$,

$$ \Rightarrow (1005) + (- 277)$$

$$ \Rightarrow 1005 - 277$$

$$ \Rightarrow 728$$

Hence, the sum is $$728.$$

Simplify : $$(-8)-(-9)$$

Given $$(-8)-(-9)$$

$$= 9-8$$

$$= 1$$

Solve:$$ \left( { - 6} \right) + \left( { - 4} \right)$$

Given that:

$$(-6)+(-4)$$

$$=(-6-4)$$

$$=(-10)$$

Hence, the value is $$-10$$.

Simplify :$$(-5)-(-5)$$

Given $$(-5) - (-5)$$

$$= 5-5$$

$$= 0$$

Simplify: $$(-8)-(+3)$$

Given $$(-8)-(+3)$$

$$= -8-3$$

$$= -11$$

Simplify :$$(-13)-(+14)$$

Given $$(-13)-(+14)$$

$$= -14-13$$

$$= -27$$

Find the sum of $$-3,8,-4,5,-11,-1$$

$$-3+8-4+5-11-1$$

$$=-6$$

Represent the following numbers as integers with appropriate signs.
A) An aeroplane is flying at a height two thousand metre above the ground.
B) A submarine is moving at a depth, eight hundred metre below the sea level.

Solution:

A) Answer: $$+2000m$$

B) Answer: $$-800m$$

Simplify:
$$(\frac{1}{2})^{-1} + (\frac{1}{3})^{-1} + (\frac{1}{4})^{-1}$$

We have,

$${{\left( \dfrac{1}{2} \right)}^{-1}}+{{\left( \dfrac{1}{3} \right)}^{-1}}+{{\left( \dfrac{1}{4} \right)}^{-1}}$$

$$ =\left( \dfrac{1}{{{2}^{-1}}} \right)+\left( \dfrac{1}{{{3}^{-1}}} \right)+\left( \dfrac{1}{{{4}^{-1}}} \right) $$

$$ =2+3+4 $$

$$ =9 $$

Hence, this is the answer.

Find the sum of :
$$-312, 39 $$ and $$192$$

$$-312+39+192$$

$$=(-312)+(39+192)$$

$$=-312+231$$

$$=-(312-231)$$

$$=-(81)$$

$$=-81$$

Write the integers that represent the following situations
a decrease of $$500.$$

The required sequence of integers is $$n,n-500,n-1000,n-1500,......$$

Where $$x$$ be an integer.

Find the sum of :
$$137$$ and $$-354$$

$$137+(-354)=137-354$$

$$=-(354-137)$$

$$=-(217)$$

$$= -217 $$

Find the sum of $$-9,+14,+21,+19,+7$$.

$$=-9+14+21+19+7=52$$

Find the sum of :
$$-52$$ and $$52$$

$$-52 + 52 = 0$$
1085927_1144033_ans_3ceebf7f47a140d2ade38922ecf201a2.jpeg

Write the integers that represent the following situations
$$40$$ SECONDS

Given that $$40 \ seconds $$

According to the question,

$$x \ seconds, x+40 \,seconds , x+80 \ seconds, x+120\ seconds, ....... $$

Where $$x$$ be an integer.

Hence, this is the answer.

Write the integers that represent the following situations
Flight at a height of $$14000\ meters$$

Given that the height $$=14000 \ m$$

According to the question,

$$x \ m, x+14000 \ m , x+28000 \ m, ....... $$

Where $$x$$ be an integer.

Hence, this is the answer.

If $$a=1$$, $$b=-1$$ and $$c=0$$, find the value of expressions.
$$a+b+c$$

$$a+b+c $$

$$= 1-1+0$$

$$=0$$

Solve :
$$ (-2) +8+(-4)$$

$$(-2)+8+(-4)$$

$$=8-2-4$$

$$=(8-2)-4$$

$$=6-4=2$$

Solve :
Mark $$7$$ and $$-1$$ on number line and hence find the difference between them.

Required no. are marked on number line in figure.

Difference between them is $$=7-(-1)=8$$

1058255_1150349_ans_953b369e7ac84c5da56d7e5e956dbcc5.jpg

Fill in the blanks with $$>,<$$ or $$=$$ sign.
$$(-3) + (-6) .......... (-3) - (-6) $$

LHS, => $$-3-6=-9$$

RHS => $$-3+6=3$$

Hence,

LHS < RHS

Write opposite of $$80\text{ m}$$ east.

Opposite of $$80\text{ m}$$ east is $$80\text{ m}$$ West.

$$50 - ( - 2 ) - ( - 5 ) - 110$$

$$50-(-2)-(-5)-110$$

$$=50+2+5-110$$

$$=57-110=-53$$

Mark $$-4$$ and $$-9$$ on No. line and fine the difference between them.

Required no. are marked on number line in figure.

Difference betweem them,

$$=-4-(-9)=5$$

1058259_1150353_ans_fd47291ca03f464fa0996158a39f92f9.png

Find the sum of :
$$-50,-200$$ and $$30$$

$$(-50)+(-200)+(30)$$

$$=-(50+200)+30$$

$$=-250+30$$

$$=-(250-30)$$

$$=-(220)$$

$$=-220$$

Fill in the blanks with $$>,<$$ or $$=$$ sign.
$$ (-21) + (-10)............(-31)+(-11) $$

LHS => $$-21-10=-31$$

RHS => $$-31-11=-42$$

Hence,

LHS > RHS

Fill in the blanks with $$>,<$$ or $$=$$ sign.
$$(-25)-(-42).........(-42)+(-25)$$

LHS => $$-25+42=17$$

RHS => $$-42-25=-67$$

Hence,

LHS > RHS

Evaluate:
$$( - 10 ) + ( + 19 )$$

$$( - 10 ) + ( + 19 )$$

$$( - 10 ) + ( 19 )$$

$$-10+19$$

$$9$$

Solve:
$$ - 6 + 7$$

Solution :

$$ -6+7 $$

$$ = 7-6 $$

$$ = 1 $$

1112613_1175518_ans_8118b8230e0c4e3c885f40f3b6d22de2.jpg

Add without using number line:
(1) $$ 1 + ( - 7 )$$ (2) $$( - 13 ) + ( 18 )$$
(3) $$( - 10 ) + ( 19 )$$ (4) $$( - 250 ) + ( 150 )$$
(5) $$( - 380 ) + ( - 270 )$$ (6) $$( - 217 ) + ( - 100 )$$

(1) $$1 + (-7)$$

Subtract smaller number from greater : $$7 - 1 = 6$$

As sign of the greater number is '$$-$$',

So, $$1 + (-7) = -6$$

(2) $$(-13)+(18)$$

$$13 < 18$$ So, subtracting smaller number from grater, we get $$18 - 13 = 5$$

Since sign of the greater number is '$$+$$',

So, $$(-13) + (18) = +5$$

(3) $$(-10) + (19)$$

On similar steps:

as $$10 < 19$$, so, $$19-10 = 9$$

& sign of $$19$$ is '$$+$$',

so, $$(-10) + 19 = +9$$

(4) $$(-250) + (150)$$

$$150 < 250$$ so, $$250 - 150 = 100$$

sign of $$250$$ is '$$-$$',

So, $$(-250) + 150 = -100$$

(5) $$(-280) + (-270)$$

sign of both numbers are '$$-$$', so add them and add a '$$-$$' sign to the result

$$(-380) + (-270) = - (380 + 270) = -(650)$$

(6) $$(-217) + (-100) = -(217 + 100)= -(317)$$

Fill in the blanks
$$(-8)+(-3)=$$_________

$$ -52 $$ Reminder

$$ -8-3 = -11 $$ because if we add same

sign we get the same

sign

1112188_1204182_ans_9b750342efb8433fa1b4af0ebbbb6bda.jpg

What is the multiplicand if the multiplier is 13 and the product is 1001 ?

Let multiplicand be $$x$$

Then $$13 \times x = 1001$$

$$x= \dfrac{1001}{13}=77 $$

Write opposites of the following :
Loss of $$Rs 700$$

Profit of $$Rs. 700$$

Solve:
$$x\% \ of\ 300$$ = $$225$$

$$\\let\>x\%\>of\>300 =225$$

$$\\ or \>(\frac{x}{100}) \cdot 300 = 225$$

$$\\\therefore x = (\frac{225 \cdot 100}{300}) = 75$$

Subtract : $$5$$ from $$-13$$

$$-13-(5)$$
$$=-13-5$$
$$=\boxed{-18}$$

Work out the addition:
$$3+(-6)$$

Given

$$3+(-6)$$

$$=3-6$$ [As product of (-) and (+) is (-)]

$$=-3$$

Evaluate:
$$\left( 3 ^ { 5 } \times 4 ^ { 7 } \times 5)\right.$$

$$\left( 3 ^ { 5 } \times 4 ^ { 7 } \times 5)\right.$$

$$\left( 3 ^ { 5 } \times 16384 \times 5)\right.$$

$$\left( 243 \times 16384 \times 5)\right.$$

$$19906560$$

The sum of two integers is $$-34$$.If one of them is $$55$$, then find the other.

1349735_1263021_ans_436dcb108d904f3e83394e5a07e75db5.PNG

What number should be added to $$57$$ to get $$-75$$?

1349729_1263404_ans_055b5b04543446528cf7aa1246c95da7.PNG

Write in descending order:$$-40,\,-20,\,-10,\,0,\,-15,\,-16,\,20,\,30$$

1349723_1265978_ans_06508142988e43cd9a0676d73fa97984.PNG

Find the sum of $$510,\,-840,\,-2010$$

1351985_1267158_ans_281b4eb25c42469aaf9f8b241469bf24.PNG

Arrange - $$9\div 3,-4+6,2(-4),|-7-3|,-5-3\times 2$$ in ascending order.

$$9\div 3$$, $$-4+6$$, $$2(-4)$$, $$|-7-3|$$, $$-5-3\times 2$$

$$=3, 2, -8, 10, -16$$

Ascending order: $$-16, -8, 2, 3, 10$$

(or)

$$-5-3\times 2$$, $$2(-4)$$, $$-4+6$$, $$9\div 3$$, $$|-7-3|$$.

1130960_1246412_ans_a591b7db4d77436da3a3fcaf6a7b25fd.jpg

Simplify:$$\left(-430\right)+\left(-30\right)+\left(-85\right)+\left(-15\right)$$

1349674_1268568_ans_263efbcebba545379ac9ea0ebf7790b4.PNG

Simplify:$$\left(-400\right)+100+\left(-2\right)+\left(-1\right)+20$$

1349672_1267435_ans_a30c379ba7a445ddbe265c71e841e39d.PNG

If $$\dfrac{p}{q}+\dfrac{3}{10}=0$$ then what can you say about $$\dfrac{p}{q}$$ to represent with $$\dfrac{3}{10}$$

Given, $$\dfrac{p}{q}+\dfrac{3}{10}=0$$

$$\dfrac{p}{q}= -\dfrac{3}{10}$$

We say that $$\dfrac{p}{q}$$ is the additive inverse of $$\dfrac{3}{10}$$ since additive inverse is a rational number when added with some other number gives the result as zero

Here $$\dfrac{p}{q}=\dfrac{-3}{10}$$

Find $$\left|-2\right|$$

1349658_1274929_ans_cde04eb6557d443ca36889224e36ee05.PNG

Simplify:$$\left(-373\right)+20+\left(-3\right)+\left(-5\right)+5$$

1349673_1267590_ans_718bda015a5a4a61a58daed7b8eb345c.PNG

Draw number lines and locate the point on them :
$$\dfrac {1}{8},\dfrac {2}{8},\dfrac {3}{8},\dfrac {7}{8}$$

1235383_1298223_ans_ba7e7d7869094eb0abb5feb91e7acb31.png

Add: $$-18, 14$$

$$Add -18, 14$$

$$-18 + 14 = -4$$

Fill in the blank:
Sum of $$-3$$ and it's additive inverse is ______

The additive inverse of $$x$$ is $$–x.$$

$$\therefore$$ additive inverse of –3 is 3.

Note: The sum of a number and its additive inverse is $$0$$.

$$\because -3 + 3 =0$$

From the sum of -38 and 40, subtract 120.

Sum of $$- 38$$ & $$40$$:

$$= -38+40 = 2$$

Now, subtracting $$120$$ from the sum:

$$2-120 = -118$$

Hence, the final answer is $$-118$$.

Find the least number that should be subtracts from $$1000$$ so that $$30$$ divides the difference exactly.

1196596_1297570_ans_be3cfd40211743a09955a70f536eed29.jpg

How many integers are there between $$-9$$ and $$-10$$

There are no integers between $$-9$$ and $$-10$$ is $$0$$

Since successor of $$-10$$ is $$-10+1=-9$$

What is the maximum and minimum number of digits in the sum of any two $$6-$$digit numbers.

a maximum number of digits would occur when we do the sum of two maximum six digits numbers ie, $$999999$$.

$$999999+999999=1999998$$

$$7$$ is the maximum number of digits

a minimum number of digits would occur when we do the sum of two minimum six digits numbers ie, $$100000$$.

$$100000+100000=200000$$

$$6$$ is the minimum number of digits

The successor of $$-91$$ is.............. .

Successor of $$-91$$ is $$-91+1=-90$$

Because Successor comes just after the number.

The predecessor of $$-380$$ is.............. .

The predecessor of $$-380$$ is $$-380-1=-381$$ because predecessor means just before.

Find the sum of:
$$-319$$ and $$-136$$

Sum of $$-319$$ and $$-136$$

$$=-319+\left(-136\right)$$

$$=-319-136=-455$$

Find two consecutive negative integers whose sum is $$-27$$.

Let $$n$$ & $$n + 1$$ be two negative integers.

Given,

$$n+n+1= -27$$

$$2n= -28$$

$$n= -14$$

Then the two negative integers are $$-14$$ and $$-13$$.

Add:
$$ (-31) +(-71)$$

$$(-31) + (-71)$$

$$(-31-71)$$

$$= -102$$

If a and b distinct integers, prove that a-b is a factor of whenever n is a positive integer.
[Hint write$$\left( a-b+b \right) ^{ n }$$ and expand]

1210061_1312604_ans_ecff9845d599496c8d84e682e1a5814c.jpg

Find the sum of:
$$-127$$ and $$300$$

Sum of $$-127$$ and $$300$$

$$=-127+300$$

$$=300-127$$

$$=173$$

Find the sum of:
$$256$$ from $$-312$$

Sum of $$256$$ and $$-312$$

$$=256+\left(-312\right)$$

$$=256-312=-56‬$$

Write the following numbers in descending order $$77890,74077,78999,79009$$ and $$73535$$.

$$77890, 74077, 78999, 79009, 73535$$

Among the given numbers, the descending order is-

$$79009 > 78999 > 77890 > 74007 > 73535$$

Subtract:
$$-73$$ from $$-80$$

$$-80-\left(-73\right)$$

$$=-80+73=-7$$

Subtract:
$$-63$$ from $$72$$

$$72-\left(-63\right)$$

$$=72+63=135$$

Write the opposite of the statement: An increase of $$8$$

The opposite statement of an increase of $$8$$ is a decrease of $$8$$

Subtract:
$$Rs.18.25$$ from $$Rs.20.75$$

$$Rs. 1 = 100 ps.$$

Therefore,

$$Rs. 18.25 = Rs. 18 + 25 ps. = 1800 ps. + 25 ps. = 1825 ps.$$

$$Rs. 20.75 = Rs. 20 + 75 ps. = 2000 ps. + 75 ps. = 2075 ps.$$

Therefore,

$$Rs. 20.75 - Rs. 18.25$$

$$= 2075 - 1825$$

$$= 250 ps.$$

$$= 200 ps. + 50 ps.$$

$$= Rs. 2 + 50 ps.$$

$$= Rs. 2.50$$

Write following in descending order
4, -7, 3, 0 .

The required order is

4, 3, 0, -7 .

$$-\dfrac { 21 }{ 2 } +15+\dfrac { 39 }{ 2 } -15$$=

$$\dfrac{-21}{2}+15+\dfrac{39}{2}-15$$

$$=\dfrac{-21+39}{2}$$

$$=\dfrac{18}{2}=9$$

Write the opposite of the following:
$$10 km$$ above sea level

The opposite statement of "$$10 km$$ above sea level" is

$$10 km$$ below sea level

Represent $$(-3)$$ on a number line.

1309483_1452076_ans_8b6658fac9f041e885662c80536d0d09.jpeg

Write the opposite of each of the statement: A loss of $$Rs.7$$

The opposite statement: a loss of $$Rs.7$$

Subtract the following integers.
$$(+12)$$ from $$(-5)$$

$$\left(-5\right)-\left(12\right)$$

$$=-5-12=-17$$

Give a positive and a negative integer whose difference is -3

Let two integers one negative and other positive whose difference is $$-3$$ be $$-1,\ 2.$$
Difference$$=-1-(2)=-1-2=-3$$

Which number is larger in the following pair?
$$-5$$, $$2$$

Since every positive integer is greater than negative integer

Hence $$2$$ is larger than $$-5$$

Which number is smaller in the following pair?
$$6$$, $$-7$$

Since every negative integer is smaller than positive integer.

