Among the fractions 2/3, 3/4, 4/5, and 5/6, what is the difference between the largest and the smallest fraction?

Introduction / Context:
This question tests your ability to compare fractions and compute the difference between them. Such comparison skills are widely used in ratio problems, percentage growth, and many quantitative aptitude topics.

Given Data / Assumptions:

• Fractions given: 2/3, 3/4, 4/5, and 5/6.
• We must find the largest fraction and the smallest fraction.
• Then we compute the difference: largest − smallest.

Concept / Approach:
To compare fractions, we can convert them to a common denominator, convert to decimals, or reason by how close they are to 1. All four fractions are less than 1 but greater than 1/2. We can use the idea that for fractions with the same numerator, a smaller denominator gives a larger fraction, and for fractions close to 1, the one with the smallest difference from 1 will be the largest.

Step-by-Step Solution:
Step 1: Convert each fraction to decimal or compare directly. Step 2: 2/3 is approximately 0.6667. Step 3: 3/4 equals 0.75. Step 4: 4/5 equals 0.8. Step 5: 5/6 is approximately 0.8333. Step 6: From these values, the smallest fraction is 2/3 and the largest fraction is 5/6. Step 7: Compute the difference: (5/6) − (2/3). First rewrite 2/3 with denominator 6: 2/3 = 4/6. Step 8: Now 5/6 − 4/6 = 1/6.

Verification / Alternative check:
We can check the difference numerically using decimals. 5/6 is about 0.8333 and 2/3 is about 0.6667. The difference is roughly 0.1666, which is the decimal value of 1/6. This confirms our fraction calculation. Also, checking with a common denominator of 60 would give 2/3 = 40/60 and 5/6 = 50/60, and again the difference is 10/60 = 1/6.

Why Other Options Are Wrong:
1/12 is approximately 0.0833, too small compared to the computed difference.
1/20 equals 0.05, also much smaller than the actual difference.
1/30 is about 0.0333, which is far below the true gap between 2/3 and 5/6.

Common Pitfalls:
One common mistake is to assume that 2/3 is larger than 3/4 because the numerator is larger, which is incorrect. Another error is to subtract numerators and denominators directly, which is not valid for fraction subtraction. Always convert to a common denominator or to decimals when comparing or subtracting fractions.

Final Answer:
The difference between the largest and smallest fractions is 1/6.