Class 11 Commerce Economics - Presentation Of Data

How much percentage is wheat more than cotton?

$$\Rightarrow$$ Percentage of cotton Ajaybhai bought from market $$=15\%$$

$$\Rightarrow$$ Percentage of wheat Ajaybhai bought from market $$=35\%$$

$$\Rightarrow$$ Percentage of wheat more than cotton $$=35\%-15\%=20\%$$

How many kilograms of sesame was bought?

$$\Rightarrow$$ Total grains Ajaybhai bought bought from market $$=3600\,kg$$

$$\Rightarrow$$ Percentage of sesame Ajaybhai bought from market $$=10\%$$

$$\Rightarrow$$ Sesame Ajaybhai bought from market $$=\left(3600\times \dfrac{10}{100}\right)=360\,kg$$

How many percentage of expenditure is done for education?

$$\Rightarrow$$ We have prepared following table from pie graph:

$$Expenses$$	$$Expenditure$$ $$(in\,\%)$$
$$Food$$	$$40\%$$
$$Education$$	$$10\%$$
$$Clothes$$	$$5\%$$
$$Other$$	$$25\%$$
$$Saving$$	$$20\%$$

$$\Rightarrow$$ From above table we can see percentage of expenditure is done for education is $$10\%$$

Use Graph paper for this question. A Survey regarding height (in cm ) of $$60$$ boys belonging to Class $$10$$ of a school was conducted. The following data was recorded.
Height in cm $$135-140$$ $$140-145$$ $$145-150$$ $$150-155$$ $$155-160$$ $$160-165$$ $$165-170$$
No. of boys $$4$$ $$8$$ $$20$$ $$14$$ $$7$$ $$6$$ $$1$$
Taking $$2$$cm $$=$$ height of $$10$$cm along one axis and $$2$$cm $$=10$$ boys along the other axis draw an ogive of the distribution. Use the graph to estimate the median.

There are $$60$$ boys.

To find the median height, we first arrange the class into increasing order of heights and then pick the height of the middle student as the median height.

To find that on the ogive plot, we have to find the mark on the y-axis which corresponds to $$30$$ boys and that would be the point $$6$$ on the y-axis.

Thus, the point on the x-axis which corresponds to the point $$6$$ on the y-axis is the median height.

We find that the number $$6$$ on the y-axis corresponds to somewhere between $$29$$ and $$30$$ on the x-axis.

That means, the median height range is $$145-150 cm$$.

This is shown on the graph attached.

Draw histogram and frequency polygon on the same graph paper for the following frequency distribution.
Class Frequency
$$15-20$$ $$20$$
$$20-25$$ $$30$$
$$25-30$$ $$50$$
$$30-35$$ $$40$$
$$35-40$$ $$25$$
$$40-45$$ $$10$$

$$Class$$	$$Class\,\, Mark$$ $$(x_i)$$	$$f_i$$
$$15-20$$	$$17.5$$	$$20$$
$$20-25$$	$$22.5$$	$$30$$
$$25-30$$	$$27.5$$	$$50$$
$$30-35$$	$$32.5$$	$$40$$
$$35-40$$	$$37.5$$	$$25$$
$$40-45$$	$$42.5$$	$$10$$

$${32}^{o}$$ $${34}^{o}$$ $${36}^{o}$$ $${37}^{o}$$ $${38}^{o}$$
$${34}^{o}$$ $${31}^{o}$$ $${34}^{o}$$ $${37}^{o}$$ $${37}^{o}$$
$${37}^{o}$$ $${35}^{o}$$ $${32}^{o}$$ $${34}^{o}$$ $${38}^{o}$$
$${39}^{o}$$ $${31}^{o}$$ $${36}^{o}$$ $${36}^{o}$$ $${39}^{o}$$
$${34}^{o}$$ $${33}^{o}$$ $${35}^{o}$$ $${34}^{o}$$ $${33}^{o}$$
$${37}^{o}$$ $${33}^{o}$$ $${32}^{o}$$ $${34}^{o}$$ $${38}^{o}$$
find the Mean Median Mode of the given table by $$DEVIATION$$ METHOD with Each step for Mean Median Mode.

Mean $$(\bar{x}) = A + \dfrac{\sum f_i d_i}{\sum f_i} = 35 + \dfrac{0}{30} = 35$$

Median = Average of $$15^{th}, 16^{th}$$ observations $$(\because n = 30)$$

$$= \dfrac{34+35}{2}$$

$$ = 34.5$$

Mode = most occurring observation $$ = 34$$

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The number of saplings planted by schools on World Tree Day is given in the table below. Draw a joint bar graph to show these figures.
School names\Name of saplings
Almond Karanj Neem Ashok Gulmohar
Nutan Vidyalaya 40 60 72 15 42
Bharat Vidyalaya 42 38 60 25 40

$$\Rightarrow$$ The above figure shows joint bar graph.

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A shop has the following number of shoe pairs of different sizes
Size$$2:20$$ Size$$3:30$$ Size$$4:28$$ Size$$5:14$$ Size$$6:8$$
Write this information in tabular form as done earlier and find the percentage of each shoe size available in the shop.

Size of shoes	No. of shoes	percentage
2	20	20%
3	30	30%
4	28	28%
5	14	14%
6	8	8%
Total	100		100%

On the basis of information given below, calculate $$AR$$ and $$MR$$ and explain their mutual relationship.
Units of a Good $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$
$$TR (Rs.)$$ $$4$$ $$7$$ $$9$$ $$10$$ $$10$$ $$9$$

Units of a Good	$$TR (Rs.)$$	$$AR (Rs.)$$	$$MR(Rs.)$$
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$	$$4$$ $$7$$ $$9$$ $$10$$ $$10$$ $$9$$	$$4 (= 4\div 1)$$ $$3.5 (= 7\div 2)$$ $$3 =(= 9\div 3)$$ $$2.5 ( = 10 \div 4)$$ $$2 (=10\div 5)$$ $$1.5 (=9\div 6)$$	$$4$$ $$3(= 9 - 7)$$ $$2(= 10 - 9)$$ $$1(= 10 - 9)$$ $$0(= 10 - 10)$$ $$-1 (= 9 - 10)$$

Relationship between $$TR, MR$$ and $$AR$$:
(i) when $$MR$$ is positive, $$TR$$ rises up to $$4th$$ units.
(ii) When $$MR = 0, TR$$ is maximum at $$5th$$ unit.
(iii) When $$MR$$ is negative, $$TR$$ starts falling, i.e., after $$5th$$ unit.
(iv) When $$AR$$ and $$MR$$ both are falling, $$MR$$ falls at a faster rate than $$AR$$ (i.e., $$MR$$ curve lies below $$AR$$ curve).

Obtain crude death rates $$(C.D.R)$$ for city $$A$$ and city $$B$$ from the given below:

$$C.D.R=\dfrac{number\,of\,deaths}{total\,population}\times 100$$

$$C.D.R\,for\,city\,A=\dfrac{134}{10,000}\times 100=13.4$$

$$C.D.R\,for\,city\,B=\dfrac{88}{7,500}\times 1000=11.73$$

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Marks No. of students
$$1-20$$ $$8$$
$$21-40$$ $$12$$
$$41-50$$ $$15$$
$$51-60$$ $$20$$
$$61-70$$ $$13$$
$$71-80$$ $$17$$
$$81-90$$ $$5$$
Number of students obtained marks between $$41$$ to $$80$$

Number of students obtained marks between 41 to 80

= sum of students from (41-50)+(51-60)+(61-70)+(71-80)

= 15+20+13+17=65

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Sanya, Aarushi, Manav and Kabir made greeting cards. Complete the table for their cards:

whose card Length width perimeter Areas

sanya 10 cm 8 cm 36 cm 80 sq.cm

Manav 11 cm 11 cm 44 cm 121 sq.cm

Aarushi 10 cm 8 cm 36 cm 80 sq.cm

Kabir 10 cm 10 cm 40 cm 100 sq.cm

sanya

we have length = 10 cm and width = 8 cm

perimeter =2(2+b)

= 2(10+8) = 2(18)

= 36 cm

Length of Manav's card =11 cm

Perimeter of his card =44 cm

Perimeter of card = sum of all its sides

=11+11+sum of other 2 sides

= 22+sum of other 2 sides

Now, sum of other two sides, 44-22=22 cm

so, width of manav's card =22/2, 11 cm

width of Arushi's card = 8 cm Area of card

$$= 8\times 10= 80cm^{2}$$

Now, we have to find length

Area of the card =80 cm

Length of Arushi's card = 80/8=10 cm.

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The minimum cost of each tablet is Rs. 10 and each capsule is Rs.If the cost of 8 tablets and 5 capsules is not less than Rs. 150, frame the inequations for the given data.

Let the cost of each tablet be x and the cost of each capsule be y.

Minimum cost of each tablet and each capsule is Rs. 10.

Therefore, $$x\geq 10$$ and $$y \geq 10$$.

The cost of 8 tablets and 5 capsules should be greater than or equal to 150, i.e., $$8x+5y \geq 150$$.

The bar graph as shown in above figure represents the heights (in cm) of 50 students of Class XI of a particular school. Study the graph and answer the following questions:
What percentage of the total number of students have their heights more than 149 cm?

The total number of students that have their height more than $$149\ cm = 16 + 15 + 5 = 31$$

The $$\%$$ of students that have their height more than $$149\ cm$$

$$=\dfrac{31}{50}\times 100= 62\%$$

What do you mean by the term sorting of data ?

Data sorting is any process that involves arranging the data into some meaningful order to make it easier to understand, analyze or visualize.Data is typically sorted based on actual values, counts or percentages, in either ascending or descending order, but can also be sorted based on the variable value labels.

In a school, $$46$$ students of $$9th$$ standard were told to measure the lengths of the pencils in their compass-boxes in centimeters. The data collected was as follows.
$$16, 15, 7, 4.5, 8.5, 5.5, 5, 6.5, 6, 10, 12, 13, 4.5, 4.9, 16, 11, 9.2, 7.3, 11.4, 12.7, 13.9, 16, 5.5, 9.9, 8.4, 11.4, 13.1, 15, 4.8, 10, 7.5, 8.5, 6.5, 7.2, 4.5, 5.7, 16, 5.7, 6.9, 8.9, 9.2, 10.2, 12.3, 13.7, 14.5, 10$$.
By taking inclusive classes $$0-5, 5-10, 10-15....$$ prepare a grouped frequency distribution table.

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The ages of ten students of a group are given below. The ages have been recorded in
$$\text{years - months:}$$
$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$
What is the lowest age?

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The following are the favorite cartoon characters of $$15$$ children.
Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey Mouse, Mickey mouse, Jerry, Jerry, Jerry, Doraemon, Doraemon, Doraemon.
Arrange the names of the cartoon characters in a table using tally marks.

Cartoon characters	Tally	Frequency
Doraemon	$$\nparallel \nparallel$$	$$6$$
Jerry	$$\nparallel \parallel$$	$$5$$
Mickey mouse	$$\nparallel$$	$$3$$
Tom	$$\|$$	$$1$$

The bar graph as shown in above figure represents the heights (in cm) of 50 students of Class XI of a particular school. Study the graph and answer the following questions:
The school wants to provide a particular type of tonic to each student below the height of 150 cm to improve his height. If the cost of the tonic for each student comes out to be Rs 55, how much amount of money is required?

Total number of students below height of $$150\ cm=7 + 12= 19$$

The cost of the tonic for each student $$=\ Rs. 55$$

The cost of the tonic for $$19$$ student $$= 55\times 19=\ Rs. 1045$$

The graph shows the maximum temperatures recorded for two consecutive weeks of a town. Study the graph and answer the questions that follow.
On which day was the temperature highest for the second week?

On $$\text{Wednesday}$$, the temperature was highest for the second week

Data available in an unorganised form is called _______ data

The correct fill up is raw.

The bar graph shown in Fig.16 represents the circulation of newspapers in 10 languages.
study the bar graph and answer the following question:
State the language other than english in which the number of newspaper published is between 2500 and 3500.

English and Hindi are the languages in which the number of newspapers were published in between $$2500$$ and $$3500$$.

The table given below shows the temperatures recorded on a day at different times.
Observe the table and answer the following questions.
During which hour was there a constant temperature?

From the graph, we can see that, the temperature is constant from 8 am to 9 am.

The bar graph drawn below shoes the number of tickets sold during a fair by $$6$$ students $$A, B, C, D, E$$ and $$F$$
Using the Bar graph, answer the following question:
What is the average number of tickets sold per student?

Average number of tickets sold per student can be calculated as below
Average tickets sold per student = ( tickets sold by $$A+B+C+D+E+F)/6$$
$$=(16+9+207+24+14)/6$$
We get
$$=90/6$$
$$=15$$
Hence, average tickets sold per student is $$15$$ tickets

How is the data of a measurement recorded in tabular form? Describe in steps.

i) The measurements corresponding to one physical quantity from one instrument must be recorded in one table. Also ensure that the instruments of different physical quantities are recorded in different tables.

ii) Always write the name of the instrument at the top of the table. The least count and the zero error of the instrument must be recorded in the table.

iii) Record each observation upto the limit of accuracy of the instrument with adding zeroes on the right side of the decimal in any observation. In case there is no digit before the decimal you should make the decimal prominent by putting a zero before the decimal.

iv) When you have recorded individual observations calculate the mean value and record it at the bottom of the column if the physical quantity measured is a constant.

A pie chart is constructed by converting the share of each component into a percentage of 360 degrees Music preferences in 13 to 19 year (teenagers) olds are clearly shown in the figure
Find the percentage occupied by the music which is least liked?

Clearly classical music is liked that least number of students and it occupies $$2$$%

Plot the frequency curve for the following histogram.

Frequency curve can also be drawn with the help of histogram by joining their mid points of rectangle.

Draw frequency curve for the following histogram.

Find Mid-point of every class and connect them by free-hand

that is frequency curve.

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Marks of $$30$$ students in mathematics test are given in the adjacent grouped frequency distribution.
Sl. No Marks No of students
$$1$$ $$0 - 5$$ $$5$$
$$2$$ $$5 - 10$$ $$7$$
$$3$$ $$10 - 15$$ $$10$$
$$4$$ $$15 - 20$$ $$6$$
$$5$$ $$20 - 25$$ $$2$$
What are the marks of $$6$$ students who are in $$4^{th}$$ class interval?

$$\Rightarrow$$ The first column here represents the marks obtained in class interval form. The lowest number in a class interval is called the lower limit and the highest number is called the upper limit. This example is a case of continuous class intervals as the upper limit of one class is the lower limit of the following class.

$$\Rightarrow$$ In continuous cases, any observation corresponding to the extreme values of a class is always included in that class where it is the lower limit. For example, if we had a student who has scored $$5$$ marks in the test, his marks would be included in the class interval $$5-10$$ and not $$0-5.$$

$$\therefore$$ Marks of $$6$$ students who are in $$4^{th}$$ class interval is between $$15-19$$.

Marks of $$30$$ students in mathematics test are given in the adjacent grouped frequency distribution.
Sl. No Marks No of students
$$1$$ $$0 - 5$$ $$5$$
$$2$$ $$5 - 10$$ $$7$$
$$3$$ $$10 - 15$$ $$10$$
$$4$$ $$15 - 20$$ $$6$$
$$5$$ $$20 - 25$$ $$2$$
If a student gets $$10$$ marks, should he be included in $$2nd$$ or $$3rd$$ class?

$$\therefore$$ A student gets $$10$$ marks, should be included in $$3^{rd}$$ group.

The following is the frequency distribution with unknown frequencies
Class $$60-70$$ $$70-80$$ $$80-90$$ $$90-100$$ Total
Frequency $$\dfrac{a}{2}$$ $$\dfrac{3a}{2}$$ $$2a$$ $$a$$ $$50$$
Find the value of $$a$$, hence find the frequencies. Draw a histogram and frequency polygon on the same coordinate system.

The value of $$a$$ can be calculated as follows
$$\dfrac{a}{2} + \dfrac{3a}{2} + 2a + a = 50$$
$$\dfrac{a + 3a + 4a + 2 a}{2} = 50$$
$$\dfrac{10a}{2} = 50$$
$$5a = 50$$
$$a = 10$$

Class	$$60-70$$	$$70-80$$	$$80-90$$	$$90-100$$
Frequency	$$5$$	$$15$$	$$20$$	$$10$$

The following data give the number of students using different modes of transport:
Modes of Transport Number of Students
Bicycle $$140$$
Bus $$100$$
Walk $$70$$
Train $$40$$
Car $$10$$
From this table, find the central angle $$(\theta)$$ for the Mode of Transport 'Bus'

Total Number of Students $$= 140 + 100 + 70 + 40 + 10$$
$$= 360$$
Let central angle $$= \theta$$
$$\dfrac {\theta}{360^{\circ}} \times \text {Total Number of students} = \text {No. of students (Transport by bus)}$$
$$\dfrac {\theta}{360^{\circ}} \times 360 = 100^{\circ}$$
$$\therefore \theta = 100^{\circ}$$

How many students scored full marks in the test?

Student Sl. No.	Test Score
$$1$$	$$3$$
$$2$$	$$4$$
$$3$$	$$1$$
$$4$$	$$3$$
$$5$$	$$5$$
$$6$$	$$2$$
$$7$$	$$4$$
$$8$$	$$1$$
$$9$$	$$3$$
$$10$$	$$2$$

Given that full marks in the test is $$5$$.

From the above table, we can observe that only one student scored $$5$$ marks.

Draw a subdivided bar graph to illustrate the given data
Years Wheat Barley Oats
1991 34 18 27
1992 43 14 24
1993 43 16 27
1994 45 13 34

$$Years$$	$$Wheat$$	$$Barley$$	$$Oats$$	$$Total$$
$$1991$$	$$34$$	$$18$$	$$27$$	$$79$$
$$1992$$	$$43$$	$$14$$	$$24$$	$$81$$
$$1993$$	$$43$$	$$16$$	$$27$$	$$86$$
$$1994$$	$$45$$	$$13$$	$$34$$	$$92$$

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How many students scored $$3$$ or more than $$3$$ marks?