Hence $$-7$$ is smaller than $$6$$

Mark the following integers on a number line:
$$-2$$

Here $$-2$$ lies on the left side of the $$0$$ on the number line

1631839_1777394_ans_ba1bd237553c475892a52b66ef9a1fd8.png

Which number is larger in the following pair?
$$0, \,-2$$

$$0$$ is greater than every negative integer

Hence $$0$$ is larger than $$-2$$

Which number is smaller in the following pair?
$$-13$$, $$-27$$

As, $$-27$$ lies on left hand side of $$-13$$ on number line.

Hence, $$-27$$ is smaller than $$-13$$

Which number is larger in the following pair?
$$-3$$, $$-5$$

$$3$$ is less than $$5$$,

Because $$5$$ is on right hand side of $$3$$ on number line.

But, $$-3$$ is on right hand side of $$-5$$ on number line.

Hence $$-3$$ is larger than $$-5$$

Indicate the following using $$"+"$$ or $$"-"$$ sign:
A Deposit of $$Rs. 200$$

Deposit means adding into amount.

So, A deposit of rupees $$200$$ indicates $$Rs.+200$$

Mark the following integer on a number line:
$$-5$$

Here, $$-5$$ lies on the left side of the $$0$$ on the number line

1631835_1777387_ans_807c42f678d6401c933b1e0ae9054f62.png

Mark the following integer on a number line:
$$0$$

Here $$0$$ lies at the indicated spot on the number line.

1631846_1777398_ans_64ffeb795dd447738a0d26621b2f85a2.png

Which number is larger in the following pair?
$$-888$$, $$8$$

Since every positive integer is greater than negative integer

Hence $$8$$ is larger than $$-888$$

Fill in the blanks:
$$(-9) + 16 = $$

$$(-9) + 16$$

$$-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5,\ 6,\ 7$$

$$-9 $$ is smaller than $$16$$.

So, we have to go $$16$$ places right from $$-9$$, which is equal to number $$7$$

Hence,$$(-9) + 16 = 7$$

Use the number line and add the following integers:
$$9 + (-6)$$

Given numbers are $$9$$ and $$-6$$.

To find out,

The sum of $$9$$ and $$-6$$ using the number line.

We know that, $$-6$$ is less than $$9$$.

We can see that we need to add a negative number to $$9$$.

Hence, we will mark $$9$$ on the number line, and move towards left.

We will move $$6$$ numbers towards left on the number line from $$9$$, as shown in the figure above.

We will to get $$+3$$.

Hence, $$9+(-6)=3$$.

2113477_1777486_ans_2d1b68db954145b4b9fa474b51789c2d.png

Use the number line and add the following integers:
$$(-3) + 7$$

$$(-3) + 7 $$

$$-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5$$

$$-3 $$ is smaller than $$7$$.

So, we have to go $$7$$ places right from $$-3$$, which is equal to number $$4$$

Hence, $$(-3) + 7=4 $$

Fill in the blanks:
$$2 + (-12) = ..$$

$$2 + (-12)$$

$$-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5,\ 6,\ 7,\ 8$$

$$-12 $$ is smaller than $$2$$.

So, we have to go $$2$$ places right from $$-12$$, which is equal to number $$-10$$

Hence,$$2 + (-12) = -10$$

Use the number line and add the following integers:
$$(-1) + (-3)$$

$$(-1) + (-3) $$

$$-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5$$

$$-3 $$ is smaller than $$-1$$.

So, we have to go $$-1$$ places right from $$-3$$, which means we have to go $$1$$ place left from $$-3$$.

Which is equal to number $$-4$$

Hence, $$(-1) + (-3) =-4$$

Fill in the blanks:
$$(-13) + 25 = $$

$$(-13) + 25$$

$$-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12$$

$$-13 $$ is smaller than $$25$$.

So, we have to go $$25$$ places right from $$-13$$, which is equal to number $$12$$

Hence,$$(-13) + 25 = 12$$

Fill in the blanks:
$$8 + (-17) =$$

$$8 + (-17)$$

$$-17,-16,-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5,\ 6,\ 7,\ 8$$

$$-17 $$ is smaller than $$8$$.

So, we have to go $$8$$ places right from $$-17$$, which is equal to number $$-9$$

Hence,$$8 + (-17) = -9$$

Subtract:
$$-28$$ from $$-43$$

$$-43 – (-28)$$

So, we have to go $$-28$$ places left from $$-43$$, which means $$28$$ places right from $$-43$$

Which is equal to number $$-15$$

Hence,$$-43 – (-28)$$ = $$-43 + 28$$ = $$-15$$

Fill in the blanks:
$$(-3) + (-9) =$$

$$(-3) + (-9) $$

$$-12,-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5$$

$$-9 $$ is smaller than $$-3$$.

So, we have to go $$-3$$ places right from $$-9$$, which means we have to go $$3$$ places left from $$-9$$

Which is equal to number $$-12$

Hence,$$(-3) + (-9) = -12$$

Fill in the blanks:
$$(-7) + (-8) = $$

$$(-7) + (-8) $$

$$-15,-14,-13,-12,-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5$$

$$-8 $$ is smaller than $$-7$$.

So, we have to go $$-7$$ places right from $$-8$$, which means we have to go $$7$$ places left from $$-8$$

Which is equal to number $$-15$$

Hence,$$(-7) + (-8) =-15$$

Find the sum of
$$-18, \,+25$$ and $$-37$$

$$-18 + 25 + (-37) =-18-37+25$$

$$=-55+25$$

$$=-30$$

Add:
$$-365$$
$$\underline{-87}$$

$$-365$$

$$\underline{+(-87)} $$

$$=\boxed{ \, \,-452 \, \, \, \,}$$

From the sum of $$33$$ and $$-47$$, Subtract $$-84$$ from it.

Finding the sum of $$33$$ and $$-47$$

$$33 + (-47) = 33 – 47$$

$$= -14$$

Subtracting $$-84$$ from $$-14$$

$$-14 – (-84) = -14 + 84$$

$$=70$$

Subtract:
$$219$$ from $$0$$

$$0- 219$$

So, we have to go $$219$$ places left from $$0$$, which is equal to number $$-219$$

Hence,$$0- 219 = -219$$

Add $$-36$$ to the difference of $$-8$$ and $$-68$$

Finding the difference of $$-8$$ and $$-68$$,

$$-8 – (-68) = – 8 + 68$$

$$= 60$$

Now add $$-36$$ to difference of $$-8$$ and $$-68$$ i.e. $$60$$

$$\therefore -36 + 60 = 24$$

Find the sum of
$$-36$$ and $$100$$

$$-36 + 100$$

$$ = 64$$

$$64$$ is the sum of $$-36$$ and $$100$$

Subtract the sum of $$-250$$ and $$138$$ from the sum of $$136$$ and $$-272$$

To find the sum of $$-250 $$ and $$138$$ we get,

$$-250 + 138 = -112$$

To find the sum of $$136$$ and $$-272$$

$$136 + (-272) = 136 - 272$$

$$= -136$$

Subtracting the sum of $$-250$$ and $$138$$ from sum of $$136$$ and $$-272$$ i.e,

$$-136 – (-112) = -136 +112$$

$$= -24$$

Subtract:
$$-135$$ from $$-250$$

$$-250 – (-135)$$

So, we have to go $$-135$$ places left from $$-250$$, which means $$135$$ places right from $$-250$$

Which is equal to number $$-115$$

Hence,$$-250 – (-135) = -250 + 135 =-115$$

$$0$$ ____ $$-2$$.

$$-2$$ lies to the left of $$0$$ on the number line hence, $$0 > -2$$.

Write the opposite of the following :
Loss of $$ Rs. 5000 $$

Opposite of the given statement is : Profit of $$ Rs. 5000 $$

Find the sum of
$$-312, \,39$$ and $$192$$

$$-312 + 39 + 192 $$

$$=[-312 + 39 ]+ 192 $$

Here we will $$1^{st}$$ solve square bracket

$$\therefore [-312+39]+192=-273+192$$

$$-81$$

$$-81$$ is the sum of $$-132, \,39$$ and $$192$$

The additive inverse of $$-1$$ is _______.

Additive Inverse of any real number $$x$$ is obtained by multiplying it by $$-1$$

$$\implies$$Additive inverse of $$x$$ is $$-x$$

$$\therefore$$ The additive inverse of $$-1$$ is $$-1\times (-1)=+1$$.

$$-60$$ _____ $$50$$.

$$-60$$ lies to the left of $$50$$ on the number line hence, $$-60 < 50$$.

Subtract :
$$ -6 $$ from $$ -9 $$

$$ -9 - (-6) = -9 + 6 = -3 $$

$$0$$ ______ $$1$$.

$$1$$ lies to the right of $$0$$ on a number line hence, $$0 < 1$$.

An atom changes to a charged particle called ion if it loses or gains electrons. The charge on an ion is the charge on electrons plus charge on protons. Now, write the missing information in the table given below:

$$\begin{array}{|c|c|c|c|}\hline \text { Name of Ion } & \begin{array}{l}\text { Proton } \\\text { Charge }\end{array} & \begin{array}{l}\text { Electron } \\\text { Charge }\end{array} & \begin{array}{l}\text { Ion } \\\text { Charge }\end{array} \\\hline \text { Hydroxide ion } & +9 & - & -1 \\\hline \text { Sodium ion } & +11 & - & +1 \\\hline \text { Aluminium ion } & +13 & -10 & - \\\hline \text { Oxide ion } & +8 & -10 & - \\\hline\end{array}$$

We know that, Ion charge = Proton charge +Electron charge
Therefore,
$$\begin{array}{|c|c|c|c|}\hline \text { Name of Ion } & \begin{array}{l}\text { Proton } \\\text { Charge }\end{array} & \begin{array}{l}\text { Electron } \\\text { Charge }\end{array} & \begin{array}{l}\text { Ion } \\\text { Charge }\end{array} \\\hline \text { Hydroxide ion } & +9 & -1-9 = -10 & -1 \\\hline \text { Sodium ion } & +11 & +1-11 = -10 & +1 \\\hline \text { Aluminium ion } & +13 & -10 & +13-10 = +3 \\\hline \text { Oxide ion } & +8 & -10 & +8-10=-2 \\\hline\end{array}$$

On the following number line, $$(-4) \times 3$$ is represented by the point_______.

$$(-4) \times 3 = -12$$
Thereofore, - 12 is represented by the point D.

1637648_1789507_ans_461320fae61840d9aa269a31e5f2d203.png

( 43) + _____ = 43

( 43) + _0_ = 43

Fill in the blanks to make the statement true:
The negative of $$1$$ is ............

The negative of$$1$$is $$-1$$

Write a pair of integers whose:
difference is $$4$$

Pair of integers whose difference is

4=3, −7 as[−3−(−7)=−3+7=4]

Compute the following:
$$30 + 25 + (-10)$$.

Given: $$30 + 25 + (-10)$$
$$= 30 + 25 - 10$$
$$= 55 - 10 = 45$$.

$$(-a)+b=b +$$ Additive inverse of ______.

b + Additive inverse of _a_

The Lowest Recorded Temperatures
Continent Temperature
(in Fahrenheit)
Africa $$-11^o$$
Antarctica $$-129^o$$
Asia $$-90^o$$
Australia $$-9^o$$
Europe $$-67^o$$
North America $$-81^o$$
South America $$-27^o$$
The table shows the lowest recorded temperatures for each continent. Write the continents in order from the lowest recorded temperature to the highest recorded temperature.

$$-129^{\circ}<-90^{\circ}<-81^{\circ}<-67^{\circ}<-27^{\circ}<-11^{\circ}<-9^{\circ}$$
Therefore, the order of the continent from the lowest to highest temperature is given below, Antarctica, Asia, North America, Europe, South America, Africa, Australia.

Use the sign $$>, <$$ or $$=$$ in the box to make the following statements true:
$$40-(-39)+(-5).....40+(-39)-5(-5)$$

$$40-(-39)+(-5).....40+(-39)-5(-5)$$
$$\Rightarrow 40+39-5.....40-39+5$$
$$\Rightarrow 79-5 .....45-39$$
$$\Rightarrow 74 > 6$$

Find sum of integers $$-72,\ 237,\ 84,\ 72,\ -184,\ -37$$

Sum of integers $$-72,\ 237,\ 84,\ 72,\ -184,\ -37$$
$$=-72+ 237+ 84+ 72+ -184+ -37$$
$$=237+84+72+(-72-184-37)$$
$$=(393)+(-293)$$
$$=393-293$$
$$=100$$

Write a pair of negative integers whose difference is $$5$$

Let two negative integers whose difference is $$5$$ be $$-7,\ -12.$$
Now difference is:

$$-7\ -(-12) $$
$$=-7+12=5$$

Write a negative integer and positive integer whose sum is $$-8$$

Let two integers one negative and other positive but sum is $$-8$$ are $$ -12, +4$$
$$=-12+4=-8$$

Use the sign $$>, <$$ or $$=$$ in the box to make the following statements true:
$$23-41+11......23-41-11$$

$$23-41+11......23-41-11$$
$$\Rightarrow 34-41 .......23-52$$
$$\Rightarrow -9 > -19$$

Use the sign $$>, <$$ or $$=$$ in the box to make the following statements true:
$$(-11)+(-7).......(-11)-(-7)$$

$$(-11)+(-7).....(-11)-(-7)$$
$$\Rightarrow -18 < -11+7=-4$$

Fill in the blanks:
$$(-a)+b=b+$$ additive inverse of ______

$$(-a)+b=b\ +$$ additive inverse of $$a$$

Use the sign $$>, <$$ or $$=$$ in the box to make the following statements true:
$$(-3)+13-(15)........25-(-2)+(-33)$$

$$(-3)+13-(15)........25-(-2)+(-33)$$
$$\Rightarrow -3+13-......25+2-33$$
$$\Rightarrow 13-18.....27-33$$
$$\Rightarrow -5 > -6$$

Fill in the blanks :
Negative of $$ 8 $$ is ..........

Negative of $$ 8 $$ is $$ -8 $$

Use the sign $$>, <$$ or $$=$$ in the box to make the following statements true:
$$(-5) \times (-7) \times (-10)......(-1400) \div (-4)$$

$$(-5) \times (-7) \times (-10)......(-1400) \div (-4)$$
$$LHS=(-5)\times (-7) \times (-10)=35\times (-10)=-350$$
$$RHS =(-1400)\div (-4)=-350$$
$$-350 < 350$$

Is $$\left\{ (-45) \div (-15) \right\} \div (-3)=(-45) \div [ (-15) \div (-3)]$$?

$$(-45) \div (-15) \div (-3)$$
$$LHS =\dfrac{-45}{-15}\div (-3)=3\div -3=-1$$
and $$RHS=-45 \div \left( \dfrac{-15}{-3}\right)$$
$$=-45\div 5$$
$$=-9$$
$$(-1) \neq -9$$
No, not equal.

Use the sign $$>, <$$ or $$=$$ in the box to make the following statements true:
$$(-13) \times 0\times (-5) ......(-7) \times (-6) \times 14$$

$$(-13) \times 0\times (-5) ......(-7) \times (-6) \times 14$$
$$LHS=-13 \times 0\times (-5)=0$$
and $$(-7) \times (-6) \times 14$$
$$RHS =42 \times 14=588$$
$$0 < 588$$

Fill in the blanks
If $$ 10 $$ represents gain of Rs. $$10 $$ then , $$ -10 $$ represents .........

If $$ 10 $$ represents gain of Rs. $$10 $$ then , $$ -10 $$ represents loss of Rs $$ 10 $$

Fill in the blanks :
Negative of $$ 0 $$ is ...........

Negative of $$ 0 $$ is $$0$$

Use the sign $$>, <$$ or $$=$$ in the box to make the following statements true:
$$(-18) \times (-3).......(-10)+(-15)+31$$

$$(-18) \times (-3).......(-10)+(-15)+31-18$$
$$LHS=-18 \div -3=6$$
$$RHS =(-10)+(-15)+31=-25+31=6$$
$$6=6$$

Use the sign $$>, <$$ or $$=$$ in the box to make the following statements true:
$$(-15)+39......27+(-50)$$

$$(-15)+39......27+(-50)$$
$$LHS=-15+38=23$$
$$RHS=27-50=-23$$
$$(23) > (-23)$$

Fill in the blanks :
Negative of $$ -20 $$ is .........

Negative of $$ -20 $$ is $$ 20 $$

Add :
$$ -13 $$ and $$ -15 $$

$$ -13 $$ and $$ -15 $$
On adding these numbers , we get
$$ (-13) + (-15) = -13 - 15 $$
$$ = -28 $$

Add:
$$ 258 $$ and $$ -473 $$

$$ 258 $$ and $$ -473 $$
On adding , we get
$$ = 258 + (-473)=258-473=-215 $$

Add:
$$ -528 $$ and $$-243 $$

$$ -528 $$ and $$-243 $$
On adding , we get
$$ = -528 + (-243) $$
$$ = -528 - 243 $$
$$ = - 771 $$

Fill in the blanks :
If $$ 3 > -2 $$ , then $$ 3 $$ is on the ........ of $$ -2 $$.

If $$ 3 > -2 $$ , then $$ 3 $$ is on the right side of $$ -2 $$.

