Question

The sum of two numbers is \[2490\]. If \[6.5\% \] of one number is equal to \[8.5\% \]of the other then the numbers are
 \[\begin{array}{*{20}{l}}
{A{\rm{ }}1411{\rm{ }} and{\rm{ }}1079}\\
{B{\rm{ }}1412{\rm{ }} and{\rm{ }}1080}\\
{C{\rm{ }}1141{\rm{ }} and{\rm{ }}1709}\\
{D{\rm{ }}1214{\rm{ }} and{\rm{ }}1800}
\end{array}\]

Answer 1

Hint:
Here we assign the variables to unknown numbers and form the equations. Then solve them to get the required answer by applying the concept of percentages and ratios.

Complete step by step solution:
Let the two numbers be $x$ and $y$ respectively.
Sum of the two numbers is $2490$ and $6.5\% $ of one number is equal to $8.5\% $ of the other. $6.5\% $ of $x$ = \dfrac{{6.5}}{{100}} \times x$
$8.5\% $ of $y$ = \dfrac{{8.5}}{{100}} \times y$
According to the question, $6.5\% $ of $x$ $8.5\% $ of $y$
$ \Rightarrow $ $\dfrac{{6.5}}{{100}} \times x$ $ = $ $\dfrac{{8.5}}{{100}} \times y$
$ \Rightarrow \dfrac{x}{y} = \dfrac{{17}}{{13}}$
From the above relation it can be inferred that $x = 17a$ and $y = 13a$
According to the question, $x + y = 2490$
$\begin{array}{l}
 \Rightarrow 17a + 13a = 2490\\
 \Rightarrow 30a = 2490\\
 \Rightarrow a = 83
\end{array}$
It implies that,
  $\begin{array}{l}
 \Rightarrow x = 17 \times a = 17 \times 83\\
\therefore x = 1411\\
 \Rightarrow y = 13 \times a = 13 \times 83\\
\therefore y = 1079
\end{array}$
Hence, Option A is the correct answer.

Note:
In such types of questions which involve relations between the terms and a value is assigned to an expression relating the terms. Assume variables to unknown terms and forms equations as per the given relation. Then solve the equations to get the values of the unknown terms.