Student Sl. No.	Test Score
$$1$$	$$3$$
$$2$$	$$4$$
$$3$$	$$1$$
$$4$$	$$3$$
$$5$$	$$5$$
$$6$$	$$2$$
$$7$$	$$4$$
$$8$$	$$1$$
$$9$$	$$3$$
$$10$$	$$2$$

Given that full marks in the test is $$5$$.

From the above table, we can observe that $$6$$ students scored $$\geq 3$$ marks.

In all how many rupees were collected as fund?

$$Students\, Name$$	$$Contribution$$ $$(in\,Rs.)$$
$$Matri$$	$$30$$
$$Dev$$	$$12$$
$$Jafar$$	$$18$$
$$Riya$$	$$24$$
$$Anuj$$	$$36$$
$$Total$$	$$120$$

$$\therefore$$ Total money collected as fund is $$Rs.120.$$

What percentage of Asopalav tree were planted?

$$\Rightarrow$$ Total number of tree $$=21+15+6+18=60$$

$$\Rightarrow$$ Number of Asopalav tree $$=21$$

$$\Rightarrow$$ Percentage of Asopalav tree $$=\dfrac{21}{60}\times 100$$

$$=35\%$$

What percentage of Gulmohar sapling were planted? in % .

$$Tree\,Name$$	$$No.\,of\,planted\,trees$$
$$Gulmohar$$	$$18$$
$$Amla$$	$$6$$
$$Neem$$	$$15$$
$$Asopalav$$	$$21$$
$$Total$$	$$60$$

$$\Rightarrow$$ Total number of planted trees $$=60$$

$$\Rightarrow$$ Number of planted Gulmohar sapling $$=18$$

$$\Rightarrow$$ $$\%$$ of Gulmohar sapling were planted $$=\dfrac{18}{60}\times 100$$

$$=30\%$$

How many runs did sachin score? in %

$$Name$$	$$Scored\,runs$$
$$Parthiv$$	$$45$$
$$Sachin$$	$$75$$
$$Ravindra$$	$$70$$
$$Sehwag$$	$$110$$
$$Total$$	$$300$$

$$\Rightarrow$$ Total runs scored during cricket match $$=300$$

$$\Rightarrow$$ Runs scored by $$Sachin=75$$

$$\therefore$$ $$\%$$ of runs scored by $$Sachin=\dfrac{75}{300}\times 100$$

$$=25\%$$

Find the degree measuremnet of the runs scored by Parthiv? in degrees.

$$Name$$	$$Scored\,runs$$
$$Parthiv$$	$$45$$
$$Sachin$$	$$75$$
$$Ravindra$$	$$70$$
$$Sehwag$$	$$110$$
$$Total$$	$$300$$

$$\Rightarrow$$ Total runs scored during cricket match $$=300$$

$$\Rightarrow$$ Runs scored by $$Parthiv=45$$

$$\therefore$$ Runs scored by $$Parthiv(in\,degree)=\dfrac{45}{300}\times 360^o$$

$$=54^o$$

Distinguish between:
Simple Bar Diagram and Multiple Bar Diagram

Simple bar diagram:

A bar chart is a type of chart which shows the values of different categories of data as rectangular barswith different lengths. A histogram is abar chart that shows how data is spread over its categories. It is another one dimensional diagram also known as pillar diagram or column graph .

Multiple bar diagram:

This example demonstrates a multi-bar chart, in which bars are clustered. It also demonstrates using alternative plot area background colors, and putting legends internal to the plot area. A multi-bar chart is a bar chart in which multiple data sets are represented by drawing the bars side by side in a cluster.

State whether True or False. Give Reasons.
Histogram is a type of diagram.

A histogram is a graphical representation of a frequency distribution with class intervals or attributes as the base and frequency as the height. Pictorial representation in the form of spaced bars to depict numerical data.

Distinguish Between.
Simple Bar Diagram and Multiple Bar Diagram.

Simple bar diagram:

Multiple bar diagram:

Define or explain the concepts:
Percentage Bar

A percentage bar chart, bars of length equal to 100 for each class are drawn in the first step and sub-divided into the proportion of the percentage of their component in the second step. The diagram so obtained is called a percentage component bar chart or percentage stacked bar chart.
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Answer in detail:
What is a Multiple Bar Diagram?
Prepare a Multiple Bar Diagram from the following data.
Product Production (in million tonnes)

2008 2009 2010
Rice 300 350 400
Wheat 200 275 300
Sugar 100 120 160

A multiple bar graph is a graph that has multiple bars for each category.A multiple bar graph compares as many sets of data you want. The process for creating a multiple bar graph is just like creating any other bar graph, only you will have more colors to represent different sets of data.

To create a multiple bar graph:

1. Draw the horizontal $$(x)$$ and vertical $$(y)$$ axis.

2. Give the graph a title.

3. Label the horizontal $$x$$ axis.

4. Label the vertical $$y$$ axis.

5. Look at the range in data and decide how the units on the vertical axis $$(y)$$ should be labeled.

For each item on the horizontal $$(x)$$ axis, draw a vertical column to the appropriate value however many times as you have sets of data.

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Do you agree with the statement? Give reasons
There is a only one type of diagram.

No.

There are various types of diagrams. A diagram is a two-dimensional geometric figure that represents information in a visual form. Some of the statistical diagrams are as follows:

i) Bar diagrams - A simple bar diagram comprises a group of rectangular bars for each class of data. They are one-dimensional diagrams in the sense, that it is only the height of the bars that matters and not the width. The height of the bars in the diagram differs according to the values of different variables while the width of different bars remains the same.

ii) Pie diagrams - A pie diagram is a circular chart or diagram that is further dissected into parts representing different components of the given data. Each segment or part represents an item of the given data set. The area of each part (or central angle) is proportional to the quantity (or information) it represents.

Answer the following:
What are the general principles for the construction of a diagram?

The diagrams should be simple. Each diagram must be given a clear, concise and suitable title without damaging clarity. A proper proportion between height and width must be maintained in order to avoid an unpleasant look. Select a proper scale; it should be in even numbers or in multiples of five or ten.

The following general rules should be observed while constructing diagrams :

$$1.$$ Title : Every diagram must be given a suitable title. The title should convey in as few a words as possible the main idea that the diagrams intent to portray. However, the brevity should not be secured at the cost of clarity or omission of essential details. The title may be given either at the top of the diagram or below it.

$$2.$$ Proportion between width and height. A proper proportion between the height and width of the diagram should be maintained. If either the height and width is too short or too long in proportion, the diagram would given an ugly look. While there are no fixed rules about the dimensions, a convenient standard as suggested by Lutz in the book entitled “Graphic Presentation” may be adopted for general use. It is known as “Root-two”, that is, a ratio of $$1$$ (short side) to $$1.414$$ (long side). Modifications may, no doubt, be made to accommodate a diagram in the space available.

$$3.$$ Selection of scale : The scale showing the values may be in even numbers or in multiples of five or ten, eg. $$25, 50, 75,$$ or $$20, 40, 60$$. Odd values like $$1, 3, 5, 7$$ may be avoided.

$$4.$$ Footnotes. In order to clarify certain points about the diagram, footnote may be given at the bottom of the diagram.

$$5.$$ Index. An index illustrating different types of lines or different shades, colours should be given so that the reader can easily make out the meaning of the diagram.

$$6.$$ Neatness and cleanliness. Diagrams should be absolutely neat and clean.

$$7.$$ Simplicity. Diagrams should be as simple as possible so that the reader can understand their meaning clearly and easily. For the sake of simplicity, it is important that too much material should not be loaded in a single diagram otherwise it may become too confusing and prove worthless. Several simple charts are often better and more effective than one or two complex ones which may present the same material in a confusing way.

An oil company has tow depots A and B with capacities of $$7000\ L$$ and $$4000\ L$$ respectively. The company is to supply oil to three petrol pumps $$D, E$$ and $$F$$ whose requirements are $$4500L, 3000L$$ and $$3500L$$ respectively. The distances (in km) between the depots and the petrol pumps is given in the following table:
Distance in km Distance in km
From/ To $$A$$ $$B$$
$$D$$ $$7$$ $$3$$
$$E$$ $$6$$ $$4$$
$$F$$ $$3$$ $$2$$
Assuming that the transportation cost of $$10$$ litres of oil is $$Rs.1$$ per km, how should the delivery be scheduled in order that the transportation cost is minimum.

Step 1:

Let $$x$$ litres of oil be supplied from depot $$A$$ to petrol pump $$D$$ and $$y$$ litres of oil be supplied from depot $$A$$ to petrol pump $$E$$,then $$7000−(x+y)\ge 0 (i.e) x+y\le 7000$$

The requirements of oil at petrol pumps $$D,E$$ and $$F$$ are $$4500l,3000l$$ and $$3500l$$

The quantity of oil transported from depot $$B$$ to petrol pump $$D,E$$ and $$F$$ are $$(4500−x),(3000−y)$$ and $$3500−[7000−(x+y)]litres$$ respectively.

$$(i.e)4500−x\ge0$$

$$\Rightarrow x\le4500$$

$$3000−y\ge0$$

$$\Rightarrow$$ $$y\le3000$$

$$3500−[7000−(x+y)]\ge0$$

$$\Rightarrow x+y\ge3500$$

Step 2:

The cost of transportation per km for $$10litre$$ oil is $$Rs.1.$$

Hence the cost of transportation/km/litre$$=Rs110=Rs 0.1$$

$$\Rightarrow$$ $$Z=0.7x+0.6y+0.3[7000−(x+y)]+0.3(4500−x)+0.4(3000−y)+0.2[(x+y)−3500]$$

On simplifying we get,

$$Z=0.3x+0.1y+3950$$

Now the corner points are $$A(500,3000),B(3500,0),E(4500,0),C(4500,2500)$$ and $$D(4000,3000)$$

Step 4:

Now let us calculate the values of the objective function as follows :

At the points $$(x,y)$$ the value of the objective function subjected to $$Z=0.3x+0.1y+3950$$

At $$A(500,3000)$$ the value of the objective function subjected to $$Z=0.3\times 500+0.1\times 3000+3950=4400$$

At $$B(3500,0)$$ the value of the objective function subjected to $$Z=0.3\times 3500+0+3950=5000$$

At $$E(4500,0)$$ the value of the objective function subjected to $$Z=0.3\times 4500+0+3950=5300$$

At $$C(4500,2500)$$ the value of the objective function subjected to $$Z=0.3\times 4500+0.1\times 2500+3950=5550$$

At $$C(4000,3000)$$ the value of the objective function subjected to $$Z=0.3\times 4000+0.1\times 3000+3950=5450$$

Step 5:

It is clear that the minimum transport charges are $$Rs.4400$$ at $$A(500,3000).$$

When $$x=500$$ and$$y=3000$$,

$$500L$$,$$3000L$$ and $$3500L$$ of oil should be transported from depot $$A$$ to petrol pumps $$D,E$$ and $$F$$ and $$4000L$$,$$0L$$ of oil be transported from depot $$B$$ to petrol pumps $$D,E$$ and $$F$$ with minimum cost of transportation of $$Rs.4400.$$

1243112_846556_ans_63c636a56558424aada37bf2dc304bb4.png

Answer in detail:
What is a Simple Bar Diagram?
Construct a Simple bar diagram from the following data
Year 2001 2002 2003 2004 2005
No of Workers 40 80 120 160 240

A bar chart or bar graph is a chart or graph that presents categorical data with rectangular bars with heights or lengths proportional to the values that they represent. The bars can be plotted vertically or horizontally.A bar graph shows comparisons among discrete categories.

1480515_773313_ans_e2483839d0cd4eabb449d0b3e9dfa08a.png

Write Short Notes on the following:
Multiple Bar Diagram

Sometimes there are more than two sets of data to be compared in a bar graph. In that case, a multiple bar graph can be used. A multiple bar graph compares as many sets of data you want. The process for creating a multiple bar graph is just like creating any other bar graph, only you will have more colors to represent different sets of data.

To create a multiple bar graph:

Draw the horizontal $$(x)$$ and vertical $$(y)$$ axis.

Give the graph a title.

Label the horizontal $$x$$ axis.

Label the vertical $$y$$ axis.

Look at the range in data and decide how the units on the vertical axis $$(y)$$ should be labeled.

For each item on the horizontal $$(x)$$ axis, draw a vertical column to the appropriate value however many times as you have sets of data. For example, if you are looking at 3 days worth of data, you will have $$3$$ bars per item.

Choose three colors to represent each different data set. Make sure to label the data sets in a key alongside the graph.

Sometimes comparing data can also be done by comparing data sets across multiple different bar graphs.

The difference is the data is split versus all being compared in one graph. Either method allows you to analyze and compare the data being displayed.

1480159_773302_ans_837eac5d61784e7bbf90eda5c301cc7f.jpg

Answer in detail:
What is a sub divided bar diagram?
Draw a sub divided bar diagram from the following data?
Stream No of Students
2009 2010 2011
Arts 50 70 75
Science 200 250 300
Commerce 100 120 130

A sub-divided or component bar chart is used to represent data in which the total magnitude is divided into different or components. In this diagram, first we make simple bars for each class taking the total magnitude in that class and then divide these simple bars into parts in the ratio of various components.

1480521_773320_ans_e58a99ab9da44ebdad4673f2f00a1e22.png

The random variable $$X$$ can take only the values $$0, 1, 2, 3$$. Given that $$P(X = 0) = P(X = 1) = p$$ and $$P(X = 2) = P(X =3)$$ such that $$\sum p_{i}x_{i}^{2} = 2\sum p_{i}x_{i}$$ find the value of $$p$$.

Let $$P(X=2)=P(X=3)=a$$

$$\sum p_{i}=1\implies a=\dfrac{1}{2}-p$$

$$\sum p_{i}x_{i}^{2}=2\sum p_{i}{x_{i}}$$

$$0(p)+1(p)+4(a)+9(a)=2(0(p)+1(p)+2(a)+3(a))$$

$$p+13{a}=2{p}+10{a}$$

$$p=3(\dfrac{1}{2}-p)\implies p=\dfrac{3}{8}$$

Complete the following table. Identify the phase of production which is relevant from the viewpoint of producer's equilibrium.
Units of Labour $$TP$$ $$AP$$ $$MP$$
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$
$$6$$
$$7$$
$$8$$ $$4$$
$$12$$
$$24$$
$$32$$
$$36$$
$$36$$
$$28$$
$$16$$ -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

Units of Labour	$$TP$$	$$AP$$	$$MP$$
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$	$$4$$ $$12$$ $$24$$ $$32$$ $$36$$ $$36$$ $$28$$ $$16$$	$$4$$ $$6$$ $$8$$ $$8$$ $$7.2$$ $$6$$ $$4$$ $$2$$	$$4$$ $$8$$ $$12$$ $$8$$ $$4$$ $$0$$ $$-8$$ $$-12$$

Relevant phase for Producer's Equilibrium: Phase of diminishing returns to a factor which is a phase of rising $$MC$$.

There are four consumers of a fruit called a smile. They are Isla, Ifraah, Ila and Ibema. Their demand curves for smile are given below. Derive the market demand curve.
Price (Rs.) Quantity Demanded by Isla (Units) Quantity Demanded by Ifraah (Units) Quantity Demanded by Ila (Units) Quantity Demanded by Ibema (Units)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$
$$6$$ $$16$$
$$11$$
$$7$$
$$4$$
$$2$$
$$1$$ $$7$$
$$6$$
$$5$$
$$4$$
$$3$$
$$2$$ $$15$$
$$12$$
$$9$$
$$6$$
$$3$$
$$0$$ $$8$$
$$6$$
$$4$$
$$2$$
$$0$$
$$0$$

In order to draw a market demand curve, we must first derive a market demand schedule, as under:
Market Demand Schedule

Price (Rs.)	Quantity (Units)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$	$$16 + 7 + 15 + 8 = 46$$ $$11 + 6 + 12 + 6 = 35$$ $$7 + 5 + 9 + 4 = 25$$ $$4 + 4 + 6 + 2 = 16$$ $$2 + 3 + 3 + 0 = 8$$ $$1 + 2 + 0 + 0 = 3$$

Note: Market demand curve is a horizontal summation of individual demand curves.

