Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the…

Concept: A ratio a : b fixes only the relative size of two quantities, never their actual values, so the quantities themselves can always be written as ax and bx for one common multiplier x. Multiplying both terms of a ratio by the same non-zero number leaves the ratio unchanged, but adding or subtracting the same amount from both terms does change it, so an altered ratio supplies a genuinely new equation. Any statement of the form p : q = r : s is a proportion, and cross-multiplication turns it into a linear equation.

Application: let the two numbers be 5x and 6x, so that their ratio is 5 : 6 for every value of x.

1. Subtract 8 from each number: the new pair is 5x - 8 and 6x - 8.

2. The new ratio is given as 4 : 5, so (5x - 8) : (6x - 8) = 4 : 5.

3. Cross-multiply the proportion: 5(5x - 8) = 4(6x - 8).

4. Expand both sides: 25x - 40 = 24x - 32.

5. Collect the x terms: 25x - 24x = 40 - 32, so x = 8.

6. Substitute the multiplier back: the numbers are 5 × 8 = 40 and 6 × 8 = 48.

Cross-check: 40 : 48 reduces to 5 : 6, which matches the original ratio, and subtracting 8 from each gives 32 : 40, which reduces to 4 : 5, matching the changed ratio. Both conditions hold together.

Contrast: the first condition on its own does not pin the numbers down, because 35 and 42 are also in the ratio 5 : 6; subtracting 8 from each of those gives 27 : 34, not 4 : 5. It is the second condition that fixes the multiplier at x = 8.

Hence the two numbers are 40 and 48.