Add:
$$ -622 $$ and $$ -254 $$

$$ -622 $$ and $$ -254 $$

On adding , we get

$$ = -622 + (-254) $$

$$ = -622 - 254 $$

$$ = -876 $$

Add:
$$ 257 $$ and $$ -254 $$

$$ 257 $$ and $$ -254 $$

On adding , we get

$$ = 257 + (-254) $$

$$ = 257 - 254 $$

$$ = 3 $$

Add:
$$ -623 $$ and $$ 326 $$

$$ -623 $$ and $$ 326 $$

On adding , we get

$$ = -623 + 326 $$

$$ = -297 $$

Add:
$$ 259 $$ and $$214$$

$$259$$ and $$214$$

On adding , we get

$$=249+214$$

$$=473$$

Add :
$$ 13 $$ and $$ -15 $$

$$ 13 $$ and $$ -15 $$
On adding these numbers, we get
$$ 13 + (-15) = 13 - 15 $$
$$ = -2 $$

Subtract :
$$ 5 $$ from $$ 8 $$

$$ 5 $$ from $$ 8 $$

On subtracting , we get

$$ = 8 - 5 $$

$$ = 3 $$

In each of the following pairs, state which integer is greater :
$$0, 8$$

8 is greater than 0, as 8 lies on the right side of 0 on the number line

Hence, 8 is greater than 0 i.e. 8>0

Subtract :
$$-8$$ from $$ -2 $$

$$-8$$ from $$ -2 $$

On subtracting , we get

$$ = -2 - (-8) $$

$$ = -2 + 8 $$

$$ = 6 $$

Subtract :
$$ -5 $$ from $$ 8 $$

$$ -5 $$ from $$ 8 $$

On subtracting , we get

$$ = 8 - (-5) $$

$$ = 8 + 5 $$

$$ = 13 $$

Subtract :
$$ 4 $$ from $$-7 $$

Given numbers are $$ 4 $$ and $$-7 $$

On subtracting$$ 4 $$ from $$-7 $$, we get,

$$ = -7 - 4 $$

$$ = -11 $$

Hence, we get $$-11$$ on subtracting $$ 4 $$ from $$-7 $$.

Subtract :
$$ -78 $$ from $$ -12 $$

$$ -78 $$ from $$ -12 $$

On subtracting , we get

$$ = -12 - (-78) $$

$$ = -12 + 78 $$

$$ = 66 $$

Subtract :
$$ -6$$ from $$ -3 $$

$$ -6$$ from $$ -3 $$

On subtracting , we get

$$ = -3 - (-6) $$

$$ =-3 + 6 $$

$$ = 3 $$

The sum of two integers is $$-56 $$ . If one integer is $$ -42$$ , find the other.

It is given that
Sum of two integers $$ = -56 $$
One integer $$ = -42 $$
Other integer $$ = -56 - (-42) $$
By further calculation
$$ = -56 + 42 $$
$$ = -14 $$

In each of the following pairs,Which number is to the right of the other on the number line?
$$2,9$$

$$9( As\ 9 > 2)$$

In each of the following pairs,Which number is to the right of the other on the number line?
$$-6 , 6 $$

2007838_1866635_ans_75ff0ba240454a9ea9efaeba69167b36.jpg

Draw a number line and answer the following :
If we at $$-8$$ on the number line , in which direction should we move to reach $$-1?$$

Clearly, $$-1$$ is to the right of $$-8$$ Therefore, we should move towards the right direction.
1879490_1866700_ans_740ca2e1168843c09e4eaea8781a30ff.png

Represent the following numbers as integers with appropriate signs
An airplane is flying at a height two thousand meters above the group

$$+2000\ m$$

In each of the following pairs,Which number is to the right of the other on the number line?
$$ -11, 10 $$

$$10( As \ 10 > -11)$$

In each of the following pairs,Which number is to the right of the other on the number line?
$$0 , -1$$

$$0 (As\ 0 > -1)$$

Compare :
$$3$$ and $$-1$$

$$3$$ and $$-1$$
P is on the right of Q
Hence, $$ 3 > -1 $$.

Compare:
$$ \dfrac{-1}{4} $$ and $$0$$

$$ \dfrac{-1}{4} $$ and $$0$$

$$ \dfrac{-1}{4} $$ is a negative rational number which is always less than $$0$$.

Hence , $$ \dfrac{-1}{4} < 0 $$.

Find the general solution in positive integers of $$29x-42y=5$$.

In converting $$\dfrac { 42 }{ 29 } $$ into a continued fraction the convergent just before $$\dfrac { 42 }{ 29 } $$ is $$\dfrac { 13 }{ 9 } $$; we have therefore
$$29\times 13-42\times 9=-1$$;
$$\therefore 29\times 65-42\times 45=-5$$;
combining this with the given equation, we obtain
$$29\left( x+65 \right) =42\left( y+45 \right) $$;
$$\therefore \dfrac { x+65 }{ 42 } =\dfrac { y+45 }{ 29 } =t$$, an integer;
hence the general solution is
$$x=42t-65$$, $$y=29t-45$$.

Arrange the following integers in descending order with the help of numberline:
$$6, 0, (-1), 15$$.

Decending order:-

$$15>6>0>-1$$

1161805_710319_ans_d725e581ea2d440bab6293a06ec7f573.JPG

Find the nonary numbers $$303478,150732,264305$$.

$$(303478)_9+(150732)_9+(264305)_9$$

When we add the $$6^{\text{th}}$$ digits $$8+2+5$$ in nonary we get $$6$$ and carry $$1$$

When we add the $$5^{\text{th}}$$ digits and carry $$1+0+3+7$$ in nonary we get $$2$$ and carry $$1$$

When we add the $$4^{\text{th}}$$ digits and carry $$1+4+7+3$$ in nonary we get $$6$$ and carry $$1$$

When we add the $$3^{\text{rd}}$$ digits and carry $$1+3+0+4$$ in nonary we get $$8$$ and carry $$0$$

When we add the $$2^{\text{nd}}$$ digits and carry $$0+0+5+6$$ in nonary we get $$2$$ and carry $$1$$

When we add the $$1^{\text{st}}$$ digits and carry $$1+3+1+2$$ in nonary we get $$7$$ and carry $$0$$

Hence,

$$(303478)_9+(150732)_9+(264305)_9=728626$$

Solve in positive integers:
$$7x+4y+19z=84$$.

$$7x+4y+19z=84$$

The maximum value of $$z=[\dfrac{84}{19}]=4$$

If $$z=4,7x+4y=8$$. This will not have any positive intergral values of $$x$$ and $$y$$

If $$z=3,7x+4y=27\Rightarrow x=1 $$ and $$y=5$$.

if $$z=2,7x+4y=46\Rightarrow x=2$$ and $$y=8$$.

if $$z=2,7x+4y=46\Rightarrow x=6$$ and $$y=1$$.

if $$z=1,7x+4y=65\Rightarrow x=3$$ and $$y=11$$.

if $$z=1,7x+4y=65\Rightarrow x=7$$ and $$y=4$$

Do following addition with the help of number line:
$$(-4) + 9$$

Referring to above number line,

We will move to the right by $$9$$ units on number line, because we are adding.

$$(−4)+9=5.$$
1497360_710332_ans_b7064ff63d32439cb715d01e66775a55.PNG

In a class test containing $$10$$ questions, '$$3$$' marks are awarded for every correct answer and $$(-1)$$ mark is for every incorrect answer and '$$0$$' for questions not attempted.
(i) Gopi gets $$5$$ correct and $$5$$ incorrect answers. What is his score?
(ii) Reshma gets $$7$$ correct answer and $$3$$ incorrect answers. What is her score?
(iii) Reshma gets $$3$$ correct and $$4$$ incorrect answers out of seven questions she attempts. What is her score?

i) Gopi gets $$5$$ correct and $$5$$ incorrect answers

To find his score

Since for every correct answer '$$3$$ marks are awarded '

For every incorrect answer '$$(-1)$$ marks are awarded '

Total Score $$ = 5 \times 3 + 5 \times (-1)$$

$$ = 15 - 5 $$

$$ = 10 $$

ii) Reshma gets $$7$$ correct answer and $$3$$ incorrect answers

To find his score

Since for every correct answer '$$3$$ marks are awarded'

For Every Incorrect Answer ' (-1) marks are awarded '

Total score $$ = 7 \times 3 + 3 \times (-1)$$

$$ = 21 - 3 $$

$$ = 18 $$

iii) Reshma gets $$3$$ correct answer and $$4$$ incorrect answers

To find his score

Since for every correct answer '$$3$$ marks are awarded '

For every incorrect answer '$$(-1)$$ marks are awarded '

Total score $$ = 3 \times 3 + 4 \times (-1)$$

$$ = 9 - 4 $$

$$ = 5 $$

Fill in the bank with < , > , = sign.

a) (-3) +(-6)(-3)-(-6)
b)(-21)-(-10)(-31)+(-11)
c)45-(-11)54+(-4)
d)(-25)-(-42)(-42)-(-25)

a) ( - 3) + ( - 6)___( - 3) - ( - 6)

$$\Rightarrow -9<3$$

b) (- 21) - ( - 10)___( - 31)+( - 11)

$$\Rightarrow -11>-42$$

c) 45 - ( - 11)___54 + ( - 4)

$$\Rightarrow 56>50$$

d) ( - 25) - ( - 42)___( - 42) - ( - 25)

$$\Rightarrow 17>-17$$

Verify $$a-(-b)=a+b$$ for $$a=131,\ b=15$$.

Given $$a=131$$ and $$b=15$$.

Now, $$a-(-b)=131-(-15)=131+15=146$$ and $$a+b=131+15=146$$.

So, $$a-(-b)=a+b$$.

$$-23+\Box=34$$

$$-23+\Box=34$$ Let $$x$$ be in the boxThen,

$$-23+\boxed{x}=34$$

On moving $$-23$$ to the right-side it will become

$$+23$$

Therefore, $$\boxed{x}=34+23=57$$

Hence, $$-23+\boxed{57}=34$$

Simplify:
$$36\times(-3)+(-36)\times 7$$

$$36\times (-3)+(-36)\times 7$$

$$=-36(3+7)$$

$$=-360$$

Prove that the square of any positive integer is of the form $$5q$$, $$ 5q+1$$, $$5q+4$$ for some integer $$ q$$.

Since, any positive integer $$n$$ is of the form $$5m$$ or $$5m+1,$$ or $$5m+2$$ or $$5m+3$$ or $$5m+4.$$

If $$n=5m,$$ then

$${ n }^{ 2 }=25{ m }^{ 2 }=5\left( 5{ m }^{ 2 } \right) =5q,$$ where $$q=5m^{2}$$

If $$n=5m+1,$$ then

$$n^{2}=(5m+1)^{2}=5m(5m+2)+1=5q+1,$$ where $$q=m(5m+2)$$

If $$n=5m+2,$$ then

$$n^{2}=(5m+1)^{2}=5m(5m+4)+4=5q+4,$$ where $$q=m(5m+4)$$

If $$n=5m+3,$$ then

$$n^{ 2 }=(5m+3)^{ 3 }=5({ m }^{ 2 }+6m+1)+4=5q+4,$$ where $$q=5{ m }^{ 2 }+6m+1$$

If $$n=5m+4,$$ then

$$n^{ 2 }=5(5{ m }^{ 2 }+8m+3)+1=5q+1,$$ where $$q=5m^{2}+8m+3$$

Hence, $$n^{2}$$ is of the form $$5q$$ or, $$5q+1$$ or, $$5q+4.$$

Work out the answers to the following calculate the number line.
$$(+5)+(-3)$$

$$\left( +5 \right) +\left( -3 \right) =5-3=2$$

Find the addition inverse $$\dfrac{8}{5}$$ and $$\dfrac{3}{8}$$

For a number $$a$$, it's additive inverse $$c$$ would be such that $$a+c=0$$ where $$0$$ is additive identity.

Therefore the addition inverse of $$\dfrac{8}{5}$$ is $$-\dfrac{8}{5}$$ and the addition inverse of $$\dfrac{3}{8}$$ is $$-\dfrac{3}{8}$$

Solve
$$ - 12 - \left( {10 \div 5} \right) + 5 - \left( { - 2 \times 3 + 5} \right) + 6 - \left\{ {\left( { - 2 \times 3} \right) \times 1 + \left( { - 3 - 5} \right)} \right\}$$

$$-12-\left(10\div \:5\right)+5-\left(-2\times \:3+5\right)+6-\left\{\left(-2\times \:3\right)\times \:1+\left(-3-5\right)\right\}$$

Applying BODMAS rule,

$$=-12-2+5-\left(-2\times \:3+5\right)+6-\left\{\left(-2\times \:3\right)\times \:1+\left(-3-5\right)\right\}$$

$$=-12-2+5-\left(-1\right)+6-\left\{\left(-2\times \:3\right)\times \:1+\left(-3-5\right)\right\}$$

$$=-12-2+5-\left(-1\right)+6-\left(-14\right)$$

$$=-9+1+6-\left(-14\right)$$

$$=-2-\left(-14\right)$$

$$=-2+14$$

$$=12$$

Find: $$(-7)-8-(-25)$$

$$\textbf{Step 1: Open the bracket and simplify.}$$

$$(-7)-8-(-25)$$ $$= -7-8-(-25)$$

$$=-7-8+25$$

$$=25-7-8$$

$$=25-(7+8)$$

$$=25-15$$

$$=10$$

$$\textbf{Hence, the answer is 10.}$$

Find the sum of the following.
i) $$-5, -4, -6$$ and $$-17$$

Adding $$ - 5, - 4, - 6, - 17$$.

$$\left( { - 5} \right) + \left( { - 4} \right) + \left( { - 6} \right) + \left( { - 17} \right) = - 5 - 4 - 6 - 17$$

$$ = - 32$$

Work out the answers to the following calculate the number line.
$$(-7)+(+5)$$

$$\left( -7 \right) +\left( +5 \right) =-7+5=-2$$

Work out the answers to the following calculate the number line.
$$(-7)+(-1)$$

$$\left( -7 \right) +\left( -1 \right) =-7-1=-8$$

Work out the answers to the following calculate the number line.
$$(-5)+(-4)$$

$$\left( -5 \right) +\left( -4 \right) =-5-4=-9$$

Arrange the following integers in descending order.
$$412,-75,-90,63,27,0,-9$$

$$412,63,27,0,-9,-75,-90$$

Fill in the blanks with $$>,<$$ or $$=$$ sign
$$(-21)-(-10)__________(-31)-(-11)$$ ?

$$-11\quad is\quad greater\quad than\quad -20\\$$

$$ >$$

Fill in the blank.
$$(-4)+$$______$$=-12$$.

Let $$x$$ be the unknown

$$(-4)+ x = -12$$

$$x= -12- (-4)$$

$$x = -12+ 4$$

$$x= -8$$

$$(-4)+ (-8)=(-12)$$

Write an integer which is $$4$$ more than $$-5$$

Write an integer which is $$4\:more\:than\:-5$$

More means moving right on the number line.

So the answer is $$-1$$.

Write down 7 pairs of negative integers whose sum is $$-8$$.

Required $$7$$ pairs of negative integrs are,

$$(-1,-7),(-2,-6),(-3,-5),(-4,-4),(-5,-3),(-6,-2),(-7,-1)$$

A TV was bought at a price of Rs. $$21000$$. After $$1$$ year the value of the TV was depreciated by $$5\%$$. (Depreciation means reduction of the value due to use and age of the item). Find the value of the TV after $$1$$ year.

$$\textbf{Step 1: Finding the depreciated amount}$$

$$\text{Initial amount}=21000$$

$$\text{Depreciation rate}=5\%$$

$$\text{Depreciated amount}=\dfrac{5}{100}\times21000=1050$$

$$\textbf{Step 2: Finding the value of TV after 1 year}$$

$$\text{Value of TV after 1 year}=\text{Initial value}-\text{Depreciated amount}$$

$$\implies\text{Value of TV after 1 year}=21000-1050=19950$$

$$\textbf{Thus, the value of TV after 1 year is Rs. 19950}$$

Fill in the blanks with $$>,<$$ or $$=$$ sign
$$-45-(-11)__________57+(-4)$$ ?

$$-34\quad is\quad less\quad than\quad 53\\$$

$$ <$$

Fill in the blanks with $$>,<$$ or $$=$$ sign
$$(-25)+(-42)__________(-42)-(-25)$$

$$-67\quad is\quad less\quad than\quad -17\\$$

$$ <$$

At Jammu temperature was $$-5^o$$C on Monday and then it dropped by $$2^o$$C on Tuesday. What was the temperature of Jammu on Tuesday

Here initially temperature $$ t_1 $$ is$$ -5^0C $$ (on Monday )

then on Tuesday temperature $$ t_1 $$ is dropped by $$ 2^0 C $$ and became $$ t_2 ^\circ C $$

so what is $$ t_2 ^\circ C $$

here,

$$ t_2= (t_1 - 2) ^0 C $$

$$ t_2 = -5 -2 ^0 C $$

$$ t_2 = -7^0 c $$

so , on Tuesday temperature was $$ t_2 = -7^0 C $$

Compare −2,−3 using number line ?

To compare -2 and -3 using the number line, remember how numbers appear on a number line:

Now, look carefully at the number line above, and see where your numbers, -3 and -2 have been drawn. There are two ways of comparing these numbers using the number line:

First way: notice that -3 is to the left of -2. This means that -3 is smaller than -2 (because smaller numbers are always to the left).

Second way: notice that -2 is to the right of -3. This means that -2 is larger than -3 (because larger numbers are always to the right).

So, there are two ways to answer your question about how these number compare:

-3 is smaller than -2.
or
-2 is larger than -3.

