Compare simple fractions: Which of the following is the greatest fraction among 7/8, 6/7, 4/5, and 5/6?

Introduction / Context:
To compare fractions with different denominators quickly, one can convert them to decimals, use cross-multiplication, or place them over a common denominator. Here the values are close to 1, so careful comparison is needed.

Given Data / Assumptions:

• Fractions: 7/8, 6/7, 4/5, 5/6.
• All are positive proper fractions less than 1.

Concept / Approach:
Use either decimal approximations or cross-multiplication pairwise. Since all denominators are small, decimals are fast and reliable to three decimal places.

Step-by-Step Solution:

Verification / Alternative check:
Cross-multiplication: compare 7/8 vs 6/7 ⇒ 7*7 = 49 vs 6*8 = 48 ⇒ 7/8 larger; similarly 7/8 vs 5/6 ⇒ 7*6 = 42 vs 5*8 = 40 ⇒ 7/8 larger; and 7/8 vs 4/5 ⇒ 7*5 = 35 vs 4*8 = 32 ⇒ 7/8 larger.

Why Other Options Are Wrong:
Each has a smaller decimal value than 0.875, so they cannot be the greatest among the four.

Common Pitfalls:
Rounding too aggressively (e.g., 0.86 vs 0.88) can reverse close comparisons. Use exact fractions or enough decimal places.

Final Answer:
7/8