Question:

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

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Equal products mean the numbers are inversely proportional to the percentages, so x : y = 8.5 : 6.5 = 17 : 13. Split 2490 into 30 parts. Also, only one option even adds to 2490.
Updated On: Jul 17, 2026
  • 1411 and 1079
  • 1412 and 1080
  • 1141 and 1709
  • 1214 and 1800
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The Correct Option is A

Solution and Explanation

Step 1: Name the numbers and write both conditions.
Let the two numbers be \( x \) and \( y \).
The sum condition gives \[ x + y = 2490 \]
The percentage condition gives \[ 6.5\% \text{ of } x = 8.5\% \text{ of } y \]
\[ \frac{6.5}{100}x = \frac{8.5}{100}y \]

Step 2: Clear the decimals and percentages.
Multiply both sides by 100 to remove the percent signs:
\[ 6.5x = 8.5y \]
Multiply by 2 to clear the halves:
\[ 13x = 17y \]
So \[ \frac{x}{y} = \frac{17}{13} \]
The smaller percentage sits on the larger number, which makes sense since the two products must be equal.

Step 3: Use the ratio with the sum.
The numbers are in the ratio 17 to 13, so write \( x = 17k \) and \( y = 13k \).
\[ 17k + 13k = 2490 \]
\[ 30k = 2490 \]
\[ k = 83 \]

Step 4: Get both numbers.
\[ x = 17 \times 83 = 1411 \]
\[ y = 13 \times 83 = 1079 \]

Step 5: Check both conditions.
Sum: \( 1411 + 1079 = 2490 \). Correct.
Percentages: \( 6.5\% \) of 1411 is \( 0.065 \times 1411 = 91.715 \), and \( 8.5\% \) of 1079 is \( 0.085 \times 1079 = 91.715 \). They are equal.

Step 6: Why the other options are wrong.
1412 and 1080 add to 2492, not 2490.
1141 and 1709 do add to 2850, which already fails the sum.
1214 and 1800 add to 3014, also wrong.
Only option (A) satisfies the sum, and it also satisfies the percentage test.

Final Answer:
The two numbers are 1411 and 1079. \[ \boxed{1411 \text{ and } 1079} \]
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