Both answers are correct.

To compare 4 and -6 using the number line, remember how numbers appear on a number line:

Now, look carefully at the number line above, and see where your numbers, -6 and 4 have been drawn. There are two ways of comparing these numbers using the number line:

First way: notice that -6 is to the left of 4. This means that -6 is smaller than 4 (because smaller numbers are always to the left).
Second way: notice that 4 is to the right of -6. This means that 4 is larger than -6 (because larger numbers are always to the right).

So, there are two ways to answer your question about how these number compare:

-6 is smaller than 4.
or
4 is larger than -6.

Both answers are correct.

1113090_1110177_ans_815eaa58e0de4f9f8b2baf8cbb62cf39.png

Represent $$-15,\,20,\,15,\,-16,\,5,\,10,\,-9$$ on a number line.

1349724_1265172_ans_21a6bc9d9f534472b6d171dccbe0d0bc.PNG

Find the number of 4 digit positive integers if the product of their digits is divisible by 3.

1179263_1241174_ans_3160d37561fe40e7b4bc577e0489c2c3.jpg

Using the number line write the integer which is:
$$(a)$$ $$ 3$$ more than $$ 5$$
$$(b)$$ $$ 5$$ more than $$- 5$$

1087851_1180172_ans_f2ac45b7a1af40b9b94d4192c89034f6.png

Mark -3, -7, -4, -8, and -1 on the number line

Solution:

Mark $$-3,\ -7,\ -4,\ -8$$ and $$-1$$ on number line

1355268_1199109_ans_89a05b1d24264678b7851a505dc0b1c1.png

Represent $$-4,\,-6,\,5$$ and $$7$$ on the number line.

1349732_1259279_ans_c5c6563610074dd6a63d5963fe09e6b9.PNG

The denominator of a fraction is 3 more than the numerator.If $$5$$ is added to both parts, the resulting fraction is equivalent to $$\dfrac { 4 } { 5 } .$$ Find the fraction.

Let number is $$\dfrac{x}{x+3}$$

Now adding $$5$$

$$\Rightarrow \dfrac{x+5}{x+8}=\dfrac{4}{5}$$

$$\Rightarrow 5x+25=4x+32$$

$$\Rightarrow x=7$$.

$$\therefore$$ the fraction is $$\dfrac{x}{x+5}=\dfrac{7}{7+5}=\dfrac {7}{12}$$

What time it would be
$$6$$ hours $$25$$ minutes after $$4:12p.m.$$

1303231_1231041_ans_f2c663b1efc34a3cb679539191b1f56f.jpg

Fill in the blank with $$>,\ <$$ or $$=$$ sign
$$(-3)+(-6)$$________$$(-3)-(-6)$$

According to question,

L.H.S= $$-3-6$$

=$$-9$$

R.H.S=$$-3+6$$

=$$3$$

L.H.S is less than R.H.S

So, sign $$<$$ is used to fill the blank

Fill in the blanks
Number which are less than zero are called_____

Number which are less than $$300$$ is called negative number

The final marks in mathematics of $$30$$ student are as follows:
$$53,61,48,60,78,68,55,100,67,90,75,88,77,37,64,58,60,48,62,56,44,58,52,64,98,$$
$$59,70,39,50,60$$
Arrange these marks in the ascending order $$30$$ or $$39$$ one group $$40$$ to $$49$$ second group ,etc.
Now answer the following:
Which observations between $$50$$ and $$60$$ have not actually appeared?

Ascending order of the scores is,

$$\rightarrow 37, 39$$

$$\rightarrow 44, 48, 48$$

$$\rightarrow 50, 52, 53, 55, 56, 58, 58, 59$$

$$\rightarrow 60, 60, 60, 61, 62, 64, 64, 67, 68$$

$$\rightarrow 70, 75, 77, 78$$

$$\rightarrow 88$$

$$\rightarrow 90, 98$$

$$\rightarrow 100$$

Hence, The observations between $$50$$ and $$60$$ have not actually appeared are $$51, 54, 57$$

Represent $$\sqrt{4.8}$$ on a number line

1912890_1273743_ans_9010b75f1984494e8cfb177efa67d5b9.jpg

Represent $$-8,\,-10,\,8$$ and $$12$$ on the number line.

1349649_1283149_ans_fdf821a449ff4d8184d331a3813f0624.PNG

Calculate:$$1-2+3-4+5-6+...-2010+2011$$

1349667_1270605_ans_d8163521abbd479a8ca49a12849cf44e.PNG

Write in ascending order:$$-100,\,-4,\,-200,\,100,\,200,\,-150,\,20$$

We know that positive integers lie to the right of zero on the number line and negative integers to the left of zero.

So positive integers are always greater than negative integers.

Now, let's compare positive integers first

$$20,~100,~200$$

$$\implies 20<100<200$$ -----(1)

Now, let's compare negative integers

$$-100,~-4~-200,~-150$$

We know that least negative number is the greatest among negative numbers

$$\implies -200<-150<-100<-4$$ -----(2)

Combining (1) and (2) we get,

$$\implies -200<-150<-100<-4<20<100<200$$

Represent $$-2,\,-12,\,6$$ and $$10$$ on the number line.

1349650_1286122_ans_7655d1f0e88c435b976eff8a9694b1c2.PNG

Represent $$-6,\,-8,\,6$$ and $$12$$ on the number line.

1349647_1290481_ans_ad2f9f595c7f4e80bc4245a2b0f6be06.PNG

Represent $$-4,\,-5,\,2$$ and $$9$$ on the number line.

1349645_1294178_ans_03f98a0e81cd4928a964d626ec421518.PNG

The difference of two integers is $$18$$.If one of them is $$-40$$.Find the other.

1349671_1270375_ans_fad06af3610e43f5a552e181e2dec28c.PNG

Represent $$-7,\,-9,\,4$$ and $$11$$ on the number line.

Point $$A$$ represent $$-7,$$ point $$B$$ represent $$-9,$$ point $$C$$ represent $$4$$ and point $$D$$ represent $$11.$$
1349646_1286394_ans_097295a50b7b4d4c9a0277dc5716c679.png

Find the number of non negative integer solutions of $$x+2y+3z=30$$

Let $$x, y, z$$ are non- negative integers.

$$\begin{array}{l} x>=0,\, \, y>=0,\, \, z>=0 \\ x+2y+3z=0 \\ x=30-2y-3z\to (i) \end{array}$$

$$10$$ possible values of $$z$$ are $$0,1,2,3,4,5,6,7,8,9,10$$

$$if\,\,z = 1,\,\,x = 30 - 2y$$, $$y$$ have value from $$0, 1.......5$$

i.e, $$16$$ possibilities

$$ \Rightarrow if\,\,z = 1,\,\,x = 27 - 2y,\,\,y$$ have $$14$$ possibilities

$$ \Rightarrow if\,\,z = 2\,\,x = 24 - 2y,\,\,y$$ have $$13$$ possibilities

$$ \Rightarrow if\,\,z = 3\,\,x = 21 - 2y,\,\,y$$ have $$11$$ possibilities

$$ \Rightarrow if\,\,z = 4\,x = 18 - 2y,\,\,y$$ have $$10$$ possibilities

Total possibilities $$ = 16 + 14 + 13 + 11 + 10 + 8 + 7 + 5 + 4 + 2 + 91$$

If $$10 ^ { 2017 } - 2017$$ is expressed as integer, what is the sum of its digits ?

$$10^{2017}-2017$$

If $$x > 1$$,

In $$10^{x}-2017 $$ last $$4$$ digit are always $$7,9,8$$ and $$3$$.

$$10^{4}-2017=7983$$

$$10^{5}-2017=97983$$

$$10^{6}-2017=997983$$

$$10^{7}-2017=9997983$$

Thus, the sum of the digits in $$10^{x}-2017 $$ $$= (x-4).9+7+9+8+3$$

Hence, for $$10^{2017}-2017 $$

Sum of the digits $$=(2017-4).9+7+9+8+3$$

$$= 2013\times9+7+9+8+3$$

$$= 18117+7+9+8+3$$

$$= 18,144$$

Which of the following are expressions with numbers only?
(a) $$y + 3$$ (b) $$(720) 8z$$ (c) $$5(217) + 72$$ (d) $$5$$

(e) $$3x$$ (f) $$55n$$ (g) $$(720)(510)45 + p$$

The expression with only numbers are

$$(c)\, 5(217)+72$$ and $$(d) \,5$$

These expressions do not contain any variables.

Draw number lines and locate the point on them :
$$\dfrac {1}{2},\dfrac {1}{4},\dfrac {3}{4},\dfrac {4}{4}$$

1230426_1298218_ans_e4107f2d44a74d0cad85eaf1475d3892.png

What is the predecessor of -9871?

Predecessor of of any number $$ =$$ number$$ - 1$$

$$\Rightarrow$$ Predecessor of $$-9871 = -9871 - 1$$

$$= - 9872$$

Fill in the blanks.
The greatest negative integer in ______.

1975310_1324027_ans_c3b89b3041e24f20a71aa79d326406de.jpg

Represent $$-5,\,-6,\,3$$ and $$8$$ on the number line.

1349644_1295062_ans_fd1f178202524598be695fb019a56bc9.PNG

Find the sum of all three-digits natural numbers which leave the remainder $$3$$, when divided by $$5$$.

1293045_1349476_ans_68d966b927f3471b96ebc4ee7bc42e82.jpg

Solve:$$-\left(x-3\right)+4>-2x+5$$ when $$x$$ is a integer.

$$-\left(x-3\right)+4>-2x+5$$

$$\Rightarrow\,-x+3+4>-2x+5$$

$$\Rightarrow\,-x+7>-2x+5$$

$$\Rightarrow\,-x+7-7>-2x+5-7$$

$$\Rightarrow\,-x>-2x-2$$

$$\Rightarrow\,-x+2x>-2x-2+2x$$

$$\Rightarrow\,x>-2$$

$$\therefore\,x\in\,\left\{-1,0,1,2,3,...\right\}$$

Represent $$\sqrt{6}$$ on the number line.

Mark a distance 6 units from a fixed point A on a given line to obtain a point B such that AB=6 units.From B mark a distance of 1 unit and mark the new point C.Find the mid-point of AC and mark the point as O.Draw a semicircle with centre O and radius OC.Draw a line perpendicular to AC passing through B and intersecting the semicircle at D .Then BD=√6

1238735_1297112_ans_26be8b42f0b945339f97aa1d106e2796.png

The difference between two numbers is $$15.$$Taking the smaller number as $$x;$$ find
$$(i)$$ the expression for larger number.
$$(ii)$$ the larger number, if the sum of these numbers is $$71$$

Difference of two numbers$$=15$$

Let smaller number$$=x$$

$$\therefore\,$$Second number$$=x+15$$

$$\therefore\,$$Larger number$$=x+15$$

If sum of two numbers$$=71$$

Then $$x+x+15=71$$

$$(i)\,2x+15=71$$

$$\Rightarrow\,2x=71-15=56$$

$$\Rightarrow\,x=\dfrac{56}{2}=28$$

$$(ii)$$Larger number$$=x+15=28+15=43$$

Subtract the sum of $$865$$ & $$-493$$ from the sum of $$-380$$ & $$675$$.

Sum of $$865$$ and $$-493=865+\left(-493\right)=865-493=372‬$$

Sum of $$-380$$ and $$675=-380+675=675-380=295‬$$

Subtracting $$372$$ from $$295$$

$$=295-372=-77‬$$

The sum of two integers is $$-278$$. If one of the integers is $$-156$$, find the other.

Sum of two integers$$=x+y=-278$$

One of them is$$y=-156$$

$$\Rightarrow$$ the other number$$=x=$$-278-y$$

$$\Rightarrow x=-278-\left(-156\right)$$

$$\Rightarrow x=-278+156=-122‬$$

$$\therefore$$ the other number $$x=-122$$

Which number in each of the following pairs is smaller?
(i) $$0, -4$$
(ii) $$-3, 12$$
(iii) $$8, 13$$
(iv) $$ -15, -27$$

(i) $$0, -4$$

The smaller number is $$-4$$

(ii) $$-3,12$$

The smaller number is $$-3$$

(iii) $$8,13$$

The smaller number is $$8$$

(iv) $$ -15,-27$$

The smaller number is $$-27$$

Indicate the following by using integers:
(i) $$25^o$$ above zero
(ii) $$5^o$$ below zero
(iii) A profit of Rs 800
(iv) A deposit of Rs 2500
(v) 3 km above sea level
(vi) 2 km below sea level.

(i) $$+25^o$$
(ii) $$-5^o$$
(iii) $$+800$$
(iv) $$+2500$$
(v) $$ +3$$
(vi) $$-2$$

At Srinagar temperature was $$-5^{\circ}C$$ on Monday and then it dropped by $$2^{\circ}C$$ on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday. It rose by $$4^{\circ}C$$. What was the temperature on this day?

On Monday, temperature at Srinagar $$= -5^{\circ}C$$
On Tuesday, temperature dropped $$= 2^{\circ}C$$
$$\therefore$$ Temperature on Tuesday $$= -5^{\circ}C - 2^{\circ}C = -7^{\circ}C$$
On Wednesday, temperature rose up $$= 4^{\circ}C$$
$$\therefore$$ Temperature on Wednesday $$= -7^{\circ}C + 4^{\circ}C = -3^{\circ}C$$.

If $$a=-10,b=-4$$, show that $$a-b\neq b-a$$.

L.H.S$$=a-b=-10-\left(-4\right)=-10+4=-6$$

R.H.S$$=b-a=-4-\left(-10\right)=-4+10=6$$

$$\therefore -6\neq 6$$

$$\Rightarrow$$L.H.S$$\neq$$R.H.S

Hence $$a-b\neq b-a$$

Hence proved.

Subtract:
$$32$$ from $$-56$$

Given numbers are $$32$$ and $$-56$$

We need to subtract $$32$$ from $$-56$$

So, $$-56-32$$

$$=-88$$

Hence, we get $$-88$$ on subtracting $$32$$ from $$-56$$.

Subtract the sum of -340 and 170 from -45.

Given numbers are $$-340,\ 170$$ and $$-45$$.

To find out,

The result when we subtract the sum of $$-340$$ and $$170$$ from $$-45$$.

Sum of $$-340$$ and $$170$$:

$$-340+170=-170$$

Subtracting $$-170 $$ from $$-45$$:

$$-45-(-170)=-45+170$$

$$=125$$

Hence, the result when we subtract the sum of $$-340$$ and $$170$$ from $$-45$$ is $$125$$.

Mark the following integers on a number line:
(i) $$7 $$
(ii) $$-4$$
(iii) $$0$$

1671111_1658688_ans_cd9504e0b59a473d91889ea8f2364989.png

Write all integers between:
(i) $$-7\, and \,3$$
(ii) $$-2\, and \,2 $$
(iii) $$-4\, and \,0$$
(iv) $$0 \,and\, 3 $$

(i) $$ -6, -5, -4, -3, -2, -1, 0, 1, 2$$

(ii) $$ -1, 0, 1$$

(iii) $$ -3, -2, -1$$

(iv) $$1,2$$,

Using the number line, write the integer which is:
(i) 2 more than 3
(ii) 5 less than 3
(iii) 4 more than -9

1671107_1658700_ans_713a29aa37184d6784bdef7f058b4380.png

Flow many integers are between?
(i) $$-4\, and \,3$$
(ii) $$5\, and\, 12$$
(iii) $$-9\, and\, -2$$
(iv) $$0\, and \,5$$

(i) $$6$$
(ii) $$ 6$$
(iii) $$ 6$$
(iv) $$4$$

Which number in each of the following pairs is larger?
(i) $$3, -4$$
(ii) $$-12, -8 $$
(iii) $$0, 7 $$
(iv) $$12, -18$$

(i) We know that $$3$$ is larger than $$-4$$ on the number line.

So we get $$3 > -4$$

Therefore, $$3$$ is larger.

(ii) We know that $$-8$$ is larger than $$-12$$ on the number line.

So we get $$-8 > -12$$

Therefore, $$-8$$ is larger.

(iii) We know that $$7$$ is larger than $$0$$ on the number line.

So we get $$7 > 0$$

Therefore, $$7$$ is larger.

(iv) We know that $$12$$ is larger than $$-18$$ on the number line.

So we get $$12 > -18$$

Therefore, $$12$$ is larger.

Write the following integers in decreasing order:
(i) $$-15, 0, -2, -9, 7, 6, -5, 8$$
(ii)$$ -154, 123, -205, -89, -74 $$

(i) $$ 8, 7, 6, 0, -2, -5, -8, -15$$
(ii) $$ 123, -74, -89, -154, -205$$

Write the following integers in increasing order:
(i) $$-8, 5, 0, -12, 1, -9, 15 $$
(ii) $$-106, 107, -320, -7, 185$$

(i) $$ -8, 5, 0, -12, 1, -9, 15$$

Ascending order of integers: $$ -12, -9, -8, 0, 1, 5, 0, 15$$

(ii) $$ -106, 107, -320, -7, 185$$

Ascending order: $$ -320, -106, -7, -107, 185$$

Subtract the sum of $$-5020$$ and $$2320$$ from $$-709$$.

$$-709-(-5020+2320)=1991$$

The sum of two integers is $$238$$. If one of the integers is $$-122$$, determine the other.

