Which of the following fractions is the largest?

Concept: To compare fractions with different denominators, first express each fraction over a common denominator — the Least Common Multiple (LCM) of all the denominators. Once every fraction shares the same denominator, the fraction with the greatest numerator is the largest, since the denominators no longer differ. Equivalently, converting each fraction to its decimal value allows a direct comparison.

1. Find the LCM of the denominators 8, 16, 40, and 80. Since 80 is a multiple of 8, 16, and 40, the LCM is 80.

2. Convert every fraction to an equivalent fraction with denominator 80: 7/8 = 70/80, 13/16 = 65/80, 31/40 = 62/80, and 63/80 remains 63/80.

3. Compare the four numerators over this common denominator: 70, 65, 62, and 63. The greatest numerator is 70, so 7/8 is the largest fraction.

Cross-check: As an independent check, convert each fraction to its decimal value: 7/8 = 0.875, 13/16 = 0.8125, 31/40 = 0.775, and 63/80 = 0.7875. The highest decimal value again confirms that 7/8 is the largest fraction.