Demand schedule for three consumers such as Mohan, Rohan and Mitu for a commodity is given below. Derive the market demand schedule.
Price (Rs.) Demand Schedule of Mohan (Units) Demand Schedule of Rohan (Units) Demand Schedule of Mitu (Units)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$60$$
$$50$$
$$40$$
$$30$$
$$20$$ $$55$$
$$40$$
$$25$$
$$10$$
$$0$$ $$24$$
$$13$$
$$5$$
$$0$$
$$0$$

Market Demand Schedule

Price (Rs.)	Quantity (Units)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$60 + 55 + 24 = 139$$ $$50 + 40 + 13 = 103$$ $$40 + 25 + 5 = 70$$ $$30 + 10 + 0 = 40$$ $$20 + 0 + 0 = 20$$

From the following table prepare demand schedule for household $$B$$, given demand schedule of household $$A, C$$ and market demand schedule.
Price (Rs.) Household $$A$$ (Units) Household $$B$$ (Units) Household $$C$$ (Units) Market Demand (Units)
$$7$$
$$6$$
$$5$$
$$4$$
$$3$$
$$6$$
$$8$$
$$12$$
$$17$$
$$24$$ -
-
-
-
-

$$11$$
$$16$$
$$22$$
$$30$$
$$42$$ $$26$$
$$36$$
$$51$$
$$71$$
$$98$$

$$B's$$ Demand Schedule

Price (Rs.)	Quantity (Units)
$$7$$ $$6$$ $$5$$ $$4$$ $$3$$	$$26 - (6 + 11) = 9$$ $$36 - (8 + 16) = 12$$ $$51 - (12 + 22) = 17$$ $$71 - (17 + 30) = 24$$ $$98 - (24 + 42) = 32$$

Calculate average physical product and marginal physical product, given the following information about total physical product of a firm.
Employment of Variable Factor (Units) Employment of Fixed Factor (Units)
Total Physical Product (Units)
$$0$$
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$
$$6$$
$$7$$
$$5$$
$$5$$
$$5$$
$$5$$
$$5$$
$$5$$
$$5$$
$$5$$
$$0$$
$$10$$
$$24$$
$$40$$
$$56$$
$$70$$
$$80$$
$$84$$

Factor (Units)	(Units)	(Units)	(Units)
$$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$	$$0$$ $$10$$ $$24$$ $$40$$ $$56$$ $$70$$ $$80$$ $$84$$	$$0$$ $$10$$ $$12$$ $$13.33$$ $$14$$ $$14$$ $$13.33$$ $$12$$	$$0$$ $$10$$ $$14$$ $$16$$ $$16$$ $$14$$ $$10$$ $$4$$

Given the following information about a firm, identify the stages of production.
Units of Labour $$1$$ $$2$$ $$3$$ $$4$$ $$5$ $$6$$ $$7$$ $$8$$
Units of output $$4$$ $$12$$ $$24$$ $$32$$ $$36$$ $$36$$ $$28$$ $$16$$

Units of Labour	Units of Output $$(TP)$$	$$MP$$	Stages
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$	$$4$$ $$12$$ $$24$$ $$32$$ $$36$$ $$36$$ $$28$$ $$16$$	$$4$$ $$8$$ $$12$$ $$8$$ $$4$$ $$0$$ $$-8$$ $$-12$$	Increasing returns to a factor - - Diminishing returns to a factor - - - - Negative returns to a factor

Complete the following table:
Output
(Units) Total Cost
(Rs.) Average Variable Cost
(Rs.) Marginal Cost
(Rs.) Average Fixed Cost
(Rs.)
$$0$$
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$30$$
-
$$68$$
$$84$$
-
$$125$$ -
-
-
$$18$$
-
$$19$$ -
$$20$$
-
-
$$18$$
- -
-
-
-
-
$$6$$

Output (Units)	Total Cost (Rs.)	Average Variable Cost (Rs.)	Marginal Cost (Rs.)	Average Fixed Cost (Rs.)	Total Fixed Cost (Rs.)	Total Variable Cost (Rs.)
$$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$30$$ $$50$$ $$68$$ $$84$$ $$102$$ $$125$$	- $$20$$ $$19$$ $$18$$ $$18$$ $$19$$	- $$20$$ $$18$$ $$16$$ $$18$$ $$23$$	$$\infty$$ $$30$$ $$15$$ $$10$$ $$7.5$$ $$6$$	$$30$$ $$30$$ $$30$$ $$30$$ $$30$$ $$30$$	$$0$$ $$20$$ $$38$$ $$54$$ $$72$$ $$95$$

Demand schedule for commodity - $$X$$ of individual $$A$$ and $$B$$ is given in the following table. Derive the market demand schedule and market demand curve.
Price of Commodity - $$X (Rs.)$$ Demand of $$A$$ (Units) Demand of $$B$$ (Units)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$9$$
$$7$$
$$6$$
$$4$$
$$3$$ $$7$$
$$6$$
$$5$$
$$4$$
$$3$$

Market Demand Schedule

Price (Rs.)	Quantity (Units)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$9 + 7 = 16$$ $$7 + 6 = 13$$ $$6 + 5 = 11$$ $$4 + 4 = 8$$ $$3 + 3 = 6$$

From the table given below, calculate total revenue, average revenue, and marginal revenue.
Units Sold Price (Rs.) Total Revenue (Rs.) Marginal Revenue (Rs.)
$$5$$
$$4$$
$$3$$ $$6$$
$$7$$
$$8$$ -
-
- -
-
-

Rewrite the table as follows.

Units Sold	Price (Rs.)	Total Revenue (Rs.)	Marginal Revenue (Rs.)
$$3$$ $$4$$ $$5$$	$$8$$ $$7$$ $$6$$	$$24$$ $$28$$ $$30$$	- $$4$$ $$2$$

Find out the average revenue and marginal revenue on the basis of the following data:
Units of Production Total Revenue (Rs.) Average Revenue (Rs.) Marginal Revenue (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$
$$6$$
$$7$$ $$20$$
$$36$$
$$48$$
$$56$$
$$60$$
$$60$$
$$56$$ - -

Units of Production	Total Revenue (Rs.)	Average Revenue (Rs.)	Marginal Revenue (Rs.)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$	$$20$$ $$36$$ $$48$$ $$56$$ $$60$$ $$60$$ $$56$$	$$20$$ $$18$$ $$16$$ $$14$$ $$12$$ $$10$$ $$8$$	$$20$$ $$16$$ $$12$$ $$8$$ $$4$$ $$0$$ $$-4$$

Completer the following table:
Output (Units) Average Fixed Cost (Rs.) Average Variable Cost (Rs.) Marginal Cost (Rs.) Total Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$120$$
$$60$$
-
$$30$$
- $$40$$
$$56$$
$$54$$
-
- -
-
-
$$54$$
- -
$$232$$
-
-
-

Output (Units)	Average Fixed Cost (Rs.)	Average Variable Cost (Rs.)	Marginal Cost (Rs.)	Total Cost (Rs.)	Total Fixed Cost (Rs.)	Total Variable Cost (Rs.)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$120$$ $$60$$ $$40$$ $$30$$ $$24$$	$$40$$ $$56$$ $$54$$ $$54$$ -	$$40$$ $$72$$ $$50$$ $$54$$ -	$$160$$ $$232$$ $$282$$ $$336$$ -	$$120$$ $$120$$ $$120$$ $$120$$ $$120$$	$$40$$ $$112$$ $$162$$ $$216$$ -

Note: Lack of 'critical' information relating to output of $$5$$ units makes it impossible to estimate the possible values.

Complete the following table:
Output (Units) Price (Rs.) Total Revenue (Rs.) Marginal Revenue (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$ -
-
-
- $$6$$
-
-
- -
$$2$$
$$-2$$
$$-2$$

Output (Units)	Price (Rs.)	Total Revenue (Rs.)	Marginal Revenue (Rs.)
$$1$$ $$2$$ $$3$$ $$4$$	$$6$$ $$4$$ $$2$$ $$1$$	$$6$$ $$4$$ $$2$$ $$1$$	$$6$$ $$2$$ $$-2$$ $$-2$$

Complete the following table:
Output (Units) Total Cost (Rs.) Average Variable Cost (Rs.) Marginal Cost (Rs.) Average Fixed Cost (Rs.)
$$0$$
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$30$$
-
$$78$$
-
-
$$150$$ -
-
-
$$23$$
-
- -
$$25$$
-
-
$$23$$
- -
$$30$$
-
$$10$$
-
$$6$$

Output (Units)	Total Cost (Rs.)	Average Variable Cost (Rs.)	Marginal Cost (Rs.)	Average Fixed Cost (Rs.)	Total Fixed Cost (Rs.)	Total Variable Cost (Rs.)
$$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$30$$ $$55$$ $$78$$ $$99$$ $$122$$ $$150$$	$$0$$ $$25$$ $$24$$ $$23$$ $$23$$ $$24$$	- $$25$$ $$23$$ $$21$$ $$23$$ $$29$$	$$\infty$$ $$30$$ $$15$$ $$10$$ $$7.5$$ $$6$$	$$30$$ $$30$$ $$30$$ $$30$$ $$30$$ $$30$$	$$0$$ $$25$$ $$48$$ $$69$$ $$92$$ $$120$$

Complete the following table:
Output (Units) Average Revenue (Rs.) Marginal Revenue (Rs.) Total Revenue (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$ -
-
$$11$$
- $$15$$
-
-
$$3$$ -
$$26$$
-
-

Output (Units)	Average Revenue (Rs.)	Marginal Revenue (Rs.)	Total Revenue (Rs.)
$$1$$ $$2$$ $$3$$ $$4$$	$$15$$ $$13$$ $$11$$ $$9$$	$$15$$ $$11$$ $$7$$ $$3$$	$$15$$ $$26$$ $$33$$ $$36$$

Complete the following table:
Output (Units) Marginal Cost (Rs.) Average Variable Cost (Rs.) Total Cost (Rs.) Average Fixed Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$60$$
-
-
$$54$$
- -
-
$$54$$
-
$$57$$ $$120$$
$$174$$
-
-
$$345$$ -
-
-
$$15$$
-

Output (Units)	Marginal Cost (Rs.)	Average Variable Cost (Rs.)	Total Cost (Rs.)	Average Fixed Cost (Rs.)	Total Fixed Cost (Rs.)	Total Variable Cost (Rs.)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$60$$ $$54$$ $$48$$ $$54$$ $$69$$	$$60$$ $$57$$ $$54$$ $$54$$ $$57$$	$$120$$ $$174$$ $$222$$ $$276$$ $$345$$	$$60$$ $$30$$ $$20$$ $$15$$ $$12$$	$$60$$ $$60$$ $$60$$ $$60$$ $$60$$	$$60$$ $$114$$ $$162$$ $$216$$ $$285$$

Complete the following table:
Output (Units) Average Fixed Cost (Rs.) Marginal Cost (Rs.) Average Variable Cost (Rs.) Average Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$60$$
-
$$20$$
-
$$12$$ $$20$$
-
-
$$18$$
-
-
$$19$$
$$18$$
-
- -
-
-
-
$$31$$

Output (Units)	Average Fixed Cost (Rs.)	Marginal Cost (Rs.)	Average Variable Cost (Rs.)	Average Cost (Rs.)	Total Fixed Cost (Rs.)	Total Variable Cost (Rs.)	Total Cost (Rs.)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$60$$ $$30$$ $$20$$ $$15$$ $$12$$	$$20$$ $$18$$ $$16$$ $$18$$ $$23$$	$$20$$ $$19$$ $$18$$ $$18$$ $$19$$	$$80$$ $$49$$ $$38$$ $$33$$ $$31$$	$$60$$ $$60$$ $$60$$ $$60$$ $$60$$	$$20$$ $$38$$ $$54$$ $$72$$ $$95$$	$$80$$ $$98$$ $$114$$ $$132$$ $$155$$

Calculate marginal revenue using following information:
Price (Rs.) $$10$$ $$9$$
Sale/ Demand (Units) $$100$$ $$110$$

Price (Rs.)	Sale/ Demand (Units)	Total Revenue (Rs.)
$$10$$ $$9$$	$$100$$ $$110$$	$$1,000$$ $$900$$

$$MR = \dfrac {\triangle TR}{\triangle Sale}$$
$$= \dfrac {990 - 1,000}{110 - 100} = \dfrac {-10}{10} = -1$$
Marginal revenue $$= -1$$.

Complete the following table:
Output (Units) Price (Rs.) Marginal Revenue (Rs.) Total Revenue (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$ -
-
$$22$$
- $$30$$
-
-
$$6$$ -
$$52$$
-
-

Output (Units)	Price (Rs.)	Marginal Revenue (Rs.)	Total Revenue (Rs.)
$$1$$ $$2$$ $$3$$ $$4$$	$$30$$ $$36$$ $$22$$ $$18$$	$$30$$ $$22$$ $$14$$ $$6$$	$$30$$ $$52$$ $$66$$ $$72$$

Find producer's equilibrium, from the following. Give logical reason.
Quantity Sold (Units) Total Revenue (Rs.) Total Cost (Rs.)
$$5$$
$$6$$
$$7$$
$$8$$
$$9$$
$$10$$ $$15$$
$$20$$
$$25$$
$$30$$
$$35$$
$$40$$ $$18$$
$$22$$
$$26$$
$$27$$
$$32$$
$$38$$

Quantity Sold (Units)	Total Revenue (TR)(Rs.)	Total Cost (TC) (Rs.)	Marginal Revenue (MR)(Rs.)	Marginal Cost (MR) (Rs.)	Profit $$(\pi =TR - TC)(Rs.)$$
$$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$	$$15$$ $$20$$ $$25$$ $$30$$ $$35$$ $$40$$	$$18$$ $$22$$ $$26$$ $$27$$ $$32$$ $$38$$	- $$5$$ $$5$$ $$5$$ $$5$$ $$5$$	- $$4$$ $$4$$ $$1$$ $$5$$ $$6$$	$$-3$$ $$-2$$ $$-2$$ $$3$$ $$3$$ $$2$$

Producer is in equilibrium at $$9th$$ unit of output.
Reason: At output levels $$8th$$ and $$9th$$ unit,the difference between $$TR$$ and $$TC$$, i.e., profit is maximum, which is equal to $$3$$ in both the cases. But the producer is in equilibrium at $$9th$$ unit only where $$MR = MC (= 5)$$, and $$MC$$ is rising.

The following distribution gives the daily income of $$50$$ workers of a factory.
Daily income (in Rupees)
$$250-300$$ $$300-350$$ $$350-400$$ $$400-450$$ $$450-500$$
Number of workers $$12$$ $$14$$ $$8$$ $$6$$ $$10$$
Convert the above distribution to a less than type cumulative frequency distribution and draw its ogive.

Daily income	Number of workers	Daily wages (x-axis)	Cumulative frequency (y-axis)
250-300	12	<300	12
300-350	14	<350	12+14=26
350-400	8	<400	26+8=34
400-450	6	<450	34+6=40
450-500	10	<500	40+10=50

1408588_967768_ans_f196fd9bd69e414ca4e4f3419f04daf7.png

Find the profit maximising level of output.
Quantity Sold (Units) Total Revenue (Rs.) Marginal Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$12$$
$$26$$
$$34$$
$$40$$
$$42$$ $$15$$
$$9$$
$$6$$
$$2$$
$$3$$

Quantity Sold (Units)	Total Revenue (Rs.)	Marginal Cost (Rs.)	Total Cost (Rs.)	Profit $$(\pi = TR - TC) (Rs.)$$
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$12$$ $$26$$ $$34$$ $$40$$ $$42$$	$$15$$ $$9$$ $$6$$ $$2$$ $$3$$	$$15$$ $$24$$ $$30$$ $$32$$ $$35$$	$$-3$$ $$2$$ $$4$$ $$8$$ $$7$$

Profit is maximised when the output level is $$4$$ units because here, profit is maximum, i.e., $$8$$ and after this starts declining.
Profit is maximim at $$4th$$ unit.

Find out the profit from the following:
Quantity Sold (Units) Price (Rs. per unit) Average Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$15$$
$$16$$
$$17$$
$$18$$
$$19$$ $$15$$
$$12$$
$$10$$
$$12$$
$$14$$

Quantity Sold (Units)	Price (Rs. per unit)	AC (Rs.)	TR (Rs.)	TC (Rs.)	Profit $$(\pi = TR - TC)(Rs.)$$
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$15$$ $$16$$ $$17$$ $$18$$ $$19$$	$$15$$ $$12$$ $$10$$ $$12$$ $$14$$	$$15$$ $$32$$ $$51$$ $$72$$ $$95$$	$$15$$ $$24$$ $$30$$ $$48$$ $$70$$	$$0$$ $$8$$ $$21$$ $$24$$ $$25$$

The annual profits earned by $$30$$ shops of a shopping complex in a locality give rise to the following distributions:
Profit (in lakhs in Rs.) Number of shops (Frequency)
More than or equal to $$5$$ $$30$$
More than or equal to $$10$$ $$28$$
More than or equal to $$15$$ $$16$$
More than or equal to $$20$$ $$14$$
More than or equal to $$25$$ $$10$$
More than or equal to $$30$$ $$7$$
More than or equal to $$35$$ $$3$$
Draw both ogives for the above data and hence obtain the median.

For more than method:

Now, we mark on x-axis lower class limits, y-axis cumulative frequency

Thus, we plot the points $$(5,30),(10,28),(15,16),(20,14),(25,10),(30,7) \,and\, (35,3)$$

Less than method:

Profit in lakhs	No. of shops	Profit less than	C.F
$$0-10$$	$$2$$	$$10$$	$$2$$
$$10-15$$	$$12$$	$$15$$	$$14$$
$$15-20$$	$$2$$	$$20$$	$$16$$
$$20-25$$	$$4$$	$$25$$	$$20$$
$$25-30$$	$$3$$	$$30$$	$$23$$
$$30-35$$	$$4$$	$$35$$	$$27$$
$$35-40$$	$$3$$	$$40$$	$$30$$

Now we mark the upper class limits along x-axis and cumulative frequency along y-axis.

Thus we plot the points $$(10,2),(15,14),(20,16),(25,20),(30,23),(35,27),(40,30)$$

We find that the two types of curves intersect of P from point L it is drawn on x-axis

The value of a profit corresponding to M is $$17.5$$. Hence median is $$17.5\,lakh$$

The following table gives the height of trees:
Height No. of trees
Less than $$7$$ $$26$$
Less than $$14$$ $$57$$
Less than $$21$$ $$92$$
Less than $$28$$ $$134$$
Less than $$35$$ $$216$$
Less than $$42$$ $$287$$
Less than $$49$$ $$341$$
Less than $$56$$ $$360$$
Draw 'less than' ogive and 'more than' ogive.

For less than method :

Plot the points $$(7,26) , (14,57) , (21,92) , (28,134) , (35,216) , (42,287) , (49,341) , (56,360)$$ by taking upper class limit over the x-axis and cumulative frequency over the y-axis.

