Question

If three times the larger of the two numbers is divided by the smaller one, we get 4 as quotient and 3 as the remainder. Also, if seven times the smaller number is divided by the larger one, we get 5 as quotient and 1 as remainder. Find the numbers.

Answer 1

- Hint: In this type of question we have to find out two numbers. so, assume that the larger number is $x $and $y$. Also assume $x>y. $Once we assume two variables, we have to form two linear equations as per the given condition of question. The question is related to the division of one number by another which gives quotient and remainder. when a number $X$ is divided by another number $Y$ such that quotient is $q$ and remainder is $r$ we can write $X= q Y + r$. The first statement can be written as
$3x=4x+3$. Similarly write down the second statement as per the question and solve them in order to find the solution.

Complete step-by-step solution -
As per question if three times the larger of two numbers is divided by smaller one, the quotient is four and remainder is three.
Let the larger number is $x$ and smaller number is $y$
So, we can write
$3x=4y+3------(a)$
Now from question, the second statement says that
If seven times of the smaller number is divided by the larger number, the quotient is five and remainder is one. So, we can form another equation as
$7y=5x+1------(b)$
Now we have to solve the equation $(a)$and $(b)$.
Equation $(a)$ can be rearranged as
$x=\dfrac{4y+3}{3}-------(c)$
Substituting the value of $x$ in equation $(b)$we can write
$\begin{align}
  & 7y=5\left( \dfrac{4y+3}{3} \right)+1 \\
 & 7y=\dfrac{5(4y+3)+3}{3} \\
\end{align}$
On cross multiplication we can write
$\begin{align}
  & 7y(3)=20y+15+3 \\
 & 21y-20y=18 \\
 & y=18 \\
\end{align}$
Now we put the value of $y=18$in $(c)$so that we can find the value of $x$
$\begin{align}
  & x=\dfrac{4(18)+3}{3} \\
 & \Rightarrow x=\dfrac{75}{3} \\
 & \Rightarrow x=25 \\
\end{align}$
Hence the larger number is 25 and the smaller number is 18.
Note: When we get the solution, we must check the solution by putting the values so obtained in the equation
It should be noted that the linear equation in two variables is a straight line when we plot the graph in XY plane. As we get two straight lines, they cut each other at a unique point. This point corresponding to X axis gives the value of x, and that corresponding to Y axis gives the value of y.
if $ax+by+c=0$ and $Ax+By+C=0$cuts each other at a unique point if $\dfrac{a}{A}\ne \dfrac{b}{B}$