1502662_1658736_ans_2561a83b942b4e81b03423f0b832a8f9.JPG

Find the sum of:
(i) $$-557\, and\, 488$$
(ii) $$-522\, and\, -160$$
(iii) $$2567\, and\, -325$$
(iv) $$-10025\, and \,139$$
(v) $$2547\, and\, -2548$$
(vi) $$ 2884\, and\, -2884$$

(i) $$-557+488 =-69$$

(ii) $$-522+(-160)=-682$$

(iii) $$2567+(-325)=-2242$$

(iv) $$-10025+139=-9886$$

(v) $$2547+(-2548)=-1$$

(vi) $$ 2884+(-2884)=0$$

The sum of two integers is $$-223$$. If one of the Integers is $$172$$, find the other.

1502666_1658737_ans_5e7dd1d72f6a4efab7f75fe88de012b1.JPG

Write all integers whose absolute values are less than 5.

$$= -4, -3, -2, -1, 0, 1, 2, 3, 4$$

Fill in the blanks:
(i) $$-7+ .... = 0$$
(ii) $$29+ ..... = 0$$
(iii) $$132+ ( -132) = ...$$
(iv) $$-14 + ..... = 22$$
(v) $$-1256 + .... = -742 $$
(vi) $$ .... -1234 = -4539$$.

(i) 7
(ii) -29
(iii) 0
(iv) 36
(v 154
(vi) - 3305

The largest negative integer is ....... .

$$= -1$$

What will be the sign of the product if we multiply together
$$199$$ negative integers and $$10$$ positive integers?

1590737_1675746_ans_25472d1173c24b25a3e16bd34bc328c6.jpg

The smallest positive integer is ....... .

$$\textbf{Step -1: Define integers.}$$

$$\text{Integers is the set containing positive numbers, negative numbers and zero.}$$

$$\textbf{Step -2: Represent integers on the number line.}$$

$$\textbf{Step -3: Find smallest positive integer.}$$

$$\text{The smallest positive interger is }1.$$

$$\textbf{Hence, the smallest positive integer is 1.}$$

What will be the sign of the product if we multiply together
$$21$$ negative integers and $$3$$ positive integers?

1590727_1675743_ans_a02d521cfc2d48308239c1e46d4a8f0c.jpg

If $$a\times (-1)=30$$, then $$a$$ is positive or negative?

Given that,

$$a\times (-1)=30$$

Now,

$$a\times (-1)=30$$

$$\Rightarrow -a=30$$

$$\Rightarrow a=-30$$

Hence, $$a$$ is a $$negative$$ integer .

If $$a\times (-1)=-30$$, is the integer a positive or negative?

$$a\times (-1) = -30$$

$$-a = -30$$

$$a = 30$$

So, $$a$$ is positive.

Write the opposite of the following:
$$24$$

The opposite of $$24$$ is $$-24$$

Because they lie on opposite side of zero at exactly same distance on number line.

Indicate the following using $$"+"$$ or $$"-"$$ sign:
$$7^{o}C$$ below the freezing point

So, $$7^{o}C$$ below the freezing point indicates $$-7^{o}C$$

Write the opposite of the following:
Earning $$Rs. 500$$

The opposite statement of earning a $$Rs. 500$$ is

Spending $$Rs. 500$$

Indicate the following using $$"+"$$ or $$"-"$$ sign:
A gain of $$Rs. 600$$

A gain means profit or addition.

So, A gain of rupees $$600$$ indicates $$+ Rs. 600$$ .

Write the opposite of the following:
A deposit of $$Rs. 100$$

The opposite statement of A deposit of $$Rs. 100$$ is

A withdrawal of $$Rs. 100$$

Indicate the following using $$"+"$$ or $$"-"$$ sign:
Decrease of $$9$$

Decrease means minus from amount.

So, decrease of $$9$$ indicates $$-9$$

Write the opposite of the following:
Going 6 km to the east.

If we are facing north, then going $$6\ km$$ to the east means we are looking at the right side of $$0$$ on the number line.

Hence, $$6\ km$$ is $$+6$$ on the number line.

The opposite of that is $$-6$$, which is in the negative direction or towards left of the $$0$$.

Towards left is west direction. So, $$-6$$ is $$6$$ units in the west of $$0$$.

Hence, the opposite statement of going $$6 \ km$$ to the east is, going $$6 \ km$$ to the west.

Write the opposite of the following:
$$5^{o}C$$ below the freezing point

The opposite statement of $$5^{o}C$$ below the freezing point is

$$5^{o}C$$ above the freezing point

Write the opposite of the following:
$$-34$$

The opposite of $$-34$$ is $$34$$.

Because they lie on opposite side of zero at exactly same distance on number line.

Indicate the following using $$"+"$$ or $$"-"$$ sign:
A loss of $$Rs. 800$$

A loss means subtraction from amount.

So, A loss of rupees $$800$$ indicates $$-Rs. 800$$

Indicate the following using $$"+"$$ or $$"-"$$ sign:
A withdrawal of $$Rs. 300$$

Withdrawal means subtraction from amount.

So, A withdrawal of rupees $$300$$ indicates $$-Rs. 300$$

Mark the following integer on a number line:
$$7$$

Here $$7$$ lies on the right side of the $$0$$ on the number line

1631848_1777402_ans_111d60c31910433fb85c58f5471199b5.png

Which number is larger in the following pair?
$$-365$$, $$-913$$

As, $$-913$$ lies on left hand side of $$-365$$ on number line.

Hence, $$-913$$ is smaller than $$-365$$

Which number is smaller in the following pair?
$$0$$, $$-1$$

Since every negative integer is smaller than $$0$$.

Hence $$-1$$ is smaller than $$0$$

Which number is smaller in the following pair?
$$-317$$, $$-603$$

As, $$-603$$ lies on left hand side of $$-317$$ on number line.

Hence, $$-603$$ is smaller than $$-317$$

Indicate the following using $$"+"$$ or $$"-"$$ sign:
$$2 \,km$$ above sea level

Above indicates addition in value.

So, $$2 \,Km$$ above sea level indicates $$+2 \,km$$

Indicate the following using $$"+"$$ or $$"-"$$ sign:
$$3 km$$ below sea level

$$3 km$$ below sea level indicates $$–3 km$$

Mark the following integers on a number line:
$$-13$$

Here $$-13$$ lies on the left side of the $$0$$ on the number line

1631860_1777407_ans_c5e9f5cb96ad404fac413f90c4900083.png

Which number is larger in the following pair?
$$-16$$, $$8$$

Since every positive integer is greater than negative integer.

Hence $$8$$ is larger than $$-16$$

Which number is smaller in the following pair?
$$-26$$, $$17$$

Since every negative integer is smaller than positive integer.

Hence, $$-26$$ is smaller than $$17$$

Subtract $$-2768$$ from $$-287$$.

To find: $$-287 – (-2768) $$

$$= -287+ 2768$$

$$=2768-287$$

$$=2481$$

Which number is smaller in the following pair?
$$-777$$, $$7$$

Since every negative integer is smaller than positive integer.

Hence, $$-777$$ is smaller than $$7$$

Use the number line and add the following integers:
$$(-4) + (-7)$$

$$(-4) + (-7) $$

$$-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5$$

$$-7 $$ is smaller than $$-4$$.

So, we have to go $$-4$$ places right from $$-7$$, which means we have to go $$4$$ place left from $$-7$$.

Which is equal to number $$-11$$

Hence, $$(-4) + (-7) =-11$$

Use the number line and add the following integers:
$$5 + (-2) + (-6)$$

$$5+ (-2) + (-6)$$

$$1^{st}$$ we will add $$5$$ and $$-2$$

$$(5) + (-2) $$

$$-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5$$

$$-2 $$ is smaller than $$5$$.

So, we have to go $$5$$ places right from $$-2$$, which is equal to number $$3$$

Hence,$$(5) + (-2) =3$$

$$\therefore 5 + (-2) + (-6) = 3+(-6)$$

$$-6 $$ is smaller than $$3$$.

So, we have to go $$3$$ places right from $$-6$$, which is equal to number $$-3$$

Hence,$$\therefore 5+ (-2) + (-6) = 3+(-6)=-3$$

Use the number line and add the following integers:
$$(-2) + (-8)$$

$$(-2) + (-8) $$

$$-11,-10,-9,-8,-7,-6,-5,-4,-3, -2, -1 \ ,0 \ ,1 \ ,2 \ ,3 \ ,4,\ 5$$

$$-8 $$ is smaller than $$-2$$.

So, we have to go $$-2$$ places right from $$-8$$, which means we have to go $$2$$ place left from $$-8$$.

Which is equal to number $$-10$$

Hence,$$(-2) + (-8) =-10$$

Use the number line and add the following integers:
$$3 + (-2) + (-4)$$

$$3 + (-2) + (-4) $$

$$1^{st}$$ we will add $$3$$ and $$-2$$

$$(3) + (-2) $$

$$-2 $$ is smaller than $$3$$.

So, we have to go $$3$$ places right from $$-2$$, which is equal to number $$1$$

Hence,$$(3) + (-2) =1$$

$$\therefore 3 + (-2) + (-4) = 1+(-4)$$

$$-4 $$ is smaller than $$1$$.

So, we have to go $$1$$ places right from $$-4$$, which is equal to number $$-3$$

$$\therefore 3 + (-2) + (-4) = 1+(-4)\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =-3$$

Add :
$$+178$$
$$\underline{-69}$$

$$178+(-69)=178-69=109$$

$$+178$$

$$\underline{-69}$$

$$\boxed{ \, \,109 \, \, \,}$$

Add :
$$-3596$$
$$\underline{-1089}$$

$$-3596+(-1089)=-(3596+1089)=-4685$$

So,

$$-3596$$

$$\underline{-1089}$$

$$\boxed{ \, \,-4685 \, \, \,}$$

Add :
$$-1065$$
$$\underline{-987}$$

$$-1065+(-987)=-(1065+987)=-2052$$

So,

$$-1065$$

$$\underline{-987}$$

$$\boxed{ \, \,-2052 \, \, \, \,}$$

Add :
$$-493$$
$$\underline{+289}$$

$$-493+289=-204$$

So,

$$-493$$

$$\underline{+289}$$

$$\boxed{ \, \,-204 \, \, \,}$$

Add :
$$-103$$
$$\underline{+312}$$

$$-103+312=209$$

So,

$$-103$$

$$\underline{+312}$$

$$\boxed{ \, \,209 \, \, \,}$$

Add :
$$-73$$
$$\underline{-687}$$

$$-73+(-687)=-(73+687)=-760$$

So,

$$-73$$

$$\underline{-687}$$

$$\boxed{ \, \,-760 \, \, \,}$$

Add :
$$-206$$
$$\underline{+98}$$

$$-206+98=-108$$

So,

$$-206$$

$$\underline{+98}$$

$$\boxed{ \, \,-108 \, \, \,}$$

Find the sum of
$$1001$$ and $$-13$$

$$1001 + (-13) $$

$$1001-13= 988$$

$$988$$ is the sum of $$1001$$ and $$-13$$

The sum of two integers is -If one of them is 53, find the other.

By assuming Closure property of addition [a+b=c],
Let,
$$a=53,\ assume\ b=x, c=-16$$
$$=53+x=-16$$
$$= x =-16-53$$
( By sending 53 from left hand side to the right hand side, it changes to -53)
$$= x = -69$$
By the calculation we got another integer is $$-69$$

The sum of two integers isIf one of them is -31, find the other

By assuming Closure property of addition [a+b=c],
Let,
$$a=-31,\ assume\ b=x, c=65$$
$$= -31 + x = 65$$
$$= x = 65+31$$
( By sending -31 from left hand side to the right hand side, it changes to 31)
$$= x = 96$$
By the calculation we got another integer is $$96$$

Write the opposite of the following :
$$ 30\,km $$ East of Delhi

$$ 30 \,km $$ West of Delhi

Find the sum of
$$-36$$ and $$1027$$

$$-36 + 1027$$

$$ = 991$$

$$991$$ is the sum of $$-36$$ and $$1027$$

Find the sum of
$$-389$$ and $$-1032$$

$$-389 + (-1032)$$

$$=-389-1032$$

$$=-(389+1032) = -1421$$

$$-1421$$ is the sum of $$-389$$ and $$-1032$$

Find the sum of $$-3057$$ and $$199$$

$$-\ 3\ 0\ 5\ 7$$

$$ +\ \ \ \ 1\ 9\ 9$$

------------------

$$ -\ 2\ 8\ 5\ 8$$

Hence $$-2858$$ is the sum of $$-3057$$ and $$199$$

Find the sum of
$$3002$$ and $$-888$$

$$3002 + (-888)$$

$$3002-888 = 2114$$

$$2114$$ is the sum of $$3002$$ and $$-888$$

Write each of the following appropriate sign '+' or '-' :
$$ 240\,m $$ below sea level.

$$ -240\,m $$ from sea level

Write the opposite of the following :
$$ 325\,BC $$.

$$ 325 \, AD $$.

Write each of the following appropriate sign '+' or '-' :
Depositing $$ Rs. 2500 $$ in a bank.

$$ +Rs. 2500 $$ in a bank

Write each of the following appropriate sign '+' or '-' :
A jet plane flying at a height of $$ 9320\,m $$.

a jet plane is flying at $$ +9320\,m $$

Write each of the following appropriate sign '+' or '-' :
Loss of $$ Rs. 470 $$.

$$ - Rs.470 $$

Write each of the following appropriate sign '+' or '-' :
Earning $$ Rs. 1340 $$.

$$ +Rs. 1340 $$

Write the opposite of the following :
$$ 200 \,m $$ above sea level.

$$ 200 \,m $$ below sea level.

Write the opposite of the following :
Spending $$ Rs. 2700 $$.

Earning $$ Rs. 2700 $$.

Write each of the following appropriate sign '+' or '-' :
$$ 20^{\circ} \,C $$ below freezing point.

$$ -20^{\circ}\,C $$ from freezing point

Write each of the following appropriate sign '+' or '-' :
Gain of $$ 3 \,kg $$ Weight.

$$ +3 \,kg $$ weight.

In each of the following pairs, which number is to the left of the other on the number line?
$$ 2 , -5 $$

As $$-5<2$$

Hence, $$ -5 $$ is in the left of $$2$$ on number line

Evaluate the following :
$$ (-8) + (-14) $$

$$ (-8) + (-14) $$
$$ = -8 - 14 = -22 $$

In each of the following pairs, which number is to the left of the other on the number line?
$$ -4 , -7 $$

From the above image, we can conclude that $$ -7$$ is on the left of $$-4$$ on the number line.

1757645_1786258_ans_1825e71a56a74bcf8c8847f201bc51f5.png

In each of the following pairs, which number is to the left of the other on the number line?
$$ -10 , -16 $$

As $$-16<-10$$

Hence, $$ -16$$ is in the left of $$-10$$ on number line

In each of the following pairs, which number is to the right of the other on the number line?
$$ -3 , -5 $$

As $$-5<-3$$

Hence, $$ -3 $$ is in the right of $$-5$$ on number line

Write each of the following appropriate sign '+' or '-' :
$$ 6\,m $$ down in the basement of a building.

$$ -6\,m $$ in the basement of a building

In each of the following pairs, which number is to the right of the other on the number line?
$$ 2 , -7 $$

As $$-7<2$$

$$\therefore 2 $$ is in the right of $$-7$$ on number line

In each of the following pairs, which number is to the left of the other on the number line?
$$ -3 , 0 $$

As $$-3<0$$

Hence, $$ -3 $$ is in the left of $$0$$ on number line

In each of the following pairs, which number is to the right of the other on the number line?
$$ 0 , -2 $$

As $$-2<0$$

Hence, $$ 0 $$ is in the right of $$-2$$ on number line

Draw a number line and answer the following questions :
(i) Which integers lie between $$ -9 $$ and $$ -2 $$ ?
(ii) Which is the largest among them ?
(iii) Which is the smallest among them ?

(i) The integers are $$ -8 , -7 , -6 , -5 , -4 , -3 $$
(ii) The largest integer among them is $$ -3 $$
(iii) The smallest integer among them is $$ -8 $$
1759624_1786277_ans_f0299e26657a42c89285f1bcb64b7823.PNG

1759624_1786277_ans_f0299e26657a42c89285f1bcb64b7823.PNG

Evaluate the following :
$$ (-203) + 501 $$

$$ (-203) + 501 $$
$$ = -203 + 501 = 298 $$

Evaluate the following :
$$ -35 + (-47) $$

$$ -35 + (-47) $$
$$ = -35 - 47 = -82 $$

Evaluate the following :
$$ -346 - (-1275) $$

1975278_1786556_ans_17500b38dc9e4982b5132b672b188692.jpg

Evaluate the following :
$$ -1083 + (-3974) $$

$$ -1083 + (-3974) $$

$$ = - 1083 - 3974 $$

$$ = -(1083 + 3974) = - 5057 $$

Evaluate the following :
$$ 91 + (-48) $$

$$ 91 + (-48) $$

$$ = 91 - 48 = 43 $$

Evaluate the following :
$$ 706 + (-394) $$

$$ 706 + (-394) $$
$$ = 706 - 394 = 312 $$

Evaluate the following :
$$ (-36) + 29 $$

$$ (-36) + 29 $$
$$ = -36 + 29 = -7 $$

Evaluate the following :
$$ 1309 + (-2811) $$

$$ 1309 + (-2811) $$

$$ = -2811 + 1309 = - 1502 $$

Subtract :
$$ -725 $$ from $$ -63 $$

$$ -63 - (-725) = -63 + 725 = +662 $$

Evaluate the following :
$$ (-131) + 97 $$

$$ (-131) + 97 $$
$$ = -131 + 97 = -34 $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ -30 $$ ......... $$ -19 $$

As $$-19$$ is greater than $$-30$$

Hence,$$ -30 $$ < $$ -19 $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ -6 $$ ......... $$ -2 $$

As $$-2$$ is greater than $$-6$$

Hence,$$ -6 $$ < $$ -2 $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ -3 $$ ......... $$ 7 $$

Since, $$7$$ is greater than $$-3$$

Hence, $$ -3 $$ < $$ 7 $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ -5 $$ ......... $$ -13 $$

As $$-5$$ is greater than $$-13$$

Hence,$$ -5 $$ > $$ -13 $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ 0 $$ ......... $$ -2 $$

As $$0$$ is greater than $$-2$$

Hence,$$ 0 $$ > $$ -2 $$

Evaluate the following :
$$ -190 - (-3512) $$

1975279_1786573_ans_c07bad1b10824a07b8cbc62cb22a10a8.jpg

The sum of two integers is $$ 17 $$ . If one of them is $$ -35 $$ , find the other.