By more than method :

Height	Frequency	Height more than	cf
$$0-7$$	$$26$$	$$0$$	$$360$$
$$7-14$$	$$31$$	$$7$$	$$334$$
$$14-21$$	$$35$$	$$14$$	$$303$$
$$21-28$$	$$42$$	$$21$$	$$268$$
$$28-35$$	$$82$$	$$28$$	$$226$$
$$35-42$$	$$71$$	$$35$$	$$144$$
$$24-49$$	$$54$$	$$42$$	$$73$$
$$49-56$$	$$19$$	$$49$$	$$19$$

Take lower class limit over the x-axis and CF over the y-axis and plot $$(0,360) , (7,334) , (14,303) , (21,268) (28,226) , (35,144) , (42,73) , (49,19).$$

Complete the following table:
Output (Units) Price (Rs.) Total Revenue (Rs.) Marginal Revenue (Rs.)
-
-
-
- $$10$$
$$7$$
$$5$$
$$3$$ -
-
-
- $$10$$
$$4$$
$$1$$
$$-3$$

Output (Units)	Price (Rs.)	Total Revenue (Rs.)	Marginal Revenue (Rs.)
$$1$$ $$2$$ $$3$$ $$4$$	$$10$$ $$7$$ $$5$$ $$3$$	$$10$$ $$14$$ $$15$$ $$12$$	$$10$$ $$4$$ $$1$$ $$-3$$

A perfectly competitive firm faces the market price of $$Rs. 20$$ per unit. Derive its $$TR$$ schedule for a range of output from $$1$$ to $$10$$ units. What will happens to market price if the firm decides to suspend its production for the time being?

Output (Units)	$$1$$	$$2$$	$$3$$	$$4$$	$$5$$	$$6$$	$$7$$	$$8$$	$$9$$	$$10$$
Price (Rs.)	$$20$$	$$20$$	$$20$$	$$20$$	$$20$$	$$20$$	$$20$$	$$20$$	$$20$$	$$20$$
Total Revenue (Rs.)	$$20$$	$$40$$	$$60$$	$$80$$	$$100$$	$$120$$	$$140$$	$$160$$	$$180$$	$$200$$

If the firm decides to suspend its production for the time being, the market price will not change. Because the supply of a single seller cannot influence the market supply in a perfectly competitive market.

Complete the following table:
Units of Output Total Revenue (Rs.) Average Revenue (Rs.) Marginal Revenue (Rs.)
$$1$$
$$2$$
$$3$$ -
-
- $$10$$
$$8$$
$$6$$ -
-
-

Units of Output	Total Revenue (Rs.)	Average Revenue (Rs.)	Marginal Revenue (Rs.)
$$1$$ $$2$$ $$3$$	$$10$$ $$16$$ $$18$$	$$10$$ $$8$$ $$6$$	$$10$$ $$6$$ $$2$$

Complete the following table:
Price (Rs.) Output (Units) Total Revenue (Rs.) Marginal Revenue (Rs.)
$$7$$
-
-
$$1$$ -
$$2$$
$$3$$
- $$7$$
$$10$$
-
- -
-
$$-1$$
$$-5$$

Price (Rs.)	Output (Units)	Total Revenue (Rs.)	Marginal Revenue (Rs.)
$$7$$ $$5$$ $$3$$ $$1$$	$$1$$ $$2$$ $$3$$ $$4$$	$$7$$ $$10$$ $$9$$ $$4$$	$$7$$ $$3$$ $$-1$$ $$-5$$

Consider the following distribution:
Marks 0 or more 10 or more 20 or more 30 or more 40 or more 50 or more
Number of Students 63 58 55 51 48 42
(a) Calculate the frequency of class 30 - 40.
(b) Calculate the class mark of the class 10 - 20.

The given table in question is more than frequency distribution table.

We are going to form frequency distribution table :

$$Marks$$	$$No.\,of\,students$$
$$0-10$$	$$63-58=5$$
$$10-20$$	$$58-55=3$$
$$20-30$$	$$55-51=4$$
$$30-40$$	$$51-48=3$$
$$40-50$$	$$48-42=6$$
$$50-60$$	$$42$$

$$(a)$$ The frequency of class $$30-40=3$$

$$(b)$$ The class mark of the class $$10-20=\dfrac{10+20}{3}=15$$

During the medical check-up of $$35$$ students of a class, their weights were recorded as follows :
Weight (in $$kg$$) Number of students
Less than $$38$$ $$0$$
Less than $$40$$ $$3$$
Less than $$42$$ $$5$$
Less than $$44$$ $$9$$
Less than $$46$$ $$14$$
Less than $$48$$ $$28$$
Less than $$50$$ $$32$$
Less than $$52$$ $$35$$
Draw less than type ogive for the given data. Hence, obtain the median weight from the graph and verify the result by using the formula.

Less than method, It is given that,

On x-axis upper class limits. Y-axis cumulative frequency

We plot the points $$(38,0) (40,3)(42,5)(44,9)(46,4)(48,28)(50,32)(52,35)$$

Here, $$N=35$$ so, $$\dfrac{N}{2}=17.5$$

Mark the point $$A$$ whose coordinate is $$17.5$$ and its x-coordinate is $$46.5$$

$$Weight$$	$$Frequency$$	$$cf$$
Less than $$38$$	$$0$$	$$0$$
$$38-40$$	$$3$$	$$3$$
$$40-42$$	$$2$$	$$5$$
$$42-44$$	$$4$$	$$9$$
$$44-46$$	$$5$$	$$14$$
$$46-48$$	$$14$$	$$28$$
$$48-50$$	$$4$$	$$32$$
$$50-52$$	$$3$$	$$35$$
	$$N=35$$

$$\Rightarrow$$ $$N=35$$ and $$\dfrac{N}{2}=17.5$$

$$\Rightarrow$$ Class interval is $$46-48$$ then median class $$=46-48$$

$$l =$$ lower limit of the modal class

$$h =$$ size of the class intervals

$$f =$$ frequency of the modal class

$$\Rightarrow$$ $$l=46,\,f=14,\,cf=14,\,h=2$$

$$\Rightarrow$$ $$Median=l+\dfrac{\dfrac{N}{2}-cf}{f}\times h$$

$$\Rightarrow$$ $$Median=46+\dfrac{17.5-14}{14}\times 2$$

$$\Rightarrow$$ $$Median=46+\dfrac{3.5}{7}=46.5$$

$$\therefore$$ Median of this data is $$46.5$$.

Hence, the value of median is verified.

Organise the following marks in a class assessment in a tabular form.
$$4,6,7,5,6,5,4,5,2,6,2,5,1,9,6,5,8,4,6,7$$

Marks	$$1$$	$$2$$	$$3$$	$$4$$	$$5$$	$$6$$	$$7$$	$$8$$	$$9$$
frequency	$$1$$	$$2$$	$$1$$	$$.3$$	$$5$$	$$4$$	$$2$$	$$1$$	$$1$$

The following table shows the details of items purchased in a school for the noon-meal.
Items Quantity Price for $$1$$ kilogram (in rupees) Total price
Tomato $$2$$ kg $$20\dfrac{1}{2}$$ $$41$$ rupees
Potato $$3\dfrac{1}{2}$$ kg $$25$$ -
Peas $$5$$ kg $$30\dfrac{1}{2}$$ -
Dal $$2\dfrac{1}{4}$$ kg $$32$$ -
What is the total weight of all the items?

As we know, we can write mixed fraction as below:

$$a\dfrac{b}{c}=\dfrac{a.c+b}{c}=a+\dfrac{b}{c}$$

So, total weight ( in $$kg$$ ) $$=2+3\dfrac{1}{2}+5+2\dfrac{1}{4}$$

$$=2+3+\dfrac{1}{2}+5+2+\dfrac{1}{4}=12+\dfrac{3}{4}=12\dfrac{3}{4}\ kg$$.

The amount collected by $$15$$ students for flood relief find is given below. Answer the following.
Amount Number of students
$$0-10$$ $$3$$
$$10-20$$ $$4$$
$$20-30$$ $$5$$
$$30-40$$ $$2$$
$$40-50$$ $$1$$
(i)How many students collected less than $$Rs.\ 20$$?

Number of student who collected less than $$Rs.20$$

$$=4+3=7$$ students

The weight of new born babies (inn kg) in a hospital on a particular day are as follows
2.3, 2.2, 2.1, 2.7, 2.6, 3.0, 2.5, 2.9, 2.8, 3.1, 2.5, 2.8, 2.7, 2.9, 2.4
(a) How many babies were born on that day?
(b) How many babies weigh below 2.5 kg?
(c) How many babies weigh more than 2.8 kg?
(D) How many babies weigh 2.8?

Given data:$$2.3,2.2,2.1,2.7,2.6,3.0,2.5,2.9,2.8,3.1,2.5,2.8,2.7,2.9,2.4$$

a) $$15$$

b) $$4$$

c) $$4$$

d) $$2$$

The following tables gives production yield per hectare (in quintals) of wheat of $$100$$ farms of a village.
Production
yield/hect
$$40-45$$
$$45-50$$
$$50-55$$
$$55-60$$
$$60-65$$
$$65-70$$
No. of farms
$$4$$ $$6$$ $$16$$
$$20$$
$$30$$
$$24$$
Change the distribution to a more than type distribution and draw its ogive

Production field	No. of Farms $$(F_i)$$
40 - 45	4
45 - 50	6
50 - 55	16
55 - 60	20
60 - 65	30
65 - 70	24
	$$\sum F_i = 100$$

Production yield (X-axis)	No. of Farms (Y -axis)
More than 40	$$100 $$
More than 45	$$100 - 4 = 96$$
More than 50	$$96 - 6 = 90$$
More than 55	$$90 - 16 = 74$$
More than 60	$$74 - 20 = 54$$
More than 65	$$54 - 30 = 24$$

Production yeild is plotted in X-axis & no of Farms in Y-axis.

Production Yield (X-axis)	No. Of Farms (Y-axis)
More than 40	100
More than 45	96
More than 50	90
More than 55	74
More than 60	54
More than 65	24

1501334_1259351_ans_ea6c887bec4f4614815654d093ef4957.png

The following is the favourite cartoon characters of $$15$$ children.
Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey mouse, Mickey mouse, Jerry,, Jerry,, Jerry, Doraemon, Doraemon, Doraemon.
Which cartoon character is liked least by the children

Cartoon characters	Tally	Frequency
Doraemon	$$\nparallel \nparallel$$	$$6$$
Jerry	$$\nparallel \parallel$$	$$5$$
Mickey mouse	$$\nparallel$$	$$3$$
Tom	$$\|$$	$$1$$

From the above table, Tom is least liked character by children.

The following are the favorite cartoon characters of $$15$$ children.
Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey Mouse, Mickey mouse, Jerry, Jerry, Jerry, Doraemon, Doraemon, Doraemon.
Which cartoon character is liked by most of the children

Cartoon characters	Tally	Frequency
Doraemon	$$\nparallel \nparallel$$	$$6$$
Jerry	$$\nparallel \parallel$$	$$5$$
Mickey mouse	$$\nparallel$$	$$3$$
Tom	$$\|$$	$$1$$

From the above table, Doraemon is most liked cartoon character by most of children.

The ages of ten students of a group are given below. The ages have been recorded in years and months:
$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$
What is the highest age?

1481799_1670793_ans_d83e18ceaaad45df98bb67833b47bc84.jpeg

The monthly pocket money of six friends is given below:
Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45
What is the highest pocket money?

1481806_1670795_ans_31c3a267b42e44f28f279f05e41644be.jpeg

The monthly pocket money of six friends is given below:
Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45
What is the lowest pocket money?

1481809_1670796_ans_f853fcd8ab9948c9b4d683a55059a546.jpeg

Read the bar graph given in Fig. 23.18 and answer the following question:
What information does it give?

1671121_1671108_ans_eb9057b1f94b46528b622e106a72e1a0.jpg

The production of oil (in lakh tonnes) in some of the refineries in India during 1982 was given below :
Refinery : Barauni Koyali Mathura Mumbai Florida
Production of oil
(in lakh tonnes ) 30 70 40 45 25
Construct a bar graph to represent the above data so that the bars are drawn horizontally.

To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let us consider that the vertical and horizontal axes represent the refineries and the production of oil in lakh tonnes respectively.

We have to draw $$5$$ bars of different lengths given in the table.

At first, we mark $$5$$ points in the vertical axis at equal distances and erect rectangles of the same wk at these points. The lengths of the rectangles are proportional to the productions of oil.

The horizontal bar graph of the given data is following:

1482624_1671318_ans_a0bc1f5332104e9e9cde555aed89e1c5.png

The ages of ten students of a group are given below. The ages have been recorded in years and months:
$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$
Determine the range?

1481803_1670794_ans_efaed38c873a40f0be44a4c5ff13917a.jpeg

The monthly pocket money of six friends is given below:
Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45
What is the range?

1481810_1670797_ans_c64dadf524b54e6ba031f530cf439e84.jpeg

The pie-chart given in Fig. shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) of
Wheat

$$\because$$Total Production $$= 81000$$ Tonnes.

Production of wheat $$=\dfrac{central\,angle\,for\,wheat\times total\,production}{{360}^{\circ}}=\dfrac{{120}^{\circ}\times\,81000}{{360}^{\circ}}=27,000$$tonnes.

The pie chart given in the figure shows the annual agricultural production of an Indian state. If the total production of all commodities is $$81000\ tonnes,$$ find the production (in tones) of
Sugar.

Production of sugar $$=\dfrac{central\,angle\,for\,sugar\times\,total\,production}{{360}^{\circ}}$$

$$=\dfrac{{100}^{\circ}\times\,81000}{{360}^{\circ}}$$

$$=22,500$$ tones.

The pie-chart given in Fig.25.18 shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) of
Gram

Given:Total Production $$=81000$$ Tonnes.

Production of rice$$=\dfrac{central\,angle\,for \, gram\times\,total\,production}{{360}^{\circ}}=\dfrac{{50}^{\circ}\times\,81000}{{360}^{\circ}}=11250$$ tonnes.

Locate the points:
$$(1,3), (2,3), (3,3), (4,3)$$

Point represent as $$(x,y)$$

First value $$x$$ is represent x-coordinates

Second value $$y$$ is represent y-coordinates

2129107_1674154_ans_12f80af462ca409283b2df6ab8425671.png

The pie-chart given in Fig.25.18 shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) of
Maize

$$\because$$Total Production $$= 81000$$ Tonnes.

Production of maize $$=\dfrac{central\,angle\,for\,maize\times total\,production}{{360}^{\circ}}=\dfrac{{30}^{\circ}\times\,81000}{{360}^{\circ}}=6,750$$tonnes.

The following pie-chart shows the number of students admitted in different faculties of a college. If $$1000$$ students are admitted in Science answer the following:
What is the total number of students?

Let the total number of students be $$x.$$

$$\text{Students in science}=\dfrac{\text{Central angle of the sector}\times\text{Total students}}{{360}^{\circ}}$$

$$1000=\dfrac{{100}^{\circ}\times x}{{360}^{\circ}}$$

$$x=1000\times \dfrac{360}{100}$$

$$=3600$$

The pie-chart given in Fig.25.18 shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) of
Rice

$$\because$$Total Production $$= 81000$$ Tonnes.

Production of rice $$=\dfrac{central\,angle\,for\,rice\times total\,production}{{360}^{\circ}}=\dfrac{{60}^{\circ}\times\,81000}{{360}^{\circ}}=13,500$$tonnes.

The following pie-chart shows the monthly expenditure of Shikha on various items. If she spends $$Rs\ 16000$$ per month, answer the following question:
How much she spend on rent ? (in Rs)

Given :

$$\text{Total money spent}\ = 16000$$

$$\text{Central angle of Education }\ = 36^{\circ}$$

$$\text{Central angle of Food }\ = 135^{\circ}$$

$$\text{Central angle of Rent }\ = 81^{\circ}$$

$$\text{Central angle of Miscellaneous }\ = 36^{\circ}\\$$

To find:

Money spent on Rent

We know that:

$$\text{Money spent on a sector}\ = \left(\dfrac{\text{Central angle of the corrosponding sector} \times \text{Total money spent} } {360^{\circ}}\right)\\$$

Solution:

$$\Rightarrow \text{Money spent on Rent}\ = \left(\dfrac{\text{Central angle of Rent} \times \text{Total money spent} } {360^{\circ}}\right)\\$$

$$\Rightarrow \text{Money spent on Rent}\ = \left(\dfrac{81^{\circ} \times 16000 } {360^{\circ}}\right)\\$$

$$\Rightarrow \text{Money spent on Rent}\ = 3600\\$$

Locate the points:
$$(1,4), (2,4), (3,4), (4,4)$$

1473258_1674155_ans_40808b4c50c54c41b1fcd31a3bfc3d00.jpg

Read the following bar graph and answer the following questions:
In which year the export is minimum?

Export is minimum in 1982-83 (800 crores) as height of bar is minimum in this year.

Read the bar graph shown in fig. and answer the following question:
Of which state, were the maximum number of tickets sold?

Maximum tickets were sold of Haryana (100) .

Read the following bar graph and answer the following questions:
In which year the import is maximum?

Import is maximum in 1986-87 (2200 crores) as height of bar is maximum in this year.

Define the following terms: data

In mathematics, data is information that has been translated into a form that is efficient for movement or processing. Relative to today's computers and transmission media, data is information converted into binary digital form. It is acceptable for data to be used as a singular subject or a plural subject.

The following bar graph (Fig.) represents the heights (in cm) of 50 students of Class XI of a particular school. Study the graph and answer the following questions:
The school wants to provide a particular type of tonic to each student below the height of 150 cm to improve his height. If the cost of the tonic for each student comes out to be Rs 55, how much amount of money is required?

The total number of students between the height range of 140cm-150 cm =5+10=15 students

Per student the cost of the tonic is Rs 55

Total cost of the tonic would be = 15*55 = Rs825

The number of stools in five rooms of a school are given below, Draw a pictograph of stools in the five rooms
Room number $$I$$ $$II$$ $$III$$ $$IV$$ $$V$$
Number of stools $$30$$ $$40$$ $$60$$ $$50$$ $$20$$

The pictograph of number of stools in five rooms of a school is given:

1623080_1778419_ans_77e52ebf33ab469cbad196ffdf77f85b.png

The number of students who were absent from the class during a week is given below, Draw the pictograph of the number of students who were absent from the class during the week.
Day Monday Tuesday Wednesday Thursday Friday
No. of absentees $$6$$ $$2$$ $$4$$ $$2$$ $$8$$

The pictograph of the number of students who were absent from the class during the week is given:

1623060_1778414_ans_ec6ebf8ce7224800a861b8361cca1e45.png

1623060_1778414_ans_ec6ebf8ce7224800a861b8361cca1e45.png

Define the term Observations

Observations: Each numerical figure in a data is called an observation.

