Which of the following fractions has the greatest value: 7/12, 9/16, 13/20 or 32/80?

Introduction:
This question asks you to compare several fractions and determine which one is the largest. Comparing fractions is a fundamental skill in arithmetic and is often required in data interpretation and percentage problems.

Given Data / Assumptions:

• Fractions to compare: 7/12, 9/16, 13/20, and 32/80.
• We must identify the fraction with the greatest numerical value.
• All fractions are positive.

Concept / Approach:
There are two main approaches: convert each fraction to a decimal or convert them to a common denominator. Because the denominators differ, converting to decimals is often quicker for multiple choice questions. Then we simply compare the decimal values to see which is the largest.

Step-by-Step Solution:
Step 1: Convert 7/12 to a decimal: 7 ÷ 12 ≈ 0.5833.Step 2: Convert 9/16 to a decimal: 9 ÷ 16 = 0.5625.Step 3: Convert 13/20 to a decimal: 13 ÷ 20 = 0.65.Step 4: Convert 32/80 to a decimal: first simplify 32/80 by dividing numerator and denominator by 16 to get 2/5; 2/5 = 0.4.Step 5: Now list the decimals: 7/12 ≈ 0.5833, 9/16 = 0.5625, 13/20 = 0.65, 32/80 = 0.4.Step 6: Compare: 0.65 is larger than 0.5833, 0.5625, and 0.4.Step 7: Therefore, 13/20 is the largest fraction.

Verification / Alternative check:
We can also compare by cross multiplication. For example, compare 13/20 and 7/12 by cross multiplying: 13 * 12 = 156 and 7 * 20 = 140. Since 156 > 140, 13/20 > 7/12. Similarly, compare 13/20 with 9/16: 13 * 16 = 208 and 9 * 20 = 180, so 13/20 > 9/16. Since 32/80 is only 2/5, it is clearly smaller than 13/20 which is 0.65. This confirms that 13/20 is the greatest.

Why Other Options Are Wrong:
7/12 and 9/16 are both less than 2/3 and have decimals below 0.6. 32/80 simplifies to 2/5, which is 0.4, much smaller than 0.65. The distractor 17/30 is about 0.5667 and is not even among the original fractions to compare. Only 13/20 has the highest decimal value among the listed fractions.

Common Pitfalls:
Students sometimes compare only numerators or only denominators without considering both together. Another mistake is miscalculating decimals or simplifying incorrectly, especially with 32/80. Always either simplify first or compute accurate decimals to compare fractions reliably.

Final Answer:
Among the given options, the largest fraction is 13/20.