Which of the following fractions is the largest in value: 5 divided by 113, 7 divided by 120, 13 divided by 145 or 17 divided by 160?

Introduction / Context:
Comparing fractions with different numerators and denominators is a classic aptitude skill. Instead of converting every fraction to decimal form with a calculator, we generally look for efficient comparison techniques such as cross multiplication or approximation. This question asks for the largest fraction among four given values.

Given Data / Assumptions:
- The fractions to be compared are 5 / 113, 7 / 120, 13 / 145 and 17 / 160. - All denominators are positive, so larger numerator to denominator ratios mean larger fractions. - We want the single fraction with the greatest value.

Concept / Approach:
- Direct decimal conversion is possible but time consuming. Instead, we can compare fractions pairwise using cross multiplication. - To compare a / b and c / d, compute a * d and c * b. The fraction with the larger product has the larger value. - Apply this method systematically to find the largest fraction among the four.

Step-by-Step Solution:
Step 1: First compare 5 / 113 and 7 / 120. Cross multiply: 5 * 120 = 600, and 7 * 113 = 791. Since 791 is greater than 600, 7 / 120 is greater than 5 / 113. Step 2: Now compare 7 / 120 and 13 / 145. Compute 7 * 145 = 1015 and 13 * 120 = 1560. Since 1560 is greater than 1015, 13 / 145 is greater than 7 / 120. Step 3: So far, 13 / 145 is larger than both 7 / 120 and 5 / 113. Step 4: Now compare 13 / 145 and 17 / 160. Compute 13 * 160 = 2080 and 17 * 145 = 2465. Since 2465 is greater than 2080, 17 / 160 is greater than 13 / 145. Step 5: Therefore, among the fractions 5 / 113, 7 / 120, 13 / 145 and 17 / 160, the largest value is 17 / 160.

Verification / Alternative check:
For additional confirmation, we can estimate decimal values. 5 / 113 is slightly larger than 5 / 115 which is about 0.043, so it is near 0.044. 7 / 120 is about 0.058. 13 / 145 is near 0.09 and 17 / 160 is about 0.10625. These approximations confirm that 17 / 160 is the largest fraction.

Why Other Options Are Wrong:
Option A (5 / 113): This is the smallest fraction because the numerator is small and the denominator is relatively large. Option B (7 / 120): This is larger than 5 / 113 but still significantly smaller than 13 / 145 and 17 / 160. Option C (13 / 145): This is fairly large but cross multiplication shows that 17 / 160 is larger. Option E (None of these): One of the listed fractions, namely 17 / 160, already satisfies the condition of being the largest, so None of these is not correct.

Common Pitfalls:
- Trying to work with approximate decimals by rough mental division without a systematic method, which can lead to mistakes. - Assuming that the fraction with the largest numerator or smallest denominator must automatically be the largest without checking the ratio. - Mixing up cross multiplication and comparing the wrong products.

Final Answer:
The largest fraction among the given options is 17/160.