In a class test, the number of students passed in various subjects are given below. Draw the pictograph of number of students passed in various subjects.
Subject English Mathematics Hindi Drawing
Number of students passes $$15$$ $$25$$ $$10$$ $$20$$

The pictograph of number of students passed in various subjects is given

1623134_1778423_ans_b78b92fb874c4c81a4e789a8e86228b7.png

Define the term:
Tabulation of data

Arranging the data in a systematic form in the form of a table is called tabulation of data.For example: Tabulation of data on the population of the world classified by one characteristic like religion is an example of a simple tabulation.

No. of children $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$
No. of families $$5$$ $$11$$ $$25$$ $$12$$ $$5$$ $$2$$
Which values is discussed here?

For progress, we should reduced the population growth.

Fill in the blanks:
Arranging the data in the form of a table is called ................ of data.

Tabulation
According to the definition, arranging the data in a systematic form in the form of a table is called tabulation of data.

Some people of society decreased with area with flags and tricolour ribbons on Republic day. the following data shown the number of person in different age group who participated in the decoration.
Age (in years) $$5-15$$ $$15-25$$ $$25-35$$ $$35-45$$ $$45-55$$ $$55-65$$
Number of patients $$6$$ $$11$$ $$21$$ $$23$$ $$14$$ $$5$$

$$h=10, f_1=23, f_0 =21, f_2=14, l=35$$
Mode $$=l+\left(\dfrac {f_1 -f_0}{2f_1 -f_0 -f_2}\right)\times h$$
Mode $$=35+\dfrac {23-21}{46+35}\times 10=35+\dfrac {20}{11}=35+1.8=36.8$$
National integrity, Unity, Beauty.

Define the term Tabulation of data

Tabulation of data: Arranging the data in systematic form in the form of a table is called Tabulation the data.

Explain the situations represented by the following distance- time graphs.

In the graph (a): The body covers equal distance in equal intervals of time.

i.e., the body moves with uniform velocity

In the graph (b): The body moves with uniform velocity for some time and after that it becomes constant.

In the graph (c): The distance covered by the body in a time interval reduces gradually.

i.e., the body moves with decreasing velocity.

The cost of a notebook is Rs $$10$$. Draw a graph after making a table showing the cost of 2, 3, 4 ... notebooks. Use it to find
(a) the cost of $$7$$ notebooks.
(b) the number of notebooks that can be purchased with Rs$$50$$.

Let $$x$$ be the number of notebooks

and $$y$$ be the cost of notebooks

Then, the table is completed as shown.

And hence, the graph is plotted from the table.

(a) The cost of $$7$$ notebooks is equal to the ordinate of the point $$(7,70)$$,

i.e. cost of $$7$$ notebooks $$=$$ Rs $$70$$

(b) The number of notebooks that can be purchased with Rs 50 is equal to the abscissa of the point $$(5,50)$$.

Hence, $$5$$ notebooks can be purchased with Rs $$50.$$

1689923_1794315_ans_c309536f0c404510ae46b14056ef941c.png

Complete the given tables and draw a graph for each.
Second table:

Given,

$$y=x-1$$

Therefore,

for $$x=2, y=(2)-1=1$$

for $$x=4, y=(4)-1=3$$

for $$x=6, y=(6)-1=5$$

The graph of the given data in table is plotted as shown.

1689992_1794346_ans_f9db5b869a5b4c0dae18d4912dc36364.png

Complete the given table and draw a graph.

Given, $$y=3x+1$$

Therefore,

for $$x=2,\ y=3(2)+1=6+1=7$$

for $$x=3,\ y=3(3)+1=9+1=10$$

The graph of the given data in table is plotted as shown.

1689989_1794344_ans_e529503706434c098fed149d5982005d.png

The following table gives the number of pages written by Sarika for completing her own book for 30 days :
Number of pages written per day 16-18 19-21 22-24 25-27 28-30
Number of days 1 3 4 9 13

The given data is not continuous.
Hence, we subtract 0.5 from the lower limit and add 0.5 in the upper limit of each class.

Class Marks	MId-Value ($$x_i$$)	Number of days ($$f_i$$)	$$f_i x_i$$
15.5-18.5	17	1	17
18.5-21.5	20	3	60
21.5-24.5	23	4	92
24.5-27.5	26	9	234
27.5-30.5	29	13	377
		$$\Sigma f_i = 30$$	$$\Sigma f_i x_i = 780$$

Mean $$(x) = \dfrac{\Sigma f_i x_i}{\Sigma f_i} = \dfrac{780}{30} = 26$$
Therefore, the mean of pages written per day = 26.

Study the bar graph given below and answer the question that follow:
(a) What information does the above bar graph represent?
(b) In which year was production the least?
(c) After which year was the maximum rise in the production?
(d) Find the average production of rice during the 5 years.
(e) Find the difference of rice production between years $$2006\, and\, 2008$$.

(a) The information about the production of rice by a country during the years of 2005 to 2009 is represented by the bar graph.

(b) The least production was in 2006.

Therefore, after 2006 the production has a maximum rise.

(d) The average production

=$$\dfrac{50+40+70+50+60}{5}$$ million tonnes

=$$\dfrac{270}{5}=54$$ million tonnes

(e) 40 tonnes rice was produced in 2006

50 tonnes rice was produced in 2008

Therefore the difference

=$$50-40=10$$ million tonnes

Below is a list of 10 tallest buildings in India.

This list ranks buildings in India that stand at least 150m (492 ft.) tall, based on standard height measurement. This includes spires and architectural details but does not include antenna marks. Following data is given as per the available information tillSince new buildings are always under construction, go on-line to check new taller buildings.

Use the information given in the table about skyscrapers to answer the following questions:
Name City Height Floors Year
Planet

UB
Tower

Ashok
Towers

The
Imperial
I

The
Imperial
II

RNA
Mirage

Oberoi
Woods
Tower I

Oberoi
Woods
Tower II

Oberoi
Woods
Tower III

MVRDC

(a) Find the height of each storey of the three tallest buildings and write them in the following table:

Building Height Number of

Stories Height of

each Storey

(b) The average height of one storey for the buildings given in (a) is ________

What is the percentage?

(d) What is the range of the data?

(e) Find the median of the data.

(f) Draw a bar graph for given data.

(a)

Building Height Number

of stories Height of

each

Storey

The

Imperial I 249 m 60 4.15 m

The

Imperial II 249 m 60 4.15 m

Ashok

Tower 193 m 49 3.9 m

(b) The average height of one storey for the buildings=$$(\dfrac{3.9+4.15+4.15}{3})$$

=$$\dfrac{122}{3}= 4.06 m$$

The number of skyscrapers =10

The total number of skyscrapers in Mumbai=9

Hence, percentage of skyscrapers in Mumbai=$$\dfrac{9}{10} \times 100= 90%$$

(d) Range= highest value- Lowest value= 249 m - 156 m=93 m

(e) Rearranging the height into ascending order $$156,170,170,170,180,181,184,193,249,249$$

Because of data are even in number

$$\therefore\, Median=(\dfrac{180+181}{2})$$

=$$\dfrac{361}{2}= 180.5 m$$

Make a table corresponding to the above graph:

Table showing student's favourite cricketer.

Cricketer	Rohit	Dhoni	Virat	Ashwin
Number of students	10	16	20	7

Draw frequency polygons for each of the following frequency distribution;
(a). Using histogram
(b). Without using histogram
(i)
$$C.I$$ $$10-30$$ $$30-50$$ $$50-70$$ $$70-90$$ $$90-110$$ $$110-130$$ $$130-150$$
$$f$$ $$4$$ $$7$$ $$5$$ $$9$$ $$5$$ $$6$$ $$4$$

$$(i)$$

$$(a)$$ Using Histrogram

$$C.I$$ $$f$$

$$10-30$$ $$4$$

$$30-50$$ $$7$$

$$50-70$$ $$5$$

$$70-90$$ $$9$$

$$90-110$$ $$5$$

$$110-130$$ $$6$$

$$130-150$$ $$4$$

Steps:

$$i.$$ Draw a histogram for the given data.

$$ii.$$ Mark the mid-point at the top of each rectangle of the histogram drawn.

$$iii.$$ Also, mark the mid-point of the immediately lower class-interval and mid-point of the immediately higher class-interval.

$$iv.$$ Join the consecutive mid-points marked by straight lines to obtain the required frequency polygon.

(b) Without using Histogram:

Steps:

$$i.$$ Find the class-mark (mid-value) of each given class-interval.

$$Class-mark=mid-value=\dfrac {Upper\ limit + Lower\ limit}{2}$$

$$ii.$$ On a graph paper, mark class-marks along $$X-$$axis and frequencies along $$Y-$$axis.

$$iii.$$ On the graph paper, mark point taking values of class-marks along $$X-$$axis and the values of their corresponding frequencies along $$Y-$$axis.

$$iv.$$ Draw the line segments joining the consecutive point marked in step $$(3)$$ above.

$$C.I.$$ Class-mark $$f$$

$$-10-10$$ $$0$$ $$0$$

$$10-30$$ $$20$$ $$4$$

$$30-50$$ $$40$$ $$7$$

$$50-70$$ $$60$$ $$5$$

$$70-90$$ $$80$$ $$9$$

$$90-110$$ $$100$$ $$5$$

$$110-130$$ $$120$$ $$6$$

$$130-150$$ $$140$$ $$4$$

$$150-170$$ $$160$$ $$0$$

1800643_1841432_ans_8c34d561e6754880aa57fef214157e47.png

Construct a combined histogram and frequency polygon for the following frequency distribution:
Class-intervals $$10-20$$ $$20-30$$ $$30-40$$ $$40-50$$ $$50-60$$
Frequency $$3$$ $$5$$ $$6$$ $$4$$ $$2$$

Steps:

(i) Draw a histogram for the given data.

(ii) Mark the mid-points at the top of each rectangle of the histogram drawn.

(iii) Also, mark the mid-point of the immediately lower class-interval and mid-point of the immediately higher class-interval.

(iv)Join the consecutive midpoints marked by straight lines to obtain the required frequency polygon.

The required combined histrogram and frequency polygon is shown in the following figure:

1800639_1841383_ans_ff5bae9316694c5392a1d291952b3dda.png

The ages of $$40$$ students are given in the following table:
Age (in yrs) 12 13 14 15 16 17 18
Frequency 2 4 6 9 8 7 4

Age in yrs $$x_i$$	Frequency $$(f_i)$$	$$f_ix_i$$
12	2	24
13	4	52
14	6	84
15	9	135
16	8	128
17	7	119
18	4	72
Total	40	614

Mean $$= \sum fi xi/ \sum fi = 614/40 = 15.35$$

C.I. $$0-5$$ $$5-10$$ $$10-15$$ $$15-20$$ $$20-25$$
Frequency $$4$$ $$8$$ $$6$$ $$4$$ $$2$$
Draw the histogram and frequency polygon of the following distribution :

The given frequency distribution is grouped and continuous. so we construct a histogram. After that we join the middle points i/e., $$P, Q, R, S, T$$ of upper horizontal lines of rectangles $$A, B, C, D, E$$ respectively by straight lines. Since nothing is said here about initial and last intervals, so join the end points with the middle points of these intervals on the x-axis. We thus, obtained the frequency polygon $$P'PQRSTT'$$ for the given frequency distribution.

1883157_1880323_ans_6093bb4ab63c46969aff148a2c8c5218.png

1883157_1880323_ans_6093bb4ab63c46969aff148a2c8c5218.png

Construct a frequency polygon with the help of histogram for the following frequency distribution. Maximum marks is $$10$$.

Marks obtained $$2-4$$ $$4-6$$ $$4-6$$ $$6-8$$ $$8-10$$
No. of students $$7$$ $$8$$ $$4$$ $$9$$ $$2$$

Here, we see that class interval $$4 - 6$$ is repeated. So, we need to make certain modifications.
1883145_1880135_ans_0ba1781f0c614d008aa9cf0ec70b6896.PNG

1883145_1880135_ans_0ba1781f0c614d008aa9cf0ec70b6896.PNG

S.No. Section Number of Girls per thousand Boys
1 Scheduled Caste (SC) 940
2 Scheduled Tribe (ST) 970
3 Non SC/ST 920
4
Backward districts 950
5 Non-backward districts 920
6 Rural 930
7 Urban 910
The following data on the number of girls (to the nearest ten) per thousand boys in different sections of Indian society is given below :
(i) Represent the information above by a bar graph.
(ii) In the class room, discuss what conclusions can be arrived from the graph.

(i) Taking different sections of the society on the x-axis and number of girls per thousand boys on the y-axis, we draw a bar graph to represent the above data.
We-take one unit of length along y-axis = 10 girls
(ii) (a) The number of girls per thousand boys under Non SC/ST and non-backward districts are same as $$920$$.
(b) Number of girls under section Scheduled Tribe (ST) per thousand boys is maximum i.e. $$970$$ while minimum $$910$$ inn Urban section.
1880779_1880123_ans_87b19e0b21e14a5b939d067374fb186a.png

1880779_1880123_ans_87b19e0b21e14a5b939d067374fb186a.png

Name the type of data which a university uses while finding out average of best four subjects out of five, from your marks sheet.

Quantitative data can be measured and not simply observed. They can be numerically represented and calculations can be performed on them. For example, data on the number of students playing different sports from your class gives an estimate of how many of the total students play which sport. This information is numerical and can be classified as quantitative. Hence, university uses quantitative data while finding out average of best four subjects out of five, from your marks sheet.

Class Interval (C.I) $$0-5$$ $$5-10$$ $$10-15$$ $$15-20$$ $$20-25$$ $$25-30$$ $$30-35$$
Frequency $$1$$ $$2$$ $$4$$ $$6$$ $$5$$ $$3$$ $$2$$
Construct a frequency polygon with the help of histogram for the following frequency distribution.

We take $$OX$$ and $$OY$$ as x-axis and y-axis respectively $$O$$ as the origin.

By taking suitable scale at the x-axis as $$1 cm = 5$$ units and y-axis as $$1 cm = 1$$ unit,

we draw graph of the given distribution and join the upper mid-point of the rectangles $$A, B, C, D, E,F$$, and $$G$$.

Hence,

$$P'PQRSTUVV'$$ from the required frequency polygon.

1882335_1880534_ans_06dfcfaa812c4078952b655c57019001.png

Height (in cm) 135 - 140 140 - 145 145 - 150 150 - 155 155 - 160
Frequency 8 20 12 11 9
Draw the histogram and frequency polygon of the following distribution :

The given frequency distribution is grouped and continuous.

so we construct a histogram. After that, we join the middle points

i.e., $$P, Q, R, S, T$$ of upper horizontal lines of rectangles $$A, B, C, D, E$$ respectively by straight lines. Since nothing is said here about initial and last intervals,

so join the endpoints with the middle points of these intervals on the x-axis.

We, thus, obtained the frequency polygon $$P'PQRSTT'$$ for the given frequency distribution.

1882381_1880324_ans_902d266e973f41b88104399ab58a448a.png

The following are marks obtained by a group of 40 class X students in an english examination:
42 88 37 75 98 93 73 62 96 80
52 76 66 54 73 69 83 62 53 79
69 56 81 75 52 65 49 80 67 59
88 80 44 71 72 87 91 82 89 79
Prepare a frequency distribution and a cumulative frequency distribution
for these data using a class interval of 5.

2125884_2014860_ans_1288e55d120a4f7fb2fbc264d2988c43.png

Change the following data into 'less than type' distribution and draw its ogive:
Clas Interval : $$30 - 40$$ $$40 - 50$$ $$50 - 60$$ $$60 - 70$$ $$70 - 80$$ $$80 - 90$$ $$90 - 100$$
Frequency : $$7$$ $$5$$ $$8$$ $$10$$ $$6$$ $$6$$ $$8$$

$$\textbf{Step 1: Obtain the cumulative frequency distribution table.}$$

$$\text{The cumulative frequency distribution table is as follows:}$$

$$\text{Class Interval}$$ $$\text{F}$$ $$\text{CF(less than type)}$$

$$30 - 40$$ $$7$$ $$7$$

$$40 - 50$$ $$5$$ $$12$$

$$50 - 60$$ $$8$$ $$20$$

$$60 - 70$$ $$10$$ $$30$$

$$70 - 80$$ $$6$$ $$36$$

$$80 - 90$$ $$6$$ $$42$$

$$90 - 100$$ $$8$$ $$50$$

$$\textbf{Step 2: Obtain the less than type ogive.}$$

$$\text{The points to be plotted are:}$$

$$\mathrm{(30,0),(40,7),(50,12),(60,20),(70,30),(80,36),(90,42)\ and\ (100,50).}$$

$$\text{These points are joined to obtain the required ogive.}$$

$$\textbf{Hence, the required less than type ogive is obtained.}$$

2173309_1985704_ans_4da7e04ee48f4f6287a89296824efa07.jpg

Draw less than ogive curve from the following data.

