Two numbers are in the ratio 8:5. If eleven is subtracted from each of them,…
2023
Two numbers are in the ratio 8:5. If eleven is subtracted from each of them, the ratio becomes 7:3. What is the product of the numbers?
Answer: B. 640 — ConceptWhen two numbers are in the ratio a:b, represent them as ak and bk for a common scale factor k. A changed-ratio condition creates an equation in k.…
- A.
460
- B.
640
- C.
560
- D.
750
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Show answer & explanation
Correct answer: B
Concept
When two numbers are in the ratio a:b, represent them as ak and bk for a common scale factor k.
A changed-ratio condition creates an equation in k. Solve that equation first, then recover the original numbers and the requested quantity.
Application
Let the numbers be 8k and 5k because their ratio is 8:5.
After subtracting 11 from each number, write the condition (8k - 11)/(5k - 11) = 7/3.
Cross-multiply: 3(8k - 11) = 7(5k - 11).
Expand: 24k - 33 = 35k - 77.
Rearrange: 77 - 33 = 35k - 24k, so 44 = 11k and k = 4.
The original numbers are 8 × 4 = 32 and 5 × 4 = 20. Their product is 32 × 20 = 640.
Cross-check
Subtracting 11 gives 21 and 9, and 21:9 simplifies to 7:3, so the scale factor satisfies the changed ratio.
Therefore, the product of the two numbers is 640.