The numerator of a fraction is increased by 150% and the denominator of the fraction is increased by 350%. The resulting fraction is 25/51. What was the original fraction?

Introduction / Context:
This question examines understanding of how fractions transform when both numerator and denominator are changed by different percentages. Instead of guessing, we must express the new fraction algebraically in terms of the original numerator and denominator. Such problems are common in aptitude tests because they connect percentages, fractions, and algebra in a single question. Mastering these ideas helps with ratio, proportion, and data interpretation tasks as well.

Given Data / Assumptions:

• The original fraction has numerator N and denominator D.
• The numerator is increased by 150%, so the new numerator becomes N plus 150% of N.
• The denominator is increased by 350%, so the new denominator becomes D plus 350% of D.
• The resulting fraction after these changes is equal to 25/51.
• We are asked to find the original fraction N/D.

Concept / Approach:
Percent increases are best handled by converting them into multiplicative factors. Increasing a quantity by 150% means the final value is 250% of the original, or 2.5 times the original. Similarly, increasing a quantity by 350% makes it 450% of the original, or 4.5 times the original. Once we express the new numerator and denominator in terms of N and D, we equate the resulting fraction to 25/51, simplify, and solve for the ratio N/D. This is an algebraic approach based on proportions and is more reliable than trial and error.

Step-by-Step Solution:
Step 1: Let the original fraction be N/D.Step 2: Increasing the numerator by 150% makes the new numerator N + 1.5N = 2.5N.Step 3: Increasing the denominator by 350% makes the new denominator D + 3.5D = 4.5D.Step 4: The new fraction is therefore (2.5N) / (4.5D).Step 5: According to the question, (2.5N) / (4.5D) = 25 / 51.Step 6: Multiply numerator and denominator by 2 to remove decimals: (5N) / (9D) = 25 / 51.Step 7: Cross-multiply to get 5N * 51 = 25 * 9D.Step 8: This simplifies to 255N = 225D.Step 9: Divide both sides by 15 to get 17N = 15D.Step 10: Therefore N / D = 15 / 17, which is the original fraction.

Verification / Alternative check:
To verify, start from the original fraction 15/17. Increase the numerator 15 by 150%. That makes the new numerator 15 + 1.5 * 15 = 15 + 22.5 = 37.5. Increase the denominator 17 by 350%. That makes the new denominator 17 + 3.5 * 17 = 17 + 59.5 = 76.5. The new fraction is 37.5 / 76.5. Multiply numerator and denominator by 2 to clear halves, giving 75 / 153. Dividing numerator and denominator by 3, we get 25 / 51, which matches the fraction given in the question. This confirms that the original fraction is indeed 15/17.

Why Other Options Are Wrong:
Option 31/25 would not yield 25/51 when the described percentage changes are applied; the resulting fraction would be very different. Option 14/25, 11/16, and 9/11 each give transformed fractions that do not simplify to 25/51 when the numerator is multiplied by 2.5 and the denominator by 4.5. Only 15/17 gives the exact required resulting fraction. Therefore, the other options are mathematically inconsistent with the given condition.

Common Pitfalls:
Many learners misinterpret "increased by 150%" as becoming 150% of the original, instead of 250% of the original, leading to incorrect factors. Another common error is to add percentage values for numerator and denominator separately without converting to proper multipliers. Some also try to guess the original fraction from the options without doing the algebra, which can be risky when options are close in value. Always convert percentage changes into multiplication factors and solve the proportional equation carefully.

Final Answer:
The original fraction is 15/17.