Marks

Students

0-10

5

10-20

12

20-30

20

30-40

11

40-50

9

50-60

7

Marks	Students
Less than 10	5
Less than 20	5 + 12 = 17
Less than 30	17 + 20 = 37
Less than 40	37 + 11 = 48
Less than 50	48 + 9 = 57
Less than 60	57 + 7 = 64

Draw (10, 5), (20, 17), (30, 37), (40, 48), (50, 57) and (60, 64) to get less than ogive curve.
2160065_2033398_ans_46f4bc66d7014f91a350f1ebafa713eb.PNG

Following data relate to the marks secured by students in their statistics paper. Graph these data in the form of less than ogive.
Marks
No. of Students
0-5
4
5-10
6
10-15
10
15-20
10
20-25
25
25-30
22
30-35
18
35-40
5

Present the data in the form of a simple bar diagram.
Year 1970 1980 1990 2000
Production of Wheat (million tonnes) 100 210 350 500

The answer is given as:
2131934_2017609_ans_f5a03c2b519c4b7b848a3d9bf2b8d0a0.png

Explain the necessity and usefulness of diagrammatic presentation of data.

The following are necessities and usefulness of the diagrammatic presentation of data:

(i) Simple and Easy to understand: The data presented in the form of diagrams is the simplest and the easiest to understand. The entire data can be easily understood even by having a single glance at the diagram.

(ii) Attractive and Impressive: Diagrammatic presentation makes the data more attractive and interesting. Diagrams tend to leave along with lasting impact on the mind.

(iii) Helpful in Making Comparisons: The presentation of data in the form of diagrams helps in making comparisons between two or more groups or two or more periods.

In the following table, data of the transport means used by students $$8th$$ standard for commutation between home and school is give. Draw a subdivided bar diagrame to show the data.

Given Table

Town mean of communication	Paithan	Yeoca	Shanapur
Cycle	$$3250$$	$$1500$$	$$1250$$
Bus & Auto	$$750$$	$$500$$	$$500$$
On Foot	$$1000$$	$$1000$$	$$500$$

We have draw subdivided bar graph
1361690_1185684_ans_0871ffca92ae4f12b180eb688186ca17.png

The final marks in mathematics of 30 students are as follows:
$$53,61,48,60,78,68,55,100,67,90$$
$$75,88,77,37,84,58,60,48,62,56$$
$$44,58,52,64,98,59,70,39,50,60$$
If 40 is the pass mark how many have failed?

Given that final marks in mathematics of 30 students are as follows:
$$53,61,48,60,78,68,55,100,67,90$$
$$75,88,77,37,84,58,60,48,62,56$$
$$44,58,52,64,98,59,70,39,50,60$$

first of all arrange the marks as groups ascending order means lowest number to highest number

Group	Ascending order
30 to 39	37, 39
40 to 49	44, 48, 48
50 to 59	50, 52, 53, 55, 56, 58, 58, 59
60 to 69	60, 60, 60, 61, 62, 64, 67, 68
70 to 79	70, 75, 77, 78
80 to 89	84, 88
90 to 100	90, 98, 100

We observe that

Highest score $$= 100$$

Lowest score $$= 37$$

We know Range $$= $$ Highest $$-$$ Lowest $$= 100-37=63$$

$$\to$$ if $$40$$ is the pass mark then

number of the failed students $$=2$$

Read the bar graph shown in Fig. and answer the following questions: What was the number of commercial banks in 1977?

Given that

In a question bar graph of the number of commercial banks in India during some years

Now we can bar graph to represent as frequency distribution table

year $$1969$$ $$1976$$ $$1977$$ $$1978$$ $$1979$$ $$1980$$ $$1981$$ $$1982$$ $$1983$$

No. of commercial banks $$90$$ $$100$$ $$130$$ $$140$$ $$140$$ $$150$$ $$190$$ $$210$$ $$230$$

Here observe that

the number of commercial banks in $$1977$$ is $$130$$

the number of commercial banks in $$1969$$ is $$90$$

the number of commercial banks in $$1980$$ is $$150$$.

Study the bar graph representing the number of persons in various age groups in a town shown in Fig. Observe the bar graph and answer the following questions:
What is the total number of persons living in the town in the age-groups 10-15 and 60-65?

Given that in a question

the bar graph representing the number of persons in various age groups in a town

the bar diagraph represented as the frequency table

Age(in years) No. of persons

$$10-15$$ $$1400$$

$$20-25$$ $$1200$$

$$30-35$$ $$1100$$

$$40-45$$ $$1000$$

$$50-55$$ $$900$$

$$60-65$$ $$800$$

$$70-75$$ $$300$$

$$\therefore$$ The number of persons in the age-group $$60-65$$ is $$800$$

No. of persons in age-group $$10-15$$ is $$1400$$

No. of persons in age-group $$30-35$$ is $$1100$$

$$\therefore$$ No. of persons are more in the age-group $$1015$$ in the age group $$30-35$$ is $$=1400-1100=300$$

$$\Rightarrow$$ The total no. of persons living in the town in the age group $$50-55$$ is $$900$$

$$\Rightarrow$$ Total no. of persons living in the town in the age group $$10-15$$ & $$60-65$$ is $$800+1400$$

$$=2200$$.

Draw less than type cumulative frequency (O give) curve for the above data.
Daily Income (in rupee) 250-300 300-350 350-400 400-450 450-500
Number of workers 14 16 10 8 12

1485630_1587437_ans_e494b207f8ef4fa7815d1f521f5512ad.jpg

The weights of new born babies (in kg) in a hospital on a particular day are as follows:
$$2.3,2.2,2.1,2.7,2.6,3.0,2.5,2.9,2.8,3.1,2.5,2.8,2.7,2.9,2.4$$
How many babies weigh 2.8 kg?

Given weights of new born babies(in kg) in a hospital on a particular day are as follows

$$2.3, 2.2, 2.1, 2.7, 2.6, 3.0, 2.5, 2.9, 2.8, 3.1, 2.5, 2.8, 2.7, 2.9, 2.4$$

First all arrange the data in as ascending order

$$2.1, 2.2, 2.3, 2.4, 2.5, 2.5, 2.6, 2.7, 2.7, 2.8, 2.8, 2.9, 2.9, 3.0, 3.1$$

$$\therefore$$ No of new born babies weigh below $$2.5$$ kg $$1$$ & $$4$$

$$\Rightarrow$$ No. of new born babies weigh more than $$2.8$$ kg $$1$$ & $$4$$

and

No. of new born babies weigh $$2.8$$ kg is $$2$$.

In a mathematics test following marks were obtained by 40 students of class VI. Arrange these marks in a table using, tally marks.
8 1 3 7 6 5 5 4 4 2
4 9 5 3 7 1 6 5 2 7
7 3 8 4 2 8 9 5 8 6
7 4 5 6 9 6 4 4 6 6
Find how many students obtained marks equal to or more than 7?

In a mathematics test following matrix were obtained by $$40$$ students of class VI.

Arrange these marks in table, tally marks

Marks Frequency Tally marks

$$1$$ $$2$$ ||

$$2$$ $$3$$ |||

$$3$$ $$3$$ |||

$$4$$ $$7$$ ||||

$$5$$ $$6$$ ||||

$$6$$ $$7$$ ||||

$$7$$ $$5$$ ||||

$$8$$ $$4$$ ||||

$$9$$ $$3$$ ||||

Total $$40$$

No. of students obtained marks equal to or more than $$7$$. $$(7, 8, 9$$ marks$$)$$

is $$=5+4+3$$

$$=12$$.

The runs scored by a cricket team in first 15 overs are given below:
Overs: $$I$$ $$II$$ $$III$$ $$IV$$ $$V$$ $$VI$$ $$VII$$ $$VIII$$ $$IX$$ $$X$$ $$XI$$ $$XII$$ $$XIII$$ $$XIV$$ $$XV$$
Runs: $$2$$ $$1$$ $$4$$ $$2$$ $$6$$ $$8$$ $$10$$ $$21$$ $$5$$ $$8$$ $$3$$ $$2$$ $$6$$ $$8$$ $$12$$
Draw the graph representing the above data in two different ways as a graph and as a bar chart:

1483215_1674141_ans_c6570759cad042918c6b2b6eff696221.jpg

The following table gives the information regarding the number of persons employed to a piece of work and time taken to complete the work:
Number of persons: $$2$$ $$4$$ $$6$$ $$8$$
Time taken(in days): $$12$$ $$6$$ $$4$$ $$3$$
Plot a graph of this information:

1483208_1674070_ans_4e777bfade9e4afd9476705ba9ec1d24.jpg

Read the bar graph shown in Fig. and answer the following questions: What is the ratio of the number of commercial banks in 1969 to that in 1980?

Given that

In a question bar graph of the number of commercial banks in India during same years

Now we can bar graph to represent as frequency distribution table

Year $$1969$$ $$1976$$ $$1977$$ $$1978$$ $$1979$$ $$1980$$ $$1981$$ $$1982$$ $$1983$$

No. of commercial banks $$90$$ $$100$$ $$130$$ $$140$$ $$140$$ $$150$$ $$190$$ $$210$$ $$230$$

Here observe that

the number of commercial banks in $$1977$$ is $$130$$

the number of commercial banks in $$1969$$ is $90$$

the number of commercial banks in $$1980$$ is $$150$$

the ratio of the number of commercial banks in $$1969$$ to that $$1980$$

$$\Rightarrow 90:150$$

$$=3:5$$.

Given below (Fig.) is the bar graph indicating the marks obtained out of 50 in mathematics paper by 100 students. Read the bar graph and answer the following question:
What is the number of students belonging to the highest mark group?

Given in a question

the bar graph indicating the marks obtained out of $$50$$ in mathematics paper by $$100$$ students

then

now we prepare frequency table

Marks No. of students

$$0-9$$ $$26$$

$$10-19$$ $$12$$

$$20-29$$ $$21$$

$$30-39$$ $$24$$

$$40-49$$ $$17$$

The no of students belonging to the highest mark group ($$40-49$$) is $$17$$.

Read the following bar graph and answer the following questions:
Which state has the total production of wheat and rice minimum?

In a given question double bar graph of wheat & Rice of different states

Now we can prepare double bar graph to frequency table

States U.P W.B M.P Maharashtra Haryana Total

Rice $$8$$ $$10$$ $$5$$ $$4$$ $$3$$ $$30$$

Wheat $$16$$ $$2$$ $$5$$ $$2$$ $$6$$ $$31$$

Total $$24$$ $$12$$ $$10$$ $$6$$ $$9$$

Now,

Here observe that

W.B state is largest producer of rice

and U.P state is largest producer of wheat

Uttar pradesh state has total production of rice and wheat as it maximum

(Rice $$+$$ Wheat)$$=24$$

Maharashtra state has total production of rice and wheat as it minimum(Rice $$+$$ Wheat)$$=4+2=6$$.

Read the following bar graph and answer the following questions:
Which state has total production of rice and wheat as its maximum?

In a given question double bar graph of wheat & Rice of different states

Now, we can prepare double bar graph to frequency table

States U.P W.B M.P Maharastra Haryana Total

Rice $$8$$ $$10$$ $$5$$ $$4$$ $$3$$ $$30$$

Wheat $$16$$ $$2$$ $$5$$ $$2$$ $$6$$

$$31$$

Total $$24$$ $$12$$ $$10$$ $$6$$ $$8$$

Now,

Here observe that

W.B state is largets producer of rice

U.P state is largest producer of wheat

Uttar pradesh state has total production of rice and wheat as it maximum(Rice + Wheat)$$=24$$

Given below Fig. is the bar graph indicating the marks obtained out of $$50$$ in mathematics paper by $$100$$ students. Read the bar graph and answer the following question:
Every student belonging to the lowest mark-group has to solve $$5$$ problems per day. How many problems, in all, will be solved by the students of this group per day?

In a given question bar graph indicating the marks obtained out of $$50$$ in mathematics paper by $$100$$ students.

We can prepare frequency table by given bar graph.

Marks Number of students

$$0-9$$ $$27$$

$$10-19$$ $$12$$

$$20-29$$ $$20$$

$$30-39$$ $$24$$

$$40-49$$ $$17$$

Given

Every student belonging to the lowest mark-group has to solve $$5$$ problems per day

we have

No. of students belonging to the lowest mark group is $$27$$

$$\therefore$$ Total no. of problem solved by the students$$=27\times 5$$

$$=135$$.

Given above Fig. is the bar graph indicating the marks obtained out of $$50$$ in mathematics paper by $$100$$ students. Read the bar graph and answer the following question:
Every student belonging to the highest mark group is entitled to get a prize of Rs. $$10.$$ How much amount of money is required for distributing the prize money?

In a given question bar graph indicating the marks obtained out of $$50$$ in mathematics paper by $$100$$ students

We can prepare frequency table by given bar graph.

Marks Number of students

$$0-9$$ $$27$$

$$10-19$$ $$12$$

$$20-29$$ $$20$$

$$30-39$$ $$24$$

$$40-49$$ $$17$$

Every student belonging to higest mark group $$(40-49)$$ is

entitled to get a prize of Rs. $$10$$

$$\therefore$$ No. of students to belonging to highest mark group is $$17$$

$$\therefore$$ The amount of money is required for distributing the prize money

$$=17\times 10$$

$$=$$Rs. $$170$$.

The following bar graph (Fig.) represents the heights (in cm) of 50 students of Class XI of a particular school. Study the graph and answer the following questions:
What percentage of the total number of students have their heights more than 149 cm?

In a given question the bar graph represents the height(in cm) of $$50$$ students of class XI of a particular school

Now we can prepare frequency table from given bar graph

Heights(in cm) No. of students

$$140-144$$ $$7$$

$$145-149$$ $$12$$

$$150-154$$ $$16$$

$$155-159$$ $$10$$

$$160-164$$ $$5$$

$$165$$ -

Total$$=50$$

Here total no. of students$$=50$$

Here observe that total no. of students have their heights more than $$149$$cm is $$=16+10+5=31$$

$$\therefore$$ Percentage of the total no. of students have their heights more than $$149$$ cm is

$$=31/50\times 100=0.62\times 100$$

$$=62\%$$.

Sonal and Amole made a sequence of tile designs from square white tiles surrounding one square purple tile. The purple tiles come in many sizes. Three of the designs are shown below.
Copy and complete the table

Refer to the figure.

When side length of purple tile is $$1$$, number of white tiles on border is $$8$$

When side length of purple tile is $$2$$, number of white tiles on border is $$12$$

When side length of purple tile is $$3$$, number of white tiles on border is $$16$$

Thus, it is clear that the series is in Arithmetic Progression.

$$\therefore { a }_{ n }={ a }_{ 1 }+\left( n-1 \right) d$$

1) When $$n=4$$,

$${ a }_{ 4 }=8+\left( 4-1 \right) \times 4$$

$$\therefore { a }_{ 4 }=8+12=20$$

2) When $$n=5$$,

$${ a }_{ 5 }=8+\left( 5-1 \right) \times \times 4$$

$$\therefore { a }_{ 5 }=8+16=24$$

3) When $$n=10$$,

$${ a }_{ 10 }=8+\left( 10-1 \right) \times 4$$

$$\therefore { a }_{ 10 }=8+36=44$$

4) When $$n=100$$,

$${ a }_{ 100 }=8+\left( 100-1 \right) \times 4$$

$$\therefore { a }_{ 100 }=8+396=404$$

Side length of purple tile	Number of white tiles on the border
1	8
2	12
3	16
4	20
5	24
10	44
100	404

Sonal and Amole made a sequence of tile designs from square white tiles surrounding one square purple tile. The purple tiles come in many sizes. Three of the designs are shown below.
Do the points lie on line?

Let $$n$$ = side length of the purple tile

Let $$m$$ = number of white tiles on the border

Refer to the figure.