Let the two numbers be $$x$$ and $$y$$.

One number $$x = -35 $$
Sum of two integers $$x+y = 17 $$
$$y=17-x$$

$$ = 17 - (-35) $$
$$ = 17 + 35 = 52 $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ -10 $$ ......... $$ -11 $$

As $$-10$$ is greater than $$-11$$

Hence,$$ -10 $$ > $$ - 11 $$

Evaluate the following.
$$ 13 - 5 $$

$$13-5=8$$

What must be added to $$ -23 $$ to get $$ -9 $$ ?

Let the number to be added $$ = x $$
$$ \therefore \, -23 + x = -9 $$
$$ \therefore \, $$ The required number $$ = -9 - (-23) $$
$$ = -9 + 23 = 14 $$

Fill in the blanks :
The sum of two negative integers is always a _____ integer.

The sum of two negative integers is always a negative integer.

Use the appropriate symbol < or > to fill in the following blanks :
$$ (-3) +(-6)$$ ..........$$ (-3) - (-6) $$

$$-3+(-6)=-3-6=-9$$

$$-3-(-6) =-3+6=3$$

Hence, $$ (-3) + (-6) < (-3) - (-6) $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ (-25) - (-42) $$ ....... $$ (-42) - (-25) $$

$$-25-(-42) =-25+42=17$$

$$-42-(-25) =-42+25=-17$$

$$ (-25) - (-42) > (-42) - (-25) $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ 45 - (-11) $$ ....... $$ (57) + (-4) $$

$$45-(-11) =45+11=56$$

$$57+(-4) =57-4=53$$

$$ 45 - (-11) > (57) + (-4) $$

Use the appropriate symbol < or > to fill in the following blanks :
$$ (-21) - (-10) $$ ....... $$ (-31) + (-11) $$

$$-21-(-10) =-21+10=-11$$

$$-31+(-11) =-31-11=-42$$

$$ (-21) - (-10) > (-31) + (-11) $$

What must be added to $$ -176 $$ to get $$ -95 $$ ?

Let the number to be added $$ = x $$

$$ \therefore \, -176 + x = -95 $$

$$ x = -95 + 176 = 81 $$

Fill in the blanks to make the statements true
The additive inverse of $$0$$ is ______.

Two integers whose sum is zero are called additive inverse of each other

Let $$'a'$$ be an integer.

Thus, the additive inverse of integer $$'a'$$ is $$-a$$ as $$a - a = 0$$

So, additive inverse of $$0 = -0$$

$$= 0$$ $$[0 - 0 = 0]$$

$$\textbf{Hence, the additive inverse of 0 is}\ \underline{\bf{0}}$$

Fill in the blanks to make the statements true
The number of integers lying between $$-5$$ and $$5$$ is _____.

$$\therefore$$ $$\textbf{The numbers of integers lying between -5 to 5 are 9}$$

Fill in the blanks to make the statement true:
On the number line, $$10$$ is to the _______ of zero.

We know that, on a number line all the positive integers lie right of zero.

Thus, $$10$$ is right of zero.

$$\textbf{Hence, On the number line, 10 is to the}\ \underline{\textbf{right}}\ \textbf{of zero}$$

Fill in the blanks to make the statement true:
On the number line, $$-15$$ is to the ______ of zero.

We know that, on a number line all the negative integers lie left of zero. As they are less than $$0$$.

Thus, $$-15$$ is left of zero.

$$\textbf{Hence, On the number line, -15 is to the}\ \underline{\textbf{left}}\ \textbf{of zero}$$.

Fill in the blanks to make the statement true:
The additive inverse of $$14$$ is ______.

Two integers whose sum is zero are called additive inverse of each other

Let $$'a'$$ be the positive integer.

Thus, additive inverse of positive integer $$'a'$$ is $$-a$$ as $$a - a = 0$$

Similarly, additive inverse of $$14$$ is $$-14$$ as $$14 - 14 = 0$$

$$\textbf{Hence, additive inverse of a positive integer 14 is}\ \underline{\bf{-14}}$$

Fill in the blanks using <, = or >
$$(-71) + (+9)$$ ______ $$(-81) + (-9)$$.

Consider,

$$(-71) + (+9) = -71 + 9$$

$$= -62$$

Now Consider,

$$(-81) + (-9) = -81 - 9$$

$$= -90$$

As $$-62 > -90$$

$$\therefore$$ $$\bf{(-71) + (+9) > (-81) + (-9)}$$

Fill in the blanks using <, = or >
$$(-2) + (-5) + (-6)$$ _______ $$(-3) + (-4) + (-6)$$.

$$(-2)+ (-5) + (-6) = (-2) + [(-5 - 6)]$$

$$= (-2) + (-11)$$

$$= -2 - 11$$

$$= -13$$

$$(-3)+ (-4) + (-6) = (-3) + [(-4 - 6)]$$

$$= (-3) + (-10)$$

$$= -3 -10$$

$$= -13$$

$$-13 = -13$$

$$\therefore$$ $$\bf{(-2) + (-5) + (-6) = (-3) + (-4) + (-6)}$$

Match the items of Column I with that of Column II:

$$1)$$ Two integers whose sum is zero are called additive inverses of each other

So, additive inverse of $$+2 = -2$$

$$\textbf{Hence, for 1) correct match is b)}$$

$$2)$$ We know that, on the number line, negative integers decrease from $$-1 , -2$$… and so on from left of $$0$$

That is, $$-1 > -2 > -3 > -4$$…

So, greatest negative integer is $$-1$$

$$\textbf{Hence, for 2) correct match is e)}$$

$$3)$$ We know that, on the number line, negative integers decrease from $$-1 , -2$$… and so on from left to right from $$0$$

That is, $$-1 > -2 > -3 > -4$$…

So, greatest even negative integer is $$-2$$

$$\textbf{Hence, for 3) correct match is b)}$$

$$4)$$ We know that, on the number line, integers decrease from $$0, -1, -2$$… and so on from left to right from $$0$$

So, $$0$$ is the smallest integer greater than every negative integer.

$$\textbf{Hence, for 4) correct match is a)}$$

$$5)$$ Predecessor is the number that precedes or comes before the given number and is found by subtracting $$1$$ from

the given number.

Predecessor of $$-1 = -1 -1$$

$$= -2$$

Successor is the number that succeeds or comes after the given number and is found by adding $$1$$ to the given

number.

Successor of $$-1 = -1 + 1$$

$$= 0$$

So, Sum of predecessor and successor of $$-1 = -2 + 0$$

$$= -2$$

$$\textbf{Hence, for 5) correct match is b)}$$

$$1 + 2 + 3$$ ____ $$(-1) + (-2) + (-3)$$.

$$1 + 2 + 3 = (-1) + (-2) + (-3)$$
$$1 + 2 + 3 = -1 - 2 - 3$$
$$+6 > -6$$
Hence, $$1 + 2 + 3 > (-1) + (-2) + (-3)$$.

Fill in the blanks using < , = or >
$$(-11) + (-15)$$ ______ $$11 + 15$$.

Consider,

$$(-11) + (-15) = -11 - 15$$

$$= -26$$

Now, Consider,

$$11 + 15 = 26$$

As, negative integers are always less than positive integers.

$$-26 < 26$$

$$\therefore$$ $$\bf{(-11) + (-15) < 11 + 15}$$

Fill in the blanks to make the statements true
________ $$- 3456 = -8910$$.

Let the missing number be $$'a'$$

$$a - 3456 = -8910$$

$$a = -8910 + 3456$$

$$= -5454$$

$$\therefore$$ $$\underline{\bf{-5454}}\ \bf{- 3456 = -8910}$$

$$-10$$ _____ $$-11$$.

Hence, $$-10 > -11$$.

Fill in the blanks using <, = or >
$$-101$$ ____ $$-102$$.

On the number line $$-101$$ lies right of $$-102$$

$$\therefore\ \bf{-101 > -102}$$

Fill in the blanks to make the statements true
______ $$+ (-11) + 111 = 130$$.

Let the missing number be $$'a'$$

$$a + (-11) + 111 = 130$$

LHS, $$a + (-11) + 111 = a - 11 + 111$$

$$= a + 100$$

$$\therefore$$ $$a + 100 = 130$$

When $$a$$ is $$30$$ and added to $$100$$, we get $$130$$

$$\textbf{So, the missing number is 30}$$

$$\therefore$$ $$\underline{\bf{30}}\ \bf{+ (-11) + 111 = 130}$$

Fill in the blanks to make the statement true
$$(-11) + (-2) + (-1) =$$ ________.

$$\textbf{Addition of three numbers:}$$

$$(-11) + (-2)+ (-1) = [(-11) + (-2)]+ (-1)$$

$$= [-11 - 2] + (-1)$$

$$= (-13) + (-1)$$

$$= (-13 - 1)$$

$$= -14$$

$$\therefore$$ $$\bf{(–11) + (–2) + (–1) =}\ \underline{\bf{-14}}$$

Write the opposite of the given statement
$$1000$$ A.D

Time period is divided into two parts just like positive and negative integers for us.

So, opposite of A.D is B.C.

$$\therefore$$ $$\textbf{Opposite of 1000 A.D is 1000 B.C}$$

Compute each of the following:
$$0 + (-5) + (-2)$$.

$$\textbf{Addition of integers:}$$

$$0 + (- 5) + (- 2) = 0 - 5 - 2 $$

$$= 0 - 7$$

$$= -7$$

$$\therefore$$ $$\bf{0 + (-5) + (-2) = -7}$$

Compute each of the following:
$$0 - (-6) - (+6)$$.

$$\textbf{Addition of integers:}$$

$$0 -(-6) - (+6) = 0 + 6 - (+6) $$

$$= 6 - 6$$

$$= 0$$

$$\therefore$$ $$\bf{0 - (-6) - (+6) = 0}$$

Compute each of the following:
$$-50 + (-60) + 50$$.

$$-50 + (-60) + 50$$
$$= -50 - 60 + 50$$

$$= -(50 + 60) + 50$$
$$\Rightarrow -110 + 50 = -60$$.

Compute each of the following:
$$70 + (-20) + (-30)$$.

$$\textbf{Addition of integers:}$$

$$70 + (-20) + (-30) = 70 - 20 - 30$$

$$= 70 - 50$$

$$= 20$$

$$\therefore$$ $$\bf{70 + (-20) + (-30) = 20}$$

Compute each of the following:
$$0 - 2 - (-2)$$.

$$\textbf{Addition of integers:}$$

$$0 -2 - (-2) = 0 - 2 + 2 $$

$$= 0 - 0$$

$$= 0$$

$$\therefore$$ $$\bf{0 - 2 - (-2) = 0}$$

Compute each of the following:
$$(-20) + (-5)$$.

$$\textbf{Addition of integers:}$$

$$(-20) + (-5) = -20 - 5$$

$$=-25$$

$$\therefore$$ $$\bf{(-20) + (-5) = -25}$$

Write the opposite of the following statement:
Profit of $$Rs. 10$$.

The given statement is:

Profit of $$Rs. 10$$ .

The required opposite statement is "Loss of $$Rs. 10$$".

Write the opposite of each of the following:
Decrease in size.

The opposite of decrease is increase.

$$\therefore$$ $$\textbf{The opposite of decrease in size is increase in size}$$.

Compute each of the following:
$$1 + (-2) + (-3) + (-4)$$.

$$\textbf{Addition of integers:}$$

$$1 + (-2) + (-3) + (-4) = 1 - 2 - 3 - 4$$

$$= 1 - 2 - 7$$

$$= 1 - 9$$

$$= -8$$

$$\therefore$$ $$\bf{1 + (-2) + (-3) + (-4) = -8}$$

Write the opposite of the given statement:
$$10\ m$$ above the danger mark of river Ganga.

As, opposite of above is below.

Diagram

Description automatically generated with medium confidence

$$\therefore\,\, \textbf{Opposite of 10 m above the danger mark of river Ganga is 10 m below the danger mark of}$$

$$\textbf{river Ganga}$$

The temperature of a place at $$12:00$$ noon was $$+5^{\circ}C$$. The temperature increased by $$3^{\circ}C$$ in the first hour and decreased by $$1^{\circ}C$$ in the second hour. What was the temperature at $$2.00\ p.m.$$?

Temperature at $$12 : 00$$ noon $$= +5^{\circ} C$$

So, Temperature at $$1$$ o'clock $$(1\ pm) =$$ Temperature at $$12$$ noon $$+3^{\circ} C$$

$$= +5^{\circ} + 3^{\circ} C$$

$$= +8^{\circ} C$$

And temperature at $$2$$ o'clock $$=$$ Temperature at $$1$$ o'clock $$-\ 1^{\circ} C$$

$$= 8^{\circ} C - 1^{\circ} C$$

$$= 7^{\circ} C$$

$$\therefore$$ $$\textbf{Temperature at 2:00 pm is}\ \bf{7^{\circ} C}$$

Write the opposite of each of the following:
$$20^{\circ}C$$ rise in temperature.

Opposite of rise is fall.

$$\therefore\ \textbf{Opposite of}\ \bf{"20^{\circ} C}\ \textbf{rise in temperature" is} \bf{"20^{\circ} C}\ \textbf{fall in temperature"}.$$

Write the digits $$0, 1, 2, 3, ..., 9$$ in this order and insert $$'+'$$ or $$'-'$$ between them to get the result $$3$$.

Given $$0, 1, 2, 3, 4, 5, 6, 7, 8$$ and $$9$$

To get $$3$$ as result $$= 0 + 1 + 2 + 3 - 4 + 5 + 6 + 7 - 8 - 9$$

$$= 6 - 4 +18 - 17$$

$$= 2 + 18 - 17$$

$$= 3$$

$$\therefore\ \textbf{Digits order whose sum is 3 is 0, +1, +2, +3, -4 ,+5, +6, +7, -8 and - 9}$$

Write six distinct integers whose sum is $$7$$.

Consider six distinct integers: $$1, 2, 3, 6, -2$$ and $$-3$$

Finding their sum we get,

$$1 + 2 + 3 + 6 + (-2) + (-3) = 7$$

$$\therefore$$ $$\textbf{Required distinct integers are 1, 2, 3, 6, -2 and -3}$$

Write the opposite of the given statement:
$$20\ m$$ below the danger mark of the river Brahmaputra.

As, opposite of below is above.

$$\therefore$$ $$\textbf{Opposite of "20 m below the danger mark of the river}$$

$$\textbf{Brahmaputra" is "20 m above the danger mark of river Brahmaputra"}$$

Write the opposite of the given statement
Rise in water level

Opposite of rise is fall.

$$\therefore$$ $$\textbf{Opposite of "Rise in water level" is "fall in water level"}$$

Write the opposite of the given statement:
Winning by a margin of $$2000$$ votes.

As, opposite of Winning is losing

$$\therefore$$ $$\textbf{Opposite of Winning by a margin of 2000 votes is losing by a margin of 2000 votes}$$

Write the integer which is its own additive inverse.

$$\bullet$$ Additive inverse of a positive integer is a negative integer.

$$\bullet$$ Additive inverse of a negative integer is a positive integer.

$$0$$ is an integer, but it is neither positive nor negative.

Additive inverse of $$0$$ is also $$0$$.

$$\therefore$$ $$\textbf{Zero is the integer which is its own additive inverse}$$.

Write the opposite of the given statement
$$60$$ km South

As, opposite direction of the South is North.

$$\therefore$$ $$\textbf{Opposite of "60 Km of South" is "60 Km of North"}$$

Write two integers whose sum is less than both the integers.

$$\bullet$$ $$\textbf{Sum of two positive integers is always greater than both the integers}$$

Eg: $$5 + 6 = 11$$ and $$11 > 5, 11 > 6$$.

$$\bullet$$ $$\textbf{Sum of two negative integers is always less than both the integers}$$

Eg: $$-5 + (-6) = -11$$ and $$-11 < -5 , -11 < -6$$.

$$\therefore$$ $$\textbf{Sum of any two negative integers is less than both the integers}$$

Write two distinct integers whose sum is equal to one of the integers.

Consider any integer, adding the integer with zero gives the same integer.

