Explanation
$${\textbf{Step -1: Writing mathematical equations according to conditions given in question}}$$
$${\text{Let the price of ticket for station A be Rs}}{\text{. x and for station B be Rs}}{\text{. y,}}$$
$${\text{According to question, }}$$
$${\text{2x + 3y = 77 }} \to {\text{ Equation 1}}$$
$${\text{3x + 5y = 124 }} \to {\text{ Equation 2}}$$
$${\textbf{Step -2: Calculating fares for station A and station B .}}$$
$${\text{Solving Equation 1 and Equation 2,}}$$
$${\text{Multiplying Equation 1 by 3,}}$$
$${\text{6x + 9y = 231 }} \to {\text{ Equation 3}}$$
$${\text{Multiplying Equation 2 by 2,}}$$
$${\text{6x + 10y = 248 }} \to {\text{ Equation 4}}$$
$${\text{Subtracting Equation 3 from Equation 4,}}$$
$$ \Rightarrow {\text{ y = 248 - 231}}$$
$$ \Rightarrow {\text{ y = 17}}$$
$${\text{Substituting value of y in Equation 1,}}$$
$$ \Rightarrow {\text{ 2x + 3(17) = 77}}$$
$$ \Rightarrow {\text{ 2x + 51 = 77}}$$
$$ \Rightarrow {\text{ 2x = 26}}$$
$$ \Rightarrow {\text{ x = 13}}$$
$${\textbf{Thus, the fare to station A is Rs}}{\textbf{. 13 and to station B is Rs}}{\textbf{.17 .}}$$
Suppose the cost of $$1$$ table $$=x$$ and cost of $$1$$ chair $$=y$$Then according to the question$$\Rightarrow 4x+3y=2250$$...........(1)$$\Rightarrow 3x+4y=1950$$.............(2)
Multiply (1) by $$4$$ and (2) by $$3$$ and Subtract both$$\Rightarrow (16x+12y=9000)- (9x+12y=5850)$$
$$\Rightarrow 7x=3150\Rightarrow x=450$$
Put $$x=450$$ in $$eq2$$
$$\Rightarrow 450\times4+3y=2250\Rightarrow 3y=2250-1800\Rightarrow 3y=450$$
$$\Rightarrow y=150$$
Cost of $$1$$ table $$=450$$Cost of $$1$$ chair $$=150$$$$\therefore$$ Cost of $$1$$ table and $$2$$ chair $$ =450+150\times2=750$$
Let no of $$20$$p coins $$=x$$No of $$25$$p coins $$=y$$According to the question$$ x+y=50$$$$\Rightarrow x=50-y.....eq1$$And $$20x+25y=1125\Rightarrow 4x+5y=225....eq2$$Put the value of $$x$$ from $$eq1$$$$\Rightarrow 4\left ( 50-y \right )+5y=225......eq3$$$$\Rightarrow 200-4y+5y=225\Rightarrow y=25$$Put $$y=25$$ in $$eq1$$$$\Rightarrow x+25=50\Rightarrow x=25 $$
No of $$20$$p coins $$=25$$
No of $$25$$p coins $$=25$$
Let the first number be $$x$$ and second number $$y.$$
According to question$$x+y=8\quad\quad\quad\dots(i)$$
$$4(x-y)=8$$
$$\Rightarrow x-y=2\quad\quad\quad\dots(ii)$$
Add equations $$(i)$$ and $$(ii),$$$$\begin{aligned}{}\left( {x + y} \right) + \left( {x - y} \right)& = 8 + 2\\2x &= 10\\x &= 5\end{aligned}$$
Substitute $$x=5$$ in $$(i),$$$$\begin{aligned}{}\left( 5 \right) + y &= 8\\y&=8-5\\y &= 3\end{aligned}$$
So, the numbers are $$5$$ and $$3.$$
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