When $$n=1$$, $$m=8$$

When $$n=2$$, $$m=12$$

When $$n=3$$, $$m=16$$

Thus, it is clear that this series is in A.P. Thus, we can use formula for A.P. to find number of white tiles on border for given side length of purple tile.

$${ a }_{ n }={ a }_{ 1 }+\left( n-1 \right) d$$

1) When $$n=4$$,

$${ a }_{ 4 }=8+\left( 4-1 \right) \times 4$$

$$\therefore { a }_{ 4 }=8+12=20$$

2) When $$n=5$$,

$${ a }_{ 5 }=8+\left( 5-1 \right) \times 4$$

$$\therefore { a }_{ 5 }=24$$

3) When $$n=10$$,

$${ a }_{ 10 }=8+\left( 10-1 \right) \times 4$$

$$\therefore { a }_{ 10 }=44$$

4) When $$n=100$$

$${ a }_{ 100 }=8+\left( 100-1 \right) \times 4$$

$$\therefore { a }_{ 100 }=404$$

If we plot first five values graphically, we get a straight line as A.P. has linear relationship

Height in cm	$$135-140$$	$$140-145$$	$$145-150$$	$$150-155$$	$$155-160$$	$$160-165$$	$$165-170$$
No. of boys	$$4$$	$$8$$	$$20$$	$$14$$	$$7$$	$$6$$	$$1$$

$${32}^{o}$$	$${34}^{o}$$	$${36}^{o}$$	$${37}^{o}$$	$${38}^{o}$$
$${34}^{o}$$	$${31}^{o}$$	$${34}^{o}$$	$${37}^{o}$$	$${37}^{o}$$
$${37}^{o}$$	$${35}^{o}$$	$${32}^{o}$$	$${34}^{o}$$	$${38}^{o}$$
$${39}^{o}$$	$${31}^{o}$$	$${36}^{o}$$	$${36}^{o}$$	$${39}^{o}$$
$${34}^{o}$$	$${33}^{o}$$	$${35}^{o}$$	$${34}^{o}$$	$${33}^{o}$$
$${37}^{o}$$	$${33}^{o}$$	$${32}^{o}$$	$${34}^{o}$$	$${38}^{o}$$

Marks	No. of students
$$1-20$$	$$8$$
$$21-40$$	$$12$$
$$41-50$$	$$15$$
$$51-60$$	$$20$$
$$61-70$$	$$13$$
$$71-80$$	$$17$$
$$81-90$$	$$5$$

Sl. No	Marks	No of students
$$1$$	$$0 - 5$$	$$5$$
$$2$$	$$5 - 10$$	$$7$$
$$3$$	$$10 - 15$$	$$10$$
$$4$$	$$15 - 20$$	$$6$$
$$5$$	$$20 - 25$$	$$2$$

Modes of Transport	Number of Students
Bicycle	$$140$$
Bus	$$100$$
Walk	$$70$$
Train	$$40$$
Car	$$10$$

Years	Wheat	Barley	Oats
1991	34	18	27
1992	43	14	24
1993	43	16	27
1994	45	13	34

Product	Production (in million tonnes)
	2008	2009	2010
Rice	300	350	400
Wheat	200	275	300
Sugar	100	120	160

	Distance in km	Distance in km
From/ To	$$A$$	$$B$$
$$D$$	$$7$$	$$3$$
$$E$$	$$6$$	$$4$$
$$F$$	$$3$$	$$2$$

Stream	No of Students
	2009	2010	2011
Arts	50	70	75
Science	200	250	300
Commerce	100	120	130

Price (Rs.)	Quantity Demanded by Isla (Units)	Quantity Demanded by Ifraah (Units)	Quantity Demanded by Ila (Units)	Quantity Demanded by Ibema (Units)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$	$$16$$ $$11$$ $$7$$ $$4$$ $$2$$ $$1$$	$$7$$ $$6$$ $$5$$ $$4$$ $$3$$ $$2$$	$$15$$ $$12$$ $$9$$ $$6$$ $$3$$ $$0$$	$$8$$ $$6$$ $$4$$ $$2$$ $$0$$ $$0$$

Price (Rs.)	Demand Schedule of Mohan (Units)	Demand Schedule of Rohan (Units)	Demand Schedule of Mitu (Units)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$60$$ $$50$$ $$40$$ $$30$$ $$20$$	$$55$$ $$40$$ $$25$$ $$10$$ $$0$$	$$24$$ $$13$$ $$5$$ $$0$$ $$0$$

Price (Rs.)	Household $$A$$ (Units)	Household $$B$$ (Units)	Household $$C$$ (Units)	Market Demand (Units)
$$7$$ $$6$$ $$5$$ $$4$$ $$3$$	$$6$$ $$8$$ $$12$$ $$17$$ $$24$$	- - - - -	$$11$$ $$16$$ $$22$$ $$30$$ $$42$$	$$26$$ $$36$$ $$51$$ $$71$$ $$98$$

Employment of Variable Factor (Units)	Employment of Fixed Factor (Units)	Total Physical Product (Units)
$$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$	$$5$$ $$5$$ $$5$$ $$5$$ $$5$$ $$5$$ $$5$$ $$5$$	$$0$$ $$10$$ $$24$$ $$40$$ $$56$$ $$70$$ $$80$$ $$84$$

Output (Units)	Total Cost (Rs.)	Average Variable Cost (Rs.)	Marginal Cost (Rs.)	Average Fixed Cost (Rs.)
$$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$30$$ - $$68$$ $$84$$ - $$125$$	- - - $$18$$ - $$19$$	- $$20$$ - - $$18$$ -	- - - - - $$6$$

Price of Commodity - $$X (Rs.)$$	Demand of $$A$$ (Units)	Demand of $$B$$ (Units)
$$1$$ $$2$$ $$3$$ $$4$$ $$5$$	$$9$$ $$7$$ $$6$$ $$4$$ $$3$$	$$7$$ $$6$$ $$5$$ $$4$$ $$3$$

Daily income (in Rupees)	$$250-300$$	$$300-350$$	$$350-400$$	$$400-450$$	$$450-500$$
Number of workers	$$12$$	$$14$$	$$8$$	$$6$$	$$10$$

Profit (in lakhs in Rs.)	Number of shops (Frequency)
More than or equal to $$5$$	$$30$$
More than or equal to $$10$$	$$28$$
More than or equal to $$15$$	$$16$$
More than or equal to $$20$$	$$14$$
More than or equal to $$25$$	$$10$$
More than or equal to $$30$$	$$7$$
More than or equal to $$35$$	$$3$$

Height	No. of trees
Less than $$7$$	$$26$$
Less than $$14$$	$$57$$
Less than $$21$$	$$92$$
Less than $$28$$	$$134$$
Less than $$35$$	$$216$$
Less than $$42$$	$$287$$
Less than $$49$$	$$341$$
Less than $$56$$	$$360$$

Marks	0 or more	10 or more	20 or more	30 or more	40 or more	50 or more
Number of Students	63	58	55	51	48	42

Weight (in $$kg$$)	Number of students
Less than $$38$$	$$0$$
Less than $$40$$	$$3$$
Less than $$42$$	$$5$$
Less than $$44$$	$$9$$
Less than $$46$$	$$14$$
Less than $$48$$	$$28$$
Less than $$50$$	$$32$$
Less than $$52$$	$$35$$

Presentation Of Data - Class 11 Commerce Economics - Extra Questions

How much percentage is wheat more than cotton?

How many kilograms of sesame was bought?

How many percentage of expenditure is done for education?

Draw histogram and frequency polygon on the same graph paper for the following frequency distribution.ClassFrequency$$15-20$$$$20$$$$20-25$$$$30$$$$25-30$$$$50$$$$30-35$$$$40$$$$35-40$$$$25$$$$40-45$$$$10$$

The number of saplings planted by schools on World Tree Day is given in the table below. Draw a joint bar graph to show these figures.School names\Name of saplingsAlmondKaranjNeemAshokGulmoharNutan Vidyalaya4060721542Bharat Vidyalaya4238602540

A shop has the following number of shoe pairs of different sizes Size$$2:20$$ Size$$3:30$$ Size$$4:28$$ Size$$5:14$$ Size$$6:8$$Write this information in tabular form as done earlier and find the percentage of each shoe size available in the shop.

On the basis of information given below, calculate $$AR$$ and $$MR$$ and explain their mutual relationship.Units of a Good$$1$$$$2$$$$3$$$$4$$$$5$$$$6$$$$TR (Rs.)$$$$4$$$$7$$$$9$$$$10$$$$10$$$$9$$

Obtain crude death rates $$(C.D.R)$$ for city $$A$$ and city $$B$$ from the given below:

MarksNo. of students $$1-20$$$$8$$$$21-40$$$$12$$$$41-50$$$$15$$$$51-60$$$$20$$$$61-70$$$$13$$$$71-80$$$$17$$$$81-90$$$$5$$Number of students obtained marks between $$41$$ to $$80$$

Sanya, Aarushi, Manav and Kabir made greeting cards. Complete the table for their cards:

The minimum cost of each tablet is Rs. 10 and each capsule is Rs.If the cost of 8 tablets and 5 capsules is not less than Rs. 150, frame the inequations for the given data.

The bar graph as shown in above figure represents the heights (in cm) of 50 students of Class XI of a particular school. Study the graph and answer the following questions:What percentage of the total number of students have their heights more than 149 cm?

What do you mean by the term sorting of data ?

The ages of ten students of a group are given below. The ages have been recorded in $$\text{years - months:}$$$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$What is the lowest age?

The following are the favorite cartoon characters of $$15$$ children. Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey Mouse, Mickey mouse, Jerry, Jerry, Jerry, Doraemon, Doraemon, Doraemon.Arrange the names of the cartoon characters in a table using tally marks.

The graph shows the maximum temperatures recorded for two consecutive weeks of a town. Study the graph and answer the questions that follow.On which day was the temperature highest for the second week?

Data available in an unorganised form is called _______ data

The bar graph shown in Fig.16 represents the circulation of newspapers in 10 languages.study the bar graph and answer the following question:State the language other than english in which the number of newspaper published is between 2500 and 3500.

The table given below shows the temperatures recorded on a day at different times.Observe the table and answer the following questions.During which hour was there a constant temperature?

The bar graph drawn below shoes the number of tickets sold during a fair by $$6$$ students $$A, B, C, D, E$$ and $$F$$Using the Bar graph, answer the following question:What is the average number of tickets sold per student?

How is the data of a measurement recorded in tabular form? Describe in steps.

A pie chart is constructed by converting the share of each component into a percentage of 360 degrees Music preferences in 13 to 19 year (teenagers) olds are clearly shown in the figureFind the percentage occupied by the music which is least liked?

Plot the frequency curve for the following histogram.

Draw frequency curve for the following histogram.

The following is the frequency distribution with unknown frequenciesClass$$60-70$$$$70-80$$$$80-90$$$$90-100$$TotalFrequency$$\dfrac{a}{2}$$$$\dfrac{3a}{2}$$$$2a$$$$a$$$$50$$Find the value of $$a$$, hence find the frequencies. Draw a histogram and frequency polygon on the same coordinate system.

The following data give the number of students using different modes of transport:Modes of Transport Number of StudentsBicycle$$140$$Bus$$100$$Walk$$70$$Train$$40$$Car$$10$$From this table, find the central angle $$(\theta)$$ for the Mode of Transport 'Bus'

How many students scored full marks in the test?

Draw a subdivided bar graph to illustrate the given data YearsWheat Barley Oats 199134 18 27 199243 14 24 199343 16 27 199445 13 34

How many students scored $$3$$ or more than $$3$$ marks?

In all how many rupees were collected as fund?

What percentage of Asopalav tree were planted?

What percentage of Gulmohar sapling were planted? in % .

How many runs did sachin score? in %

Find the degree measuremnet of the runs scored by Parthiv? in degrees.

Distinguish between:Simple Bar Diagram and Multiple Bar Diagram

State whether True or False. Give Reasons.Histogram is a type of diagram.

Distinguish Between.Simple Bar Diagram and Multiple Bar Diagram.

Define or explain the concepts:Percentage Bar

Answer in detail:What is a Multiple Bar Diagram? Prepare a Multiple Bar Diagram from the following data.ProductProduction (in million tonnes)200820092010Rice300350400Wheat200275300Sugar100120160

Do you agree with the statement? Give reasonsThere is a only one type of diagram.

Answer the following:What are the general principles for the construction of a diagram?

Answer in detail:What is a Simple Bar Diagram?Construct a Simple bar diagram from the following dataYear20012002200320042005No of Workers4080120160240

Write Short Notes on the following:Multiple Bar Diagram

Answer in detail:What is a sub divided bar diagram? Draw a sub divided bar diagram from the following data?StreamNo of Students200920102011Arts507075Science200250300Commerce100120130

The random variable $$X$$ can take only the values $$0, 1, 2, 3$$. Given that $$P(X = 0) = P(X = 1) = p$$ and $$P(X = 2) = P(X =3)$$ such that $$\sum p_{i}x_{i}^{2} = 2\sum p_{i}x_{i}$$ find the value of $$p$$.

Complete the following table. Identify the phase of production which is relevant from the viewpoint of producer's equilibrium.Units of Labour$$TP$$$$AP$$$$MP$$$$1$$$$2$$$$3$$$$4$$$$5$$$$6$$$$7$$$$8$$$$4$$$$12$$$$24$$$$32$$$$36$$$$36$$$$28$$$$16$$----------------

Given the following information about a firm, identify the stages of production.Units of Labour$$1$$$$2$$$$3$$$$4$$$$5$$$6$$$$7$$$$8$$Units of output$$4$$$$12$$$$24$$$$32$$$$36$$$$36$$$$28$$$$16$$

Complete the following table:Output(Units)Total Cost(Rs.)Average Variable Cost(Rs.)Marginal Cost(Rs.)Average Fixed Cost(Rs.)$$0$$$$1$$$$2$$$$3$$$$4$$$$5$$$$30$$-$$68$$$$84$$-$$125$$---$$18$$-$$19$$-$$20$$--$$18$$------$$6$$

From the table given below, calculate total revenue, average revenue, and marginal revenue.Units SoldPrice (Rs.)Total Revenue (Rs.)Marginal Revenue (Rs.)$$5$$$$4$$$$3$$$$6$$$$7$$$$8$$------

Find out the average revenue and marginal revenue on the basis of the following data:Units of ProductionTotal Revenue (Rs.)Average Revenue (Rs.)Marginal Revenue (Rs.)$$1$$$$2$$$$3$$$$4$$$$5$$$$6$$$$7$$$$20$$$$36$$$$48$$$$56$$$$60$$$$60$$$$56$$--

Completer the following table:Output (Units)Average Fixed Cost (Rs.)Average Variable Cost (Rs.)Marginal Cost (Rs.)Total Cost (Rs.)$$1$$$$2$$$$3$$$$4$$$$5$$$$120$$$$60$$-$$30$$-$$40$$$$56$$$$54$$-----$$54$$--$$232$$---

Complete the following table:Output (Units)Price (Rs.)Total Revenue (Rs.)Marginal Revenue (Rs.)$$1$$$$2$$$$3$$$$4$$----$$6$$----$$2$$$$-2$$$$-2$$

Complete the following table:Output (Units)Total Cost (Rs.)Average Variable Cost (Rs.)Marginal Cost (Rs.)Average Fixed Cost (Rs.)$$0$$$$1$$$$2$$$$3$$$$4$$$$5$$$$30$$-$$78$$--$$150$$---$$23$$---$$25$$--$$23$$--$$30$$-$$10$$-$$6$$

Complete the following table:Output (Units)Average Revenue (Rs.)Marginal Revenue (Rs.)Total Revenue (Rs.)$$1$$$$2$$$$3$$$$4$$--$$11$$-$$15$$--$$3$$-$$26$$--

Complete the following table:Output (Units)Marginal Cost (Rs.)Average Variable Cost (Rs.)Total Cost (Rs.)Average Fixed Cost (Rs.)$$1$$$$2$$$$3$$$$4$$$$5$$$$60$$--$$54$$---$$54$$-$$57$$$$120$$$$174$$--$$345$$---$$15$$-

Complete the following table:Output (Units)Average Fixed Cost (Rs.)Marginal Cost (Rs.)Average Variable Cost (Rs.)Average Cost (Rs.)$$1$$$$2$$$$3$$$$4$$$$5$$$$60$$-$$20$$-$$12$$$$20$$--$$18$$--$$19$$$$18$$------$$31$$

Calculate marginal revenue using following information:Price (Rs.)$$10$$$$9$$Sale/ Demand (Units)$$100$$$$110$$

Complete the following table:Output (Units)Price (Rs.)Marginal Revenue (Rs.)Total Revenue (Rs.)$$1$$$$2$$$$3$$$$4$$--$$22$$-$$30$$--$$6$$-$$52$$--

Find producer's equilibrium, from the following. Give logical reason.Quantity Sold (Units)Total Revenue (Rs.)Total Cost (Rs.)$$5$$$$6$$$$7$$$$8$$$$9$$$$10$$$$15$$$$20$$$$25$$$$30$$$$35$$$$40$$$$18$$$$22$$$$26$$$$27$$$$32$$$$38$$

Find the profit maximising level of output.Quantity Sold (Units)Total Revenue (Rs.)Marginal Cost (Rs.)$$1$$$$2$$$$3$$$$4$$$$5$$$$12$$$$26$$$$34$$$$40$$$$42$$$$15$$$$9$$$$6$$$$2$$$$3$$

Find out the profit from the following:Quantity Sold (Units)Price (Rs. per unit)Average Cost (Rs.)$$1$$$$2$$$$3$$$$4$$$$5$$$$15$$$$16$$$$17$$$$18$$$$19$$$$15$$$$12$$$$10$$$$12$$$$14$$

The following table gives the height of trees:HeightNo. of treesLess than $$7$$$$26$$Less than $$14$$$$57$$Less than $$21$$$$92$$Less than $$28$$$$134$$Less than $$35$$$$216$$Less than $$42$$$$287$$Less than $$49$$$$341$$Less than $$56$$$$360$$Draw 'less than' ogive and 'more than' ogive.

Complete the following table:Output (Units)Price (Rs.)Total Revenue (Rs.)Marginal Revenue (Rs.)----$$10$$$$7$$$$5$$$$3$$----$$10$$$$4$$$$1$$$$-3$$

A perfectly competitive firm faces the market price of $$Rs. 20$$ per unit. Derive its $$TR$$ schedule for a range of output from $$1$$ to $$10$$ units. What will happens to market price if the firm decides to suspend its production for the time being?

Complete the following table:Units of OutputTotal Revenue (Rs.)Average Revenue (Rs.)Marginal Revenue (Rs.)$$1$$$$2$$$$3$$---$$10$$$$8$$$$6$$---

Complete the following table:Price (Rs.)Output (Units)Total Revenue (Rs.)Marginal Revenue (Rs.)$$7$$--$$1$$-$$2$$$$3$$-$$7$$$$10$$----$$-1$$$$-5$$

Consider the following distribution:Marks0 or more10 or more20 or more30 or more40 or more50 or moreNumber of Students635855514842(a) Calculate the frequency of class 30 - 40.(b) Calculate the class mark of the class 10 - 20.

Organise the following marks in a class assessment in a tabular form.$$4,6,7,5,6,5,4,5,2,6,2,5,1,9,6,5,8,4,6,7$$

The amount collected by $$15$$ students for flood relief find is given below. Answer the following. Amount Number of students$$0-10$$$$3$$$$10-20$$$$4$$$$20-30$$$$5$$$$30-40$$$$2$$$$40-50$$$$1$$(i)How many students collected less than $$Rs.\ 20$$?

The following is the favourite cartoon characters of $$15$$ children. Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey mouse, Mickey mouse, Jerry,, Jerry,, Jerry, Doraemon, Doraemon, Doraemon. Which cartoon character is liked least by the children

The following are the favorite cartoon characters of $$15$$ children. Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey Mouse, Mickey mouse, Jerry, Jerry, Jerry, Doraemon, Doraemon, Doraemon. Which cartoon character is liked by most of the children

The ages of ten students of a group are given below. The ages have been recorded in years and months:$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$What is the highest age?

The monthly pocket money of six friends is given below:Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45What is the highest pocket money?

The monthly pocket money of six friends is given below:Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45What is the lowest pocket money?

Read the bar graph given in Fig. 23.18 and answer the following question:What information does it give?