Let the number be $$x$$

So sum of $$x$$ and $$0 = x + 0$$

$$= x$$

$$\therefore\ \textbf{Two distinct integers whose sum is equal to one of the integers is any integer with 0}$$

Write the integer which is $$2$$ less than its additive inverse.

Two integers whose sum is zero are called additive inverses of each other.

We are given that the difference of integer and its additive inverse is $$2$$

As, $$1 - (-1) = 1 + 1 = 2$$

So $$-1$$ is $$2$$ less than $$1$$

$$\therefore$$ $$\textbf{-1 is the integer which is 2 less than its additive inverse}$$

Write five integers which are less than $$-100$$ but greater than $$-150$$.

Numbers between $$-100$$ and $$-150$$

$$-101 > -102> -103 > -104 > -105$$ ..… $$-149 > -150$$.

$$\therefore\ \textbf{Five integers which are less than -100 but greater than -150 are -101 , -102, -103, -104 and -105}$$

Write two integers whose sum is $$6$$ and difference is also $$6$$.

If we add an integer to zero, then the result will be the integer only.

Sum of $$6$$ and $$0 = 6 + 0$$

$$= 6$$

Difference of $$6$$ and $$0 = 6 - 0$$

$$= 6$$

$$\therefore$$ $$\textbf{Two integers whose sum is 6 and the difference is also 6 are 0 and 6}$$

Arrange the following integers in the descending order: $$0, -1, -4, -3, -6$$.

Negative integers are smaller than $$0$$ and $$0$$ is smaller tyan positive integers.

Also, numbers lying on left side of number line are smaller than right side.

For descending order we go from higher number to smaller number.

So, descending order of given integer$$0, -1, -3, -4, -6$$.

$$0>-1>-3>-4>-6$$

Using number line, how do you compare:
(a) Two negative integers?
(b) Two positive integers?
(c) One positive and one negative integer?

Since, the integer lying on the left is lesser than the integer lying right side.
$$\therefore$$ With the help of the above statement we verify all $$(a), (b)$$ and $$(c)$$ statement by observing which one of given integer lie on the right or left.

Observe the following:
$$1 + 2 - 3 + 4 + 5 - 6 - 7 + 8 - 9 = -5$$
Change one $$'-'$$ sign as $$'+'$$ sign to get the sum $$9$$.

$$1 + 2 - 3 + 4 + 5 - 6 - 7 + 8 - 9 = -5$$

If we replace $$-7$$ by $$7$$, we get the sum $$9$$

$$1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9 = (1+ 2 + 4 + 5 + 7 + 8) - 3 - 6 - 9$$

$$= (27) - (3 + 6 + 9) $$

$$= 27 - 18$$

$$= 9 $$

$$\therefore$$ $$\textbf{in the given expression replacing -7 with +7, we get the required sum}$$.

Write the integer which is $$4$$ more than its additive inverse.

$$\textbf{Integers whose sum is zero are called additive inverses of each other}$$.

We are given that the difference of integer and its additive inverse is $$4$$

We have $$2 - (-2) = 2 + 2 = 4$$

So $$2$$ is $$4$$ more than $$-2$$

$$\therefore$$ $$\textbf{2 is the integer which is 4 more than its additive inverse}$$

Arrange the following integers in the ascending order:
$$-2, 1, 0, -3, +4, -5$$.

Positive integers lie right of zero and negative integers lie left of the zero.

Therefore, positive integers are greater than negative integers.

$$\therefore$$ $$\textbf{Ascending order of integers is -5 < -3 < -2 < 0 < 1 < 4}$$

If we denote the height of a place above sea level by a positive integer and depth below the sea level by a negative integer, write the following using integers with the appropriate signs:
(a) $$200\ m$$ above sea level
(b) $$100\ m$$ below sea level
(c) $$10\ m$$ above sea level
(d) Sea level.

Given that,

Height of a place above sea level is denoted by a positive integer and

Depth below the sea level is denoted by a negative integer

Therefore,

(a) $$200\ m$$ above sea level $$= +200$$
(b) $$100\ m$$ below sea level $$= -100$$
(c) $$10\ m$$ above sea level $$= +10$$
(d) Sea level $$= 0$$.

If we are at $$8$$ on the number line, in which direction should we move to reach the integer.
(a) $$-5$$ (b) $$11$$ (c) $$0$$?

To get to lesser number we have to go left and to go for higher number we have to go right on number line.

(a)$$-5<8$$

So, we go left to go from $$8$$ to $$-5$$ on the number line.

(b)$$11>8$$

So, we go right to go from $$8$$ to $$11$$ on the number line.

So, we go left to go from $$8$$ to $$0$$ on the number line.

Find the value of $$49 - (-40) - (-3) + 69$$.

$$\textbf{Simplification of given expression:}$$

$$49 - (-40) - (-3) + 69 = 49 + 40 + 3 + 69$$

$$= 89 + 72$$

$$= 161$$

$$\therefore$$ $$\bf{49 - 40 - 3 + 69 = 161}$$

Write four pairs of integers which are at the same distance from $$2$$ on the number line.

$$\bullet$$ From $$2$$, move $$1$$ unit toward right and left. We will reach to 1 and $$3$$.

$$\textbf{Thus, 1 and 3 are at unit distance from 2}$$.

$$\bullet$$ From $$2$$, move $$2$$ units towards right and left. We will reach to $$0$$ and $$4$$

$$\textbf{Thus, 0 and 4 are at 2 unit distance from 2}$$.

$$\bullet$$ From $$2$$, move $$3$$ units towards right and left. We will reach to $$-1$$ and $$5$$

$$\textbf{Thus, -1 and 5 are at 3 units distance from 2}$$.

$$\bullet$$ From $$2$$, move $$4$$ units towards right and left. We will reach to $$-2$$ and $$6$$.

$$\textbf{Thus, -2 and 6 are 4 units away from 3}$$.

$$\therefore$$ $$\textbf{Four pairs of integers which are at the same distance from 2 on the number line are 1 and 3;}$$

$$\textbf{0 and 4; -1 and 5; -2 and 6}$$

Using the number line, write the integer which is $$2$$ less than $$-2$$.

$$\bullet$$ To reach larger numbers, we have to move to the right on the number line.

$$\bullet$$ To reach lesser numbers, we always have to move to the left on the number line.

So, to reach a small number from $$-2$$ by $$2$$, move towards the left by $$2$$ units. We will reach $$-4$$

$$\therefore$$ $$\textbf{Integer 2 less than -2 is -4}$$

Using the number line, write the integer which is $$3$$ less than $$2$$.

$$\bullet$$ To reach larger numbers, we have to move to the right on the number line.

$$\bullet$$ To reach lesser numbers, we always have to move to the left on the number line.

So, to reach a smaller number from $$2$$ by $$3$$ units, move towards the left by $$3$$ units. We will reach $$-1$$

$$\therefore$$ $$\textbf{Integer which is 3 less than 2 is -1}$$

Sum of two integers is $$-80$$. If one of the integers is $$-90$$, then find the other.

Given, Sum of two integers $$= -80$$

Let the other integer be $$x$$

$$-90 + x = -80$$

$$x = -80 + 90$$

$$x = 10$$

$$\therefore$$ $$\textbf{The other integer is 10}$$

The sum of two integers is $$30$$. If one of the integers is $$-42$$, then find the other.

Given, Sum of two integers $$= 30$$

Let the other integer be $$x$$

$$-42 + x = 30$$

$$x = 30 + 42$$

$$x = 72$$

$$\therefore$$ $$\textbf{The other integer is 72}$$

Division is the inverse operation of ____________.

Multiplication is the inverse operation of the division.

We get the additive inverse of an integer a when we multiply it by_________.

By the multiplication of (-1) will give the additive inverse of an integer a.

$$\Rightarrow a \times(-1)=-a$$

Social Studies Application: Remembering that1AD came immediately after 1BC, while solving these problems takes 1BC as 1 and 1AD as +
(a) The Greeco-Roman era, when Greece and Rome ruled Egypt started in the year 330 BC and ended in the year 395 AD. How long did this era last?
(b) Bhaskaracharya was born in the year 1114 AD and died in the year 1185 AD. What was his age when he died?
(c) Turks ruled Egypt in the year 1517 AD and QueenNefertis ruled Egypt about 2900 years before the Turks ruled. In what year did she rule?
(d) Greek mathematician Archimedes lived between287 BC and 212 BC and Aristotle lived between 380BC and 322 BC. Who lived during an earlier period?

Given, 1 BC as -1 and 1 AD as +1
(a) Year when it starts $$=330 \mathrm{BC} =(-330) \mathrm{AD}$$
Ending Year $$=395 \mathrm{AD}$$
The era lasted for $$=395-(-330)$$
$$=395+330=725$$ Years

(b) Year of Born = 1114 Ad
Year of death=1185 AD
Therefore, total age = $$1185-1114=71$$ years

(c) When Turks ruled the Egypt = 1517 AD
Queen Nefertis ruled Egypt in (1517-2900) AD
$$=-1383 \mathrm{ AD } \text { or } 1383 \mathrm{ BC}$$

(d) Archimedes lived between 287 BC and 212 BC.
Aristotle lived between 380 BC and 322 BC.
Therefore, an earlier period is the lived duration by Aristotle.

If $$x$$ is a negative integer, $$-x$$ is a positive integer.

1902929_1789787_ans_c944b509637a4a00aca4f49ea26daac3.jpg

Height of a place A is 1800 m above sea level.Another place B is 700 m below sea level. What is the difference between the levels of these two places?

Height of $$A$$ can be written as $$1800\ m$$

and of $$B$$ is written as $$-700\ m$$, because it is below see level, that is on opposite side of $$A$$ with respect to sea level.

So, difference between the levels of places A and B=

$$\begin{array}{l}{[1800-(-700)] m} \\=1800+700 \\=2500\ m\end{array}$$

1673728_1790147_ans_55dfe5db65ee48e49c846720250b5b96.png

The additive inverse of an integer $$x$$ is $$2x$$.

1902928_1789786_ans_0bcba98709db41a5ae75423fc5a9c6f7.jpg

You are at an elevation 380 m above sea level as you start a motor ride. During the ride, your elevation changes by the following meters: 540 m, -268 m, 116 m, -152 m, 490 m, -844 m, 94 m. What is your elevation relative to the sea level at the end of the ride?

To get final elevation we have to add all the elevations in original elevation.

So, elevation relative to the sea level at the end of the ride

$$\begin{array}{l}=[380+540+(-268)+116+(-152)+490+(-844)+94]\ m\\=[380+540-268+116-152+490-844+94]\ m \\=[380+540+116+490+94-268-152-844]\ m \\=[1620-1264] m \\=356\ m\end{array}$$

So, elevation at the end of ride will be $$356$$ m.

Taking today as zero on the number line, if the day before yesterday is 17 January, what is the date 3days after tomorrow?

A day before yesterday is $$17\text{ January}$$.

Therefore, today is $$17+2=19\text{ January}$$.

$$3$$ days after tomorrow will be

$$19+1+3=20 + 3 = 23\text{ January}$$

Science Application: An atom consists of charged particles called electrons and protons. Each proton has a charge of +1 and each electron has a charge ofRemember the number of electrons is equal to the number of protons while answering these questions:
(a) What is the charge on an atom?
(b) What will be the charge on an atom if it loses an electron?
(c) What will be the charge on an atom if it gains an electron?

(a) Total charge on the atom $$+1 -1 = 0$$
(b) If atom losses an atom the charge becomes $$+1.$$
(c) If atom gains an atom, the charge becomes $$-1.$$

The table given below shows the elevations relative to sea level of four locations. Taking sea level as zero, answer the following questions:

$$\begin{array}{|l|l|}\hline \text { Location } & \text { Elevation (in m) } \\\hline \mathbf{A} & -\mathbf{1 8 0} \\\hline \mathbf{B} & \mathbf{1 6 0 0} \\\hline \mathbf{C} & -\mathbf{5 5} \\\hline \hline \mathbf{D} & \mathbf{3200} \\\hline \end{array}$$

(a) Which location is closest to the sea level?
(b) Which location is farthest from sea level?
(c) Arrange the locations from the least to the greatest elevation.

$$(\mathbf{a})$$$$C=-55$$ is the closest location from the sea level,

Because its distance from sea level is $$55\ m$$ which is minimun among given list.

$$(\mathbf{b})$$$$D=3200$$ is the farthest location from the sea level,

Because its distance from sea level is $$3200\ m$$ which is maximum among given list.

$$\mathbf{(c)}$$ As per given data, $$-180<-55<1600<3200$$

Therefore, the location form the least to the greatest elevation is $$A<C<B<D$$

Sana and Fatima participated in an apple race. The race was conducted in 6 parts. In the first part, Sana won by 10 seconds. In the second part she lost by 1 minute, then won by 20 seconds in the third part and lost by 25 seconds in the fourth part, she lost by37 seconds in the fifth part and won by 12 seconds in the last part. Who won the race finally?

Let winning time be a positive integer and losing time is a negative integer, as it is opposite of winning.

$$1$$ minute = $$60$$ seconds

Therefore, Sana’s total difference time with Fatima

$$\begin{array}{l}=10-60+20-25-37+12 \\=42-122 \\=-80 \text { Seconds }\end{array}$$

Therefore, Sana lost the race by $$80$$ seconds or by $$1$$ min $$20$$ seconds.

Which means Fatima, won the race.

A water tank has steps inside it. A monkey is sitting on the first step. The water level is at the ninth step.
After drinking water, he wants to go back. For this, he jumps $$5$$ steps up and then jumps back $$3$$ steps down in every move. In how many jumps will he reach back the top of the tank?

In a tank,
Monkey is at first step and water level is at $$9^{th}$$ step.
After drinking water, he jump up $$5$$ steps and then back $$3$$ steps. So, he covers $$2$$ steps in one jump.
So, $$9-5=4+1=5$$ jumps, he will come back at the first step of the tank.

Fill in the blanks, using the following number line:
An integer, on the given number line, is . than every number on its left.

As we moves toward right on the number line the value of integer increases.

$$\textbf{Hence, an integer on the given number line is}$$ $$\underline{\textbf{greater}} \textbf{ than every number on its left.}$$

In each of the following pairs, state which integer is greater :
$$ 12, 15$$

We know as we move towards right value of numbers gets increases.

15 is greater than 12 ,as 15 lies on the right of 12 on the number line

$$\textbf{Hence, 15 is greater than 12 i.e 15 > 12}$$

A water tank has steps inside it. A monkey is sitting on the first step. The water level is at the ninth step.
He jump $$3$$ step down and then jumps back $$2$$ steps up. In how many jumps will he reach the water level?

In a tank,
Monkey is at first step and water level is at $$9^{th}$$ step.
Monkey is jumps $$3$$ steps down jump back $$2$$ step
So, he jumps $$3-2=1$$ step in one attempt.

In each of the following pairs, state which integer is greater :
$$0, 3$$

$$0,-3$$

$$0$$ lies on the right side of $$-3$$ on the number line. Hence, $$0$$ is greater than $$-3$$

The highest point measured above sea level is the summit of Mt. Everest which is 8,848m above sea level and the lowest point is Challenger Deep at the bottom of Mariana Trench which is 10911m below sea level. What is the vertical distance between these two points?

The lowest point =$$ 10,911\ m$$

The highest point =$$ 8,848\ m$$

Let, distance above sea level be positive, so distance below sea level will be negative, as they are on opposite side of sea level.

Therefore, the vertical distance between two points

$$\begin{array}{l}=(8848-(-10911)) m \\=[8848+10911] m \\=19759 m\end{array}$$

In each of the following pairs, state which integer is greater :
$$ 15, 23$$

$$-15, -23$$
$$-15$$ lies on the right side of $$-23$$ on the number line. Therefore, $$-15$$ is greater than $$-23$$

Fill in the blanks, using the following number line:
$$-6$$ is . than $$3$$ and the opposite of $$-6$$ is . than opposite of $$3$$.

$$-6$$ is less than $$3$$ and the opposite of $$-6$$ is greater than opposite of $$3$$

In each of the following pairs, replace * with $$<$$ or $$>$$ to make the statement true:
$$ 3 * 3$$

$$-3$$ lies on the left side of $$3$$ on the number line
Therefore, $$-3$$ is smaller than $$3$$

$$-3<3$$

In each of the following pairs, state which integer is smaller :
$$15, 51$$

We know as we move to the left on number line value of numbers get decreases.

-51 is smaller than 15, as -51 lies on the left of 15 on the number line

$$\textbf{Hence, -51 is smaller than 15 i.e. -51<15}$$

In each of the following pairs, replace * with $$<$$ or $$>$$ to make the statement true:
$$5 * 1$$

$$5$$ lies on right side of $$-1$$ on the number line
Hence, $$5$$ is greater than $$-1$$

$$5>-1$$

In each of the following pairs, replace $$$$ with $$<$$ or $$>$$ to make the statement true:
$$3 0$$

$$3$$ lies on the right side of $$0$$ on the number line
Hence, $$3$$ is greater than $$0$$

$$3>0$$

In each of the following pairs, replace $$$$ with $$<$$ or $$>$$ to make the statement true:
$$ 9 3$$

$$-9$$ lies on the left side of $$-3$$ on the number line
Hence, $$-9$$ is smaller than $$-3$$

$$-9<-3$$

In each of the following pairs, state which integer is smaller :
$$0, 6$$

0 is greater than -3, as 0 lies on the right of the -3 on the number line.

$$\textbf{Hence, 0 is greater than -3 i.e 0>3}$$

In each of the following pairs, state which integer is smaller :
$$2, -3$$

We know as we move to the left on number line value of numbers get decreases.

