Question

The sum of two numbers is 2490. If $6.5\%$ of one number is equal to $8.5\%$ of the other. Find the numbers.

Answer 1

Hint: We first assume the numbers as variables. We got two unknowns. From the sum of two numbers we get one linear equation. Then we have the condition that $6.5\%$ of one number is equal to $8.5\%$ of the other. We put that in mathematical form to find the other linear equation. We solve those two equations to find the solution of the problem.

Complete step-by-step answer:
The sum of two numbers is 2490. Let’s assume the numbers are a and b. We can express this condition in the form of mathematical expressions.
So, $a+b=2490......(i)$.
It’s also given that $6.5\%$ of one number is equal to $8.5\%$ of the other one.
Let’s assume $6.5\%$ of a is equal to $8.5\%$ of b.
So, $6.5\%$ of a will be $a\times \dfrac{6.5}{100}=\dfrac{65a}{1000}$ and $8.5\%$ of b is $b\times \dfrac{8.5}{100}=\dfrac{85b}{1000}$.
The mathematical expression of this condition is $\dfrac{85b}{1000}=\dfrac{65a}{1000}$.
Simplifying the second equation we get
$\begin{align}
  & \dfrac{85b}{1000}=\dfrac{65a}{1000} \\
 & \Rightarrow 17b=13a........(ii) \\
\end{align}$
We have two linear equations to find out the value of two unknowns.
Now we take the value of from equation (ii) to put in the equation (i).
$\begin{align}
  & 17b=13a \\
 & \Rightarrow b=\dfrac{13a}{17} \\
\end{align}$
We put this value in $a+b=2490......(i)$.
$\begin{align}
  & a+b=2490 \\
 & \Rightarrow a+\dfrac{13a}{17}=2490 \\
\end{align}$
We solve it to find the value of a.
$\begin{align}
  & a+\dfrac{13a}{17}=2490 \\
 & \Rightarrow \dfrac{17a+13a}{17}=2490 \\
 & \Rightarrow \dfrac{30a}{17}=2490 \\
 & \Rightarrow a=\dfrac{2490\times 17}{30}=1411 \\
\end{align}$
So, one number is 1411. The other number is $2490-1411=1079$.
The numbers are 1411 and 1079.

Note: We can also work with one variable. We assume one number being a and then we can use the summation to find the other number. As the summation is 2490 and one number is a, that’s why the other number will be $\left( 2490-a \right)$. We use a and $\left( 2490-a \right)$ to form the second condition and solve that to find the solution.