The production of oil (in lakh tonnes) in some of the refineries in India during 1982 was given below : Refinery :BarauniKoyaliMathuraMumbaiFloridaProduction of oil (in lakh tonnes )3070404525Construct a bar graph to represent the above data so that the bars are drawn horizontally.

The ages of ten students of a group are given below. The ages have been recorded in years and months:$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$Determine the range?

The monthly pocket money of six friends is given below:Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45What is the range?

The pie-chart given in Fig. shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) ofWheat

Draw histogram and frequency polygon on the same graph paper for the following frequency distribution.
Class Frequency
$$15-20$$ $$20$$
$$20-25$$ $$30$$
$$25-30$$ $$50$$
$$30-35$$ $$40$$
$$35-40$$ $$25$$
$$40-45$$ $$10$$

The number of saplings planted by schools on World Tree Day is given in the table below. Draw a joint bar graph to show these figures.
School names\Name of saplings
Almond Karanj Neem Ashok Gulmohar
Nutan Vidyalaya 40 60 72 15 42
Bharat Vidyalaya 42 38 60 25 40

A shop has the following number of shoe pairs of different sizes
Size$$2:20$$ Size$$3:30$$ Size$$4:28$$ Size$$5:14$$ Size$$6:8$$
Write this information in tabular form as done earlier and find the percentage of each shoe size available in the shop.

On the basis of information given below, calculate $$AR$$ and $$MR$$ and explain their mutual relationship.
Units of a Good $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$
$$TR (Rs.)$$ $$4$$ $$7$$ $$9$$ $$10$$ $$10$$ $$9$$

Marks No. of students
$$1-20$$ $$8$$
$$21-40$$ $$12$$
$$41-50$$ $$15$$
$$51-60$$ $$20$$
$$61-70$$ $$13$$
$$71-80$$ $$17$$
$$81-90$$ $$5$$
Number of students obtained marks between $$41$$ to $$80$$

The bar graph as shown in above figure represents the heights (in cm) of 50 students of Class XI of a particular school. Study the graph and answer the following questions:
What percentage of the total number of students have their heights more than 149 cm?

The ages of ten students of a group are given below. The ages have been recorded in
$$\text{years - months:}$$
$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$
What is the lowest age?

The following are the favorite cartoon characters of $$15$$ children.
Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey Mouse, Mickey mouse, Jerry, Jerry, Jerry, Doraemon, Doraemon, Doraemon.
Arrange the names of the cartoon characters in a table using tally marks.

The graph shows the maximum temperatures recorded for two consecutive weeks of a town. Study the graph and answer the questions that follow.
On which day was the temperature highest for the second week?

The bar graph shown in Fig.16 represents the circulation of newspapers in 10 languages.
study the bar graph and answer the following question:
State the language other than english in which the number of newspaper published is between 2500 and 3500.

The table given below shows the temperatures recorded on a day at different times.
Observe the table and answer the following questions.
During which hour was there a constant temperature?

The bar graph drawn below shoes the number of tickets sold during a fair by $$6$$ students $$A, B, C, D, E$$ and $$F$$
Using the Bar graph, answer the following question:
What is the average number of tickets sold per student?

A pie chart is constructed by converting the share of each component into a percentage of 360 degrees Music preferences in 13 to 19 year (teenagers) olds are clearly shown in the figure
Find the percentage occupied by the music which is least liked?

The following data give the number of students using different modes of transport:
Modes of Transport Number of Students
Bicycle $$140$$
Bus $$100$$
Walk $$70$$
Train $$40$$
Car $$10$$
From this table, find the central angle $$(\theta)$$ for the Mode of Transport 'Bus'

Draw a subdivided bar graph to illustrate the given data
Years Wheat Barley Oats
1991 34 18 27
1992 43 14 24
1993 43 16 27
1994 45 13 34

Distinguish between:
Simple Bar Diagram and Multiple Bar Diagram

State whether True or False. Give Reasons.
Histogram is a type of diagram.

Distinguish Between.
Simple Bar Diagram and Multiple Bar Diagram.

Define or explain the concepts:
Percentage Bar

Answer in detail:
What is a Multiple Bar Diagram?
Prepare a Multiple Bar Diagram from the following data.
Product Production (in million tonnes)

2008 2009 2010
Rice 300 350 400
Wheat 200 275 300
Sugar 100 120 160

Do you agree with the statement? Give reasons
There is a only one type of diagram.

Answer the following:
What are the general principles for the construction of a diagram?

Answer in detail:
What is a Simple Bar Diagram?
Construct a Simple bar diagram from the following data
Year 2001 2002 2003 2004 2005
No of Workers 40 80 120 160 240

Write Short Notes on the following:
Multiple Bar Diagram

Answer in detail:
What is a sub divided bar diagram?
Draw a sub divided bar diagram from the following data?
Stream No of Students
2009 2010 2011
Arts 50 70 75
Science 200 250 300
Commerce 100 120 130

Complete the following table. Identify the phase of production which is relevant from the viewpoint of producer's equilibrium.
Units of Labour $$TP$$ $$AP$$ $$MP$$
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$
$$6$$
$$7$$
$$8$$ $$4$$
$$12$$
$$24$$
$$32$$
$$36$$
$$36$$
$$28$$
$$16$$ -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

Given the following information about a firm, identify the stages of production.
Units of Labour $$1$$ $$2$$ $$3$$ $$4$$ $$5$ $$6$$ $$7$$ $$8$$
Units of output $$4$$ $$12$$ $$24$$ $$32$$ $$36$$ $$36$$ $$28$$ $$16$$

Complete the following table:
Output
(Units) Total Cost
(Rs.) Average Variable Cost
(Rs.) Marginal Cost
(Rs.) Average Fixed Cost
(Rs.)
$$0$$
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$30$$
-
$$68$$
$$84$$
-
$$125$$ -
-
-
$$18$$
-
$$19$$ -
$$20$$
-
-
$$18$$
- -
-
-
-
-
$$6$$

From the table given below, calculate total revenue, average revenue, and marginal revenue.
Units Sold Price (Rs.) Total Revenue (Rs.) Marginal Revenue (Rs.)
$$5$$
$$4$$
$$3$$ $$6$$
$$7$$
$$8$$ -
-
- -
-
-

Find out the average revenue and marginal revenue on the basis of the following data:
Units of Production Total Revenue (Rs.) Average Revenue (Rs.) Marginal Revenue (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$
$$6$$
$$7$$ $$20$$
$$36$$
$$48$$
$$56$$
$$60$$
$$60$$
$$56$$ - -

Completer the following table:
Output (Units) Average Fixed Cost (Rs.) Average Variable Cost (Rs.) Marginal Cost (Rs.) Total Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$120$$
$$60$$
-
$$30$$
- $$40$$
$$56$$
$$54$$
-
- -
-
-
$$54$$
- -
$$232$$
-
-
-

Complete the following table:
Output (Units) Price (Rs.) Total Revenue (Rs.) Marginal Revenue (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$ -
-
-
- $$6$$
-
-
- -
$$2$$
$$-2$$
$$-2$$

Complete the following table:
Output (Units) Total Cost (Rs.) Average Variable Cost (Rs.) Marginal Cost (Rs.) Average Fixed Cost (Rs.)
$$0$$
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$30$$
-
$$78$$
-
-
$$150$$ -
-
-
$$23$$
-
- -
$$25$$
-
-
$$23$$
- -
$$30$$
-
$$10$$
-
$$6$$

Complete the following table:
Output (Units) Average Revenue (Rs.) Marginal Revenue (Rs.) Total Revenue (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$ -
-
$$11$$
- $$15$$
-
-
$$3$$ -
$$26$$
-
-

Complete the following table:
Output (Units) Marginal Cost (Rs.) Average Variable Cost (Rs.) Total Cost (Rs.) Average Fixed Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$60$$
-
-
$$54$$
- -
-
$$54$$
-
$$57$$ $$120$$
$$174$$
-
-
$$345$$ -
-
-
$$15$$
-

Complete the following table:
Output (Units) Average Fixed Cost (Rs.) Marginal Cost (Rs.) Average Variable Cost (Rs.) Average Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$60$$
-
$$20$$
-
$$12$$ $$20$$
-
-
$$18$$
-
-
$$19$$
$$18$$
-
- -
-
-
-
$$31$$

Calculate marginal revenue using following information:
Price (Rs.) $$10$$ $$9$$
Sale/ Demand (Units) $$100$$ $$110$$

Complete the following table:
Output (Units) Price (Rs.) Marginal Revenue (Rs.) Total Revenue (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$ -
-
$$22$$
- $$30$$
-
-
$$6$$ -
$$52$$
-
-

Find producer's equilibrium, from the following. Give logical reason.
Quantity Sold (Units) Total Revenue (Rs.) Total Cost (Rs.)
$$5$$
$$6$$
$$7$$
$$8$$
$$9$$
$$10$$ $$15$$
$$20$$
$$25$$
$$30$$
$$35$$
$$40$$ $$18$$
$$22$$
$$26$$
$$27$$
$$32$$
$$38$$

Find the profit maximising level of output.
Quantity Sold (Units) Total Revenue (Rs.) Marginal Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$12$$
$$26$$
$$34$$
$$40$$
$$42$$ $$15$$
$$9$$
$$6$$
$$2$$
$$3$$

Find out the profit from the following:
Quantity Sold (Units) Price (Rs. per unit) Average Cost (Rs.)
$$1$$
$$2$$
$$3$$
$$4$$
$$5$$ $$15$$
$$16$$
$$17$$
$$18$$
$$19$$ $$15$$
$$12$$
$$10$$
$$12$$
$$14$$

Complete the following table:
Output (Units) Price (Rs.) Total Revenue (Rs.) Marginal Revenue (Rs.)
-
-
-
- $$10$$
$$7$$
$$5$$
$$3$$ -
-
-
- $$10$$
$$4$$
$$1$$
$$-3$$

Complete the following table:
Units of Output Total Revenue (Rs.) Average Revenue (Rs.) Marginal Revenue (Rs.)
$$1$$
$$2$$
$$3$$ -
-
- $$10$$
$$8$$
$$6$$ -
-
-

Complete the following table:
Price (Rs.) Output (Units) Total Revenue (Rs.) Marginal Revenue (Rs.)
$$7$$
-
-
$$1$$ -
$$2$$
$$3$$
- $$7$$
$$10$$
-
- -
-
$$-1$$
$$-5$$

Consider the following distribution:
Marks 0 or more 10 or more 20 or more 30 or more 40 or more 50 or more
Number of Students 63 58 55 51 48 42
(a) Calculate the frequency of class 30 - 40.
(b) Calculate the class mark of the class 10 - 20.

Organise the following marks in a class assessment in a tabular form.
$$4,6,7,5,6,5,4,5,2,6,2,5,1,9,6,5,8,4,6,7$$

The amount collected by $$15$$ students for flood relief find is given below. Answer the following.
Amount Number of students
$$0-10$$ $$3$$
$$10-20$$ $$4$$
$$20-30$$ $$5$$
$$30-40$$ $$2$$
$$40-50$$ $$1$$
(i)How many students collected less than $$Rs.\ 20$$?

The following is the favourite cartoon characters of $$15$$ children.
Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey mouse, Mickey mouse, Jerry,, Jerry,, Jerry, Doraemon, Doraemon, Doraemon.
Which cartoon character is liked least by the children

The following are the favorite cartoon characters of $$15$$ children.
Doraemon, Jerry, Mickey mouse, Jerry, Tom, Doraemon, Doraemon, Mickey Mouse, Mickey mouse, Jerry, Jerry, Jerry, Doraemon, Doraemon, Doraemon.
Which cartoon character is liked by most of the children

The ages of ten students of a group are given below. The ages have been recorded in years and months:
$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$
What is the highest age?

The monthly pocket money of six friends is given below:
Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45
What is the highest pocket money?

The monthly pocket money of six friends is given below:
Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45
What is the lowest pocket money?

Read the bar graph given in Fig. 23.18 and answer the following question:
What information does it give?

The production of oil (in lakh tonnes) in some of the refineries in India during 1982 was given below :
Refinery : Barauni Koyali Mathura Mumbai Florida
Production of oil
(in lakh tonnes ) 30 70 40 45 25
Construct a bar graph to represent the above data so that the bars are drawn horizontally.

The ages of ten students of a group are given below. The ages have been recorded in years and months:
$$8-6,9-0,8-4,9-3,7-8,8-11,8-7,9-2,7-10,8-8$$
Determine the range?

The monthly pocket money of six friends is given below:
Rs. 45, Rs. 30, Rs. 40, Rs. 50, Rs. 25, Rs. 45
What is the range?

The pie-chart given in Fig. shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) of
Wheat

The pie chart given in the figure shows the annual agricultural production of an Indian state. If the total production of all commodities is $$81000\ tonnes,$$ find the production (in tones) of
Sugar.

The pie-chart given in Fig.25.18 shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) of
Gram

Locate the points:
$$(1,3), (2,3), (3,3), (4,3)$$

The pie-chart given in Fig.25.18 shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) of
Maize

The following pie-chart shows the number of students admitted in different faculties of a college. If $$1000$$ students are admitted in Science answer the following:
What is the total number of students?

The pie-chart given in Fig.25.18 shows the annual agricultural production of an Indian state. If the total production of the all commodities in $$81000\ tonnes,$$ find the production (in tonnes) of
Rice

The following pie-chart shows the monthly expenditure of Shikha on various items. If she spends $$Rs\ 16000$$ per month, answer the following question:
How much she spend on rent ? (in Rs)

Locate the points:
$$(1,4), (2,4), (3,4), (4,4)$$

Read the following bar graph and answer the following questions:
In which year the export is minimum?

Read the bar graph shown in fig. and answer the following question:
Of which state, were the maximum number of tickets sold?

Read the following bar graph and answer the following questions:
In which year the import is maximum?

The number of stools in five rooms of a school are given below, Draw a pictograph of stools in the five rooms
Room number $$I$$ $$II$$ $$III$$ $$IV$$ $$V$$
Number of stools $$30$$ $$40$$ $$60$$ $$50$$ $$20$$

The number of students who were absent from the class during a week is given below, Draw the pictograph of the number of students who were absent from the class during the week.
Day Monday Tuesday Wednesday Thursday Friday
No. of absentees $$6$$ $$2$$ $$4$$ $$2$$ $$8$$

In a class test, the number of students passed in various subjects are given below. Draw the pictograph of number of students passed in various subjects.
Subject English Mathematics Hindi Drawing
Number of students passes $$15$$ $$25$$ $$10$$ $$20$$

Define the term:
Tabulation of data

No. of children $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$
No. of families $$5$$ $$11$$ $$25$$ $$12$$ $$5$$ $$2$$
Which values is discussed here?

Fill in the blanks:
Arranging the data in the form of a table is called ................ of data.

The cost of a notebook is Rs $$10$$. Draw a graph after making a table showing the cost of 2, 3, 4 ... notebooks. Use it to find
(a) the cost of $$7$$ notebooks.
(b) the number of notebooks that can be purchased with Rs$$50$$.

Complete the given tables and draw a graph for each.
Second table:

The following table gives the number of pages written by Sarika for completing her own book for 30 days :
Number of pages written per day 16-18 19-21 22-24 25-27 28-30
Number of days 1 3 4 9 13

Draw frequency polygons for each of the following frequency distribution;
(a). Using histogram
(b). Without using histogram
(i)
$$C.I$$ $$10-30$$ $$30-50$$ $$50-70$$ $$70-90$$ $$90-110$$ $$110-130$$ $$130-150$$
$$f$$ $$4$$ $$7$$ $$5$$ $$9$$ $$5$$ $$6$$ $$4$$

Construct a combined histogram and frequency polygon for the following frequency distribution:
Class-intervals $$10-20$$ $$20-30$$ $$30-40$$ $$40-50$$ $$50-60$$
Frequency $$3$$ $$5$$ $$6$$ $$4$$ $$2$$

The ages of $$40$$ students are given in the following table:
Age (in yrs) 12 13 14 15 16 17 18
Frequency 2 4 6 9 8 7 4

Items	Quantity	Price for $$1$$ kilogram (in rupees)	Total price
Tomato	$$2$$ kg	$$20\dfrac{1}{2}$$	$$41$$ rupees
Potato	$$3\dfrac{1}{2}$$ kg	$$25$$	-
Peas	$$5$$ kg	$$30\dfrac{1}{2}$$	-
Dal	$$2\dfrac{1}{4}$$ kg	$$32$$	-

Amount	Number of students
$$0-10$$	$$3$$
$$10-20$$	$$4$$
$$20-30$$	$$5$$
$$30-40$$	$$2$$
$$40-50$$	$$1$$

Production yield/hect	$$40-45$$	$$45-50$$	$$50-55$$	$$55-60$$	$$60-65$$	$$65-70$$
No. of farms	$$4$$	$$6$$	$$16$$	$$20$$	$$30$$	$$24$$

Refinery :	Barauni	Koyali	Mathura	Mumbai	Florida
Production of oil (in lakh tonnes )	30	70	40	45	25

Room number	$$I$$	$$II$$	$$III$$	$$IV$$	$$V$$
Number of stools	$$30$$	$$40$$	$$60$$	$$50$$	$$20$$

Day	Monday	Tuesday	Wednesday	Thursday	Friday
No. of absentees	$$6$$	$$2$$	$$4$$	$$2$$	$$8$$

Subject	English	Mathematics	Hindi	Drawing
Number of students passes	$$15$$	$$25$$	$$10$$	$$20$$

No. of children	$$0$$	$$1$$	$$2$$	$$3$$	$$4$$	$$5$$
No. of families	$$5$$	$$11$$	$$25$$	$$12$$	$$5$$	$$2$$

Age (in years)	$$5-15$$	$$15-25$$	$$25-35$$	$$35-45$$	$$45-55$$	$$55-65$$
Number of patients	$$6$$	$$11$$	$$21$$	$$23$$	$$14$$	$$5$$

Number of pages written per day	16-18	19-21	22-24	25-27	28-30
Number of days	1	3	4	9	13