Suppose there is a fraction whose numerator as well as denominator both are…

Concept: A percentage change acts as a multiplier. Increasing a quantity by 20% multiplies it by 120/100 = 6/5, and decreasing a quantity by 20% multiplies it by 80/100 = 4/5. When the numerator and the denominator of a fraction are scaled by different factors, the fraction as a whole is multiplied by the ratio of those two factors, so the two separate percentage changes collapse into one multiplier acting on the fraction.

Application: Let the required fraction be n/d, with n and d positive integers.

1. Increase the numerator by 20%: n becomes (6/5)n.

2. Decrease the denominator by 20%: d becomes (4/5)d.

3. The new fraction is ((6/5)n) / ((4/5)d) = (6/5) x (5/4) x (n/d) = (3/2) x (n/d), so the single multiplier on the fraction is 3/2.

4. The question states that this new value equals 6/9, which reduces to 2/3. Hence (3/2) x (n/d) = 2/3.

5. Multiply both sides by 2/3: n/d = (2/3) x (2/3) = 4/9.

Cross-check: take the fraction 4/9. Its numerator 4 increased by 20% is 4.8, and its denominator 9 decreased by 20% is 7.2. Then 4.8/7.2 = 48/72 = 2/3 = 6/9, exactly the value stated in the question, and both 4 and 9 are positive integers as required.

Contrast: the same transformation applied to every offered value gives:

Only the fraction 4/9 is carried to 6/9 by this change, so the required fraction is 4/9. A common slip is to multiply by 4/5 over 6/5 (that is, by 2/3 instead of 3/2), which reverses the roles of the numerator and the denominator; another is to forget that 6/9 reduces to 2/3 before the reverse step.