When 4 is added to the numerator and 10 is subtracted from the denominator of…

Concept: When a fraction is given only as a ratio (here 8/11), its actual numerator and denominator are unknown -- treat them as 8k and 11k for some positive whole number k. A further condition that ties the numerator and denominator together (after a stated change, the fraction equals another known ratio) becomes one linear equation in k, solved by cross-multiplication.

Application:

1. Let the original fraction be 8k/11k, since it must reduce to 8/11 for some whole number k.

2. Adding 4 to the numerator and subtracting 10 from the denominator changes it to 3/4: (8k + 4)/(11k − 10) = 3/4.

3. Cross-multiplying: 4(8k + 4) = 3(11k − 10), which simplifies to 32k + 16 = 33k − 30.

4. Solving gives k = 46, so the original fraction is 368/506, and after the first change it is (368 + 4)/(506 − 10) = 372/496, which equals 3/4.

5. Now subtract 4 from the numerator and 10 from the denominator of this changed fraction: (372 − 4)/(496 − 10) = 368/486.

6. Dividing 368/486 by their greatest common factor (2) gives the new fraction, 184/243.

Cross-check: 368/506 divided by 46 gives 8/11, confirming the original ratio, and 372/496 divided by 124 gives 3/4, confirming the stated change. Both given conditions hold, so the new fraction after the second subtraction is 184/243.