-3 is smaller than 2, as -3 lies on the left of 2 on the number line

$$\textbf{Hence, -3 is smaller than 2 i.e. -3 < 2}$$

In each of the following pairs, replace $$$$ with $$<$$ or $$>$$ to make the statement true:
$$0 8$$

$$0$$ lies on the right side of $$-8$$ on the number line
Therefore, $$0$$ is greater than $$-8$$

$$0>-8$$

In each of the following pairs, replace * with $$<$$ or $$>$$ to make the statement true:
$$ 13 * 0$$

$$-13$$ lies left side of $$0$$ on the number line
Therefore, $$-13$$ is smaller than $$0$$

$$-13<0$$

In each of the following pairs, state which integer is smaller :
$$2, 3$$

$$-3$$ lies on the left side of $$2$$ on the number line. Therefore, $$-3$$ is smaller than $$2$$

The sum of two integers is $$28$$. If one integer is $$-45$$, find the other.

It is given that
Sum of two integers $$ = 28 $$
One integer $$ = -45 $$
Other integer $$ = 28 - (-45) $$
By further calculation
$$ = 28 + 45 $$
$$ = 73 $$

Represent the following numbers on a number line: $$-1$$

$${\textbf{We have represented }}\mathbf{-1}{\textbf{ on number line}}$$

2166245_1866500_ans_773bcdcf7ba44355b33dfd1c88a50df3.jpeg

Find which is greater ?
$$ 12 \times (6 - 8) $$ or $$ 12 \times 6 - 8 $$

$$ 12 \times (6 - 8) $$ or $$ 12 \times 6 - 8 $$
We know that
$$ 12 \times (6 - 8) = 12 (-2) = -24 $$
$$ 12 \times 6 - 8 = 72 - 8 = 64 $$
Therefore , $$12 \times (6 - 8) > 12 \times 6 - 8 $$.

In each of the following pairs, replace * with $$<$$ or $$>$$ to make the statement true:
$$ 8 * 18$$

$$-8$$ lies on the right side of $$-18$$ on the number line
Hence, $$-8$$ is greater than $$-18$$

$$-8>-18$$

The difference between integers $$x$$ and $$-6$$ is $$-5$$ . Find the values of $$x$$.
$$ x - (-6) = -5 $$ or $$ -6 -x = -5 $$

It is given that
The difference between integers
$$ x - (-6) = -5 $$ or $$ -6 -x = -5 $$
So the value of $$x$$ is
$$ x + 6 = -5 $$ or $$ -x = -5 + 6 $$
By further calculation
$$ x = -5 - 6 $$ or $$ -x = 1 $$
$$ x = -11 $$ or $$ x = -1 $$

Compare the fractions and put an appropriate sign
i) $$\dfrac{6}{7} ........ \dfrac{8}{7}$$

ii) $$\dfrac{5}{8} ........ \dfrac{3}{8}$$

iii) $$\dfrac{6}{13}......... \dfrac{6}{17}$$

iv) $$\dfrac{5}{22} ......... \dfrac{3}{22}$$

v) $$\dfrac{9}{47} ......... \dfrac{9}{42}$$

i) $$\dfrac{6}{7} < \dfrac{8}{7}$$

ii) $$\dfrac{5}{8} > \dfrac{3}{8}$$

iii) $$\dfrac{6}{13} > \dfrac{6}{17}$$

iv) $$\dfrac{5}{22} > \dfrac{3}{22}$$

v) $$\dfrac{9}{47} < \dfrac{9}{42}$$

Find which is greater ?
$$ \left \{(-3) - 4 \right \} \times (-5) $$ or $$ (-3) - 4 \times (-5) $$

$$ \left \{(-3) - 4 \right \} \times (-5) $$ or $$ (-3) - 4 \times (-5) $$
We know that
$$ \left \{(-3) - 4 \right \} \times (-5) = \left \{-3 - 4 \right \} \times (-5) $$
By further calculation
$$ = -7 \times -5 = 35 $$
Similarly
$$ (-3) - 4 \times (-5) = -3 + 20 = 17 $$
Therefore , $$ \left \{(-3) - 4 \right \} \times (-5) > (-3) - 4 \times (-5) $$

Find which is greater ?
$$ (8 + 10) \times 15 $$ or $$ 8 + 10 \times 15 $$

$$ (8 + 10) \times 15 $$ or $$ 8 + 10 \times 15 $$
We know that
$$ (8 + 10) \times 15 = 18 \times 15 = 270 $$
$$ 8 + 10 \times 15 = 8 + 150 = 158 $$
Therefore , $$ (8 + 10) \times 15 > 8 + 10 \times 15 $$ .

The sum of two integers is $$ -15$$ . If one of them is $$9$$ , find the other.

It is given that
Sum of integers $$ = -15 $$
One integer $$ = 9 $$
Other integer $$ = -15 - 9 $$
Taking negative sign as common
$$ = -(15 + 9) $$
$$ = -24 $$

Draw a number line and answer the following :
If we are at $$-6$$ on the number line, in which direction should we move to reach $$-1$$?

REF image

Clearly, $$-1$$ is to the right of $$-6$$. Therefore, we should move towards the right direction.
1879491_1866701_ans_ecb81a39ca6d4261a0996918459886a1.png

Find the sum of the first $$1000$$ positive integers.

Sum of n positive integers$$=\cfrac{n(n+1)}{2}$$

For 1000 integers, $$n=1000$$

$$=\cfrac{1000(1000+1)}{2}$$

$$=500\times 1001$$

$$=500500$$

The Lowest Recorded Temperatures
Continent	Temperature (in Fahrenheit)
Africa	$$-11^o$$
Antarctica	$$-129^o$$
Asia	$$-90^o$$
Australia	$$-9^o$$
Europe	$$-67^o$$
North America	$$-81^o$$
South America	$$-27^o$$

Integers - Class 6 Maths - Extra Questions

Verify: $$-(-x) = x$$ for $$x = \dfrac{2}{15}$$.

Write the following in increasing order:$$21, -8, -26, 85, 33, -333, -210, 0, 2011$$

Write the following in decreasing order$$85, 210, -58, 2011, -1024, 528, 364, -10000, 12$$

Subtract $$-6$$ from $$3$$.

Find the greatest value of $$c$$ in order that the equation $$7x+9y=c$$ may have exactly six solutions in positive integers.

Subtract $$1732765$$ from $$3673124$$ in the scale of eight.

Solve in positive integers:$$23x+17y+11z=130$$.

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.Which city has the highest temperature?

Simplify:$$(-6) + (-10) + 5 + 17$$.

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.Write the temperatures of the cities marked on it?

Some integers are marked on the number line. Which is the biggest and which is the smallest?

Find the following using number line. - $$6 + 7 + 7=? $$

Simplify:$$(-6) - (5) - (+2)$$.

The temperature of 55 cities in India on a particular day are shown on the number line given. Name the cities with temperature more than 0oC0oC?

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.Which city has the lowest temperature?

Represent $$6,-4,0$$ and $$4$$ on a number line.

Represent the following subtractions on the number line:$$7-2$$

Represent the following addition on a number line: $$5+(-2)$$

Represent the following addition on a number line: $$5+(-7)$$

Represent the following addition on a number line: $$5+2$$

Compute the following:$$9+(-7)$$

Compute the following:$$4+(-4)$$

Solve the following:$$17-(-14)$$

Compute the following:$$14+(-6)$$

Compute the following:$$8+(-3)$$

Show the following numbers on a numberline:$$2, (-4), (-1)$$.

Show the following numbers on a numberline:$$(-3), 5, 8$$.

Show the following numbers on a numberline:$$0, 3$$

Show the following numbers on a numberline:$$4, 6, (-6)$$.

Express '$$-6$$' as the difference between a negative integer and a whole number.

Solve the following:$$13-(-8)$$

Solve the following:$$19-(-5)$$

Find:$$(-2) + 9$$

Arrange the following integers in descending order with the help of numberline:$$(-18), (-3), (-2), (-7)$$.

Do following addition with the help of numberline:$$2 + (-5)$$.

Do following addition with the help of numberline:$$(-4) + (-5)$$.

Arrange the following integers in an ascending order with the help of numberline:$$(-5), 4, (-1), 12$$.

Do following addition with the help of numberline:$$0 + (-4)$$.

Find the following:$$8 + (-6)$$

Do following addition with the help of number line:$$(-4) + 5$$.

Do following addition with the help of numberline:$$(-5)+ (-7)$$.

Subtract the following with the help of numberline:$$(-3) - 6$$.

Subtract the following with the help of numberline:$$4 - 1$$.

Subtract the following with the help of numberline:$$0 - 4$$.

What is the sum of $$4 + (-10)$$?

Do following addition with the help of number line:$$5 + (8)$$.

Subtract the following with the help of numberline:$$3 - (-6)$$.

Subtract the following with the help of numberline:$$(-3) - (-2)$$.

Subtract the following with the help of number line:$$(-2) - (-5)$$.

Arrange the following integers into ascending order with the help of numberline:$$3, (-2), (-6), (-3)$$.

Arrange the following integers in descending order with the help of numberline:$$13, (-3), 4, (-2)$$.

Do the following addition with the help of numberline:$$(-2) + (-5)$$.

Subtract the following with the help of numberline:$$(-3) - (-4)$$.

Subtract the following with the help of numberline:$$5 - 9$$.

Arrange the following integers into ascending order with the help of numberline:$$1, (-5), 2, (-9)$$.

Arrange the following integers in descending order with the help of numberline:$$(-6), 10, 0, (-1)$$.

Arrange the following integers in descending order with the help of numberline:$$7, (-5), (-1), 4$$.

Arrange the following integers into ascending order with the help of numberline:$$(-9), (-1), (-6), (-4)$$.

Arrange the following integers into ascending order with the help of numberline:$$(-5), 8, (-13), 0$$.

Do the following subtraction with the help of number line:$$8 - (-4)$$.

Do the following addition with the help of numberline:$$(-7) + 4$$.

Do the following subtraction with the help of number line:$$(-3) - (-9)$$.

The value of $$7 + (-10)$$ is

Do the following subtraction with the help of number line:$$8-2$$.

The value of $$0 + (-7)$$ is0+(−7)==-==yyju

Do the following addition with the help of numberline:$$(-3) + 8$$.

In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack's scores in five successive rounds were $$ 25, -5, -10, 15$$ and $$10$$, what was his total at the end?

Simplify :$$21-27+3+12\times(-1)$$

$$-23+\Box =34$$

Find the sum of $$-23$$ and $$3$$.

Add(-7)+(-8)+(-90)

Find: $$(-13)+32-8-1$$

Write the additive inverse of $$\dfrac {8}{5}$$

Add without using number line: $$11 + (-7)$$

Using the numbers $$7, 9, 0, 1, 2, 5$$ the biggest number that can be formed is_______

Find $$(-32)-(-50)$$

Solve$$50-(-40)-(-2)$$

Solve(-7)-8-(-25)

Show that every positive integer is either even or odd..

Write the following in increasing order:
$$21, -8, -26, 85, 33, -333, -210, 0, 2011$$

Write the following in decreasing order
$$85, 210, -58, 2011, -1024, 528, 364, -10000, 12$$

Solve in positive integers:
$$23x+17y+11z=130$$.

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.
Which city has the highest temperature?

Simplify:
$$(-6) + (-10) + 5 + 17$$.

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.
Write the temperatures of the cities marked on it?

Find the following using number line.
- $$6 + 7 + 7=? $$

Simplify:
$$(-6) - (5) - (+2)$$.

The temperature of 55 cities in India on a particular day are shown on the number line given. Name the cities with temperature more than 0oC?

The temperature of $$5$$ cities in India on a particular day are shown on the given number line.
Which city has the lowest temperature?

Represent the following subtractions on the number line:
$$7-2$$

Represent the following addition on a number line:
$$5+(-2)$$

Represent the following addition on a number line:
$$5+(-7)$$

Represent the following addition on a number line:
$$5+2$$

Compute the following:
$$9+(-7)$$

Compute the following:
$$4+(-4)$$

Solve the following:
$$17-(-14)$$

Compute the following:
$$14+(-6)$$

Compute the following:
$$8+(-3)$$

Show the following numbers on a numberline:
$$2, (-4), (-1)$$.

Show the following numbers on a numberline:
$$(-3), 5, 8$$.

Show the following numbers on a numberline:
$$0, 3$$

Show the following numbers on a numberline:
$$4, 6, (-6)$$.

Solve the following:
$$13-(-8)$$

Solve the following:
$$19-(-5)$$

Find:
$$(-2) + 9$$

Arrange the following integers in descending order with the help of numberline:
$$(-18), (-3), (-2), (-7)$$.

Do following addition with the help of numberline:
$$2 + (-5)$$.

Do following addition with the help of numberline:
$$(-4) + (-5)$$.

Arrange the following integers in an ascending order with the help of numberline:
$$(-5), 4, (-1), 12$$.

Do following addition with the help of numberline:
$$0 + (-4)$$.

Find the following:
$$8 + (-6)$$

Do following addition with the help of number line:
$$(-4) + 5$$.

Do following addition with the help of numberline:
$$(-5)+ (-7)$$.

Subtract the following with the help of numberline:
$$(-3) - 6$$.

Subtract the following with the help of numberline:
$$4 - 1$$.

Subtract the following with the help of numberline:
$$0 - 4$$.

Do following addition with the help of number line:
$$5 + (8)$$.

Subtract the following with the help of numberline:
$$3 - (-6)$$.

Subtract the following with the help of numberline:
$$(-3) - (-2)$$.

Subtract the following with the help of number line:
$$(-2) - (-5)$$.

Arrange the following integers into ascending order with the help of numberline:
$$3, (-2), (-6), (-3)$$.

Arrange the following integers in descending order with the help of numberline:
$$13, (-3), 4, (-2)$$.

Do the following addition with the help of numberline:
$$(-2) + (-5)$$.

Subtract the following with the help of numberline:
$$(-3) - (-4)$$.

Subtract the following with the help of numberline:
$$5 - 9$$.

Arrange the following integers into ascending order with the help of numberline:
$$1, (-5), 2, (-9)$$.

Arrange the following integers in descending order with the help of numberline:
$$(-6), 10, 0, (-1)$$.

Arrange the following integers in descending order with the help of numberline:
$$7, (-5), (-1), 4$$.

Arrange the following integers into ascending order with the help of numberline:
$$(-9), (-1), (-6), (-4)$$.

Arrange the following integers into ascending order with the help of numberline:
$$(-5), 8, (-13), 0$$.

Do the following subtraction with the help of number line:
$$8 - (-4)$$.

Do the following addition with the help of numberline:
$$(-7) + 4$$.

Do the following subtraction with the help of number line:
$$(-3) - (-9)$$.

Do the following subtraction with the help of number line:
$$8-2$$.

The value of $$0 + (-7)$$ is

Do the following addition with the help of numberline:
$$(-3) + 8$$.

Simplify :
$$21-27+3+12\times(-1)$$

Add
(-7)+(-8)+(-90)

Solve
$$50-(-40)-(-2)$$

Solve
(-7)-8-(-25)

Fill the appropriate sign in given box. $$('<', '=', '>')$$ .
$$\frac{1}{2} \square \frac{1}{5}$$

Eighteen integers are multiplied together. What will be the sign of their product, if :
(i) 15 of them are negative and 3 are positive ?
(ii) 12 of them are negative and 6 are positive ?
(iii) 9 of them are positive and the remaining are negative ?
(iv) all are negative ?

Arrange the following integers in ascending order.
$$-4,7,6,0,-8,-11,2,14,-15,9$$.

Use the sign of $$>, <$$ or $$=$$ in the box to make the statement true.
$$23-41+11\square 23-41-11$$

Use the sign of $$>, <$$ or $$=$$ in the box to make the statement true.
$$39+(-24)-(15)\square 36+(-52)-(-36)$$

Fill in the blank.
$$12+(-12)=$$ ______.

Fill in the blank.
$$13+$$ _______$$=0$$.

Use the sign of $$>, <$$ or $$=$$ in the box to make the statement true.
$$-231+79+51\square -399+159+81$$

Solve:
$$\left( { - 4} \right) + 6 - 2 + 3 + 2 = $$

Represent the following numbers as integers with appropriate signs.
A) An aeroplane is flying at a height two thousand metre above the ground.
B) A submarine is moving at a depth, eight hundred metre below the sea level.

Simplify:
$$(\frac{1}{2})^{-1} + (\frac{1}{3})^{-1} + (\frac{1}{4})^{-1}$$

Find the sum of :
$$-312, 39 $$ and $$192$$

Write the integers that represent the following situations
a decrease of $$500.$$

Find the sum of :
$$137$$ and $$-354$$

Find the sum of :
$$-52$$ and $$52$$

Write the integers that represent the following situations
$$40$$ SECONDS

Write the integers that represent the following situations
Flight at a height of $$14000\ meters$$

If $$a=1$$, $$b=-1$$ and $$c=0$$, find the value of expressions.
$$a+b+c$$

Solve :
$$ (-2) +8+(-4)$$