Two numbers are in the ratio 8:5. If eleven is subtracted from each of them,…

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

1. Let the numbers be 8k and 5k because their ratio is 8:5.

2. After subtracting 11 from each number, write the condition (8k - 11)/(5k - 11) = 7/3.

3. Cross-multiply: 3(8k - 11) = 7(5k - 11).

4. Expand: 24k - 33 = 35k - 77.

5. Rearrange: 77 - 33 = 35k - 24k, so 44 = 11k and k = 4.

6. 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.