If the numerator of a fraction is increased by 200% and the denominator is…

Concept: A percentage increase of x% multiplies a quantity by a factor of (1 + x/100). So a 200% increase scales a number by 3, and a 250% increase scales a number by 3.5. When both the numerator and denominator of a fraction are scaled this way, the new fraction becomes (3 × numerator) divided by (3.5 × denominator).

1. Let the original fraction be P/Q.

2. A 200% increase on the numerator gives 3P, since 1 + 200/100 = 3.

3. A 250% increase on the denominator gives 3.5Q, since 1 + 250/100 = 3.5.

4. The resultant fraction is 3P / 3.5Q, which is given to equal 3/14.

5. So, 3P / 3.5Q = 3/14.

6. Cross-multiplying: 3P × 14 = 3 × 3.5Q, that is, 42P = 10.5Q.

7. Dividing both sides by 42Q: P/Q = 10.5/42 = 1/4.

Cross-check: Check: take P = 1, Q = 4 (so P/Q = 1/4). A 200% increase on 1 gives 3, and a 250% increase on 4 gives 14. The new fraction is 3/14, exactly matching the given resultant fraction, confirming the answer.

Therefore, the original fraction is 1/4.