Question

A brother asks his sister for $Rs100$ so that he can be $3$times as rich as her sister, but the sister tells her brother to give her $Rs400$, so that she can be $7$times as rich as his brother. How much money do the brother and sister have now?

Answer 1

Hint: First, let the money do the brother and sister have and make the linear equation by using the given condition in the question. Solve the linear equation by any method substitution or elimination.

Complete step by step answer:
Let the brother have $Rs.x$ and sister have $Rs.y$.
We have to evaluate the value of $x$ and $y$.
To evaluate these values, we have two conditions given in the question. We will discuss one by one.
Condition-$1$
Brother asks his sister for $Rs100$ so that he can be $3$times as rich as her sister. If the sister gives him $Rs100$, the money left with her is $y - 100$and now brother has the money $x + 100$ and he is now $3$times as rich as her sister.
Write the condition in the mathematical form and solve.
$
  x + 100 = 3(y - 100) \\
   \Rightarrow x + 100 = 3y - 300 \\
   \Rightarrow x = 3y - 300.........(1) \\
 $
Condition-$2$
Sister tells her brother to give her $Rs400$, so that she can be $7$times as rich as her brother. If the brother gives her $Rs400$, the money left with him is $x - 400$and now sister have the money $y + 400$ and she is now $7$times as rich as his brother.
Write the condition in the mathematical form and solve.
$
  y + 400 = 7(x - 400) \\
   \Rightarrow y + 400 = 7x - 2800 \\
   \Rightarrow y = 7x - 3200........(2) \\
 $
Substitute the value of $x$ from equation $(1)$ to equation $(2)$.
$
  \therefore y = 7(3y - 400) - 3200 \\
   \Rightarrow y = 21y - 2800 - 3200 \\
   \Rightarrow 20y = 6000 \\
   \Rightarrow y = 300 \\
 $
Substitute the value of $y$ in equation $(1)$
$
  \therefore x = 3(300) - 400 \\
   \Rightarrow x = 900 - 400 \\
   \Rightarrow x = 500 \\
 $
Therefore, brother and sister have $Rs.500$ and $Rs.300$ respectively.
Note: We can also solve the linear equation by elimination method.
The linear equation in the condition-$1$ is,
$
  x = 3y - 400 \\
  x - 3y = - 400....(3) \\
 $
 The linear equation in the condition-$2$ is,
$
  y + 400 = 7x - 2800 \\
  7x - y = 3200.....(4) \\
 $
Multiply equation $(4)$ by $3$ and subtract from equation $(3)$.
$
  \therefore x - 3y - 3(7x - y) = - 400 - 3(3200) \\
   \Rightarrow x - 21x = - 400 - 9600 \\
   \Rightarrow 20x = 10000 \\
   \Rightarrow x = 500 \\
 $
Similarly, we can find the value of $y$by eliminating $x$ from both the equations.