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The sum of two numbers is 2490. If 6.5% of one number is equal to 8.5% of the other, find the numbers.

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Hint: Assume the numbers to be some variables. Form simultaneous equations as per the information given in the question and then solve them.

According to the question, the sum of two numbers is 2490.
Let the two numbers be a and b. Then we have:
$ \Rightarrow a + b = 2490 .....(i)$
Next, it is given in the question that 6.5% of one number is equal to 8.5% of other. From this we’ll get:
$
   \Rightarrow \dfrac{{6.5}}{{100}} \times a = \dfrac{{8.5}}{{100}} \times b, \\
   \Rightarrow 65a = 85b, \\
   \Rightarrow 13a = 17b, \\
   \Rightarrow a = \dfrac{{17}}{{13}}b \\
$
Putting the value of a in equation $(i)$, we’ll get:
\[
   \Rightarrow \dfrac{{17}}{{13}}b + b = 2490, \\
   \Rightarrow \dfrac{{30}}{{13}}b = 2490, \\
   \Rightarrow b = \dfrac{{2490 \times 13}}{{30}}, \\
   \Rightarrow b = 1079 \\
\]
Putting the value of b in equation $(i)$, we’ll get:
$
   \Rightarrow a + 1079 = 2490, \\
   \Rightarrow a = 2490 - 1079, \\
   \Rightarrow a = 1411 \\
$
Thus the numbers are 1411 and 1079 respectively.

Note: Instead of assuming variables, we can directly solve the question using ratios. It is given that 6.5% of one number is equal to 8.5% of the other. So, we can conclude that the numbers are in the ratio 6.5:8.8 i.e. 65:85. This ratio can be written as 13:17.
So, sum of two numbers is 2490 and their ratio is 13:17. Thus, we have to divide 2490 in the ratio 13:17.
So, our numbers will be $\dfrac{{13}}{{13 + 17}} \times 2490$ and $\dfrac{{17}}{{13 + 17}} \times 2490$ .
If we solve them, we’ll get 1079 and 1411.
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