Amongst the following fractions, the largest and second largest fractions, respectively are
$\frac{5}{6}, \frac{3}{4}, \frac{1}{2}, \frac{3}{5}$

Question

Accepted Answer

To determine the largest and second largest fractions among the given options, we need to compare the fractions: $\frac{5}{6}$, $\frac{3}{4}$, $\frac{1}{2}$, and $\frac{3}{5}$.

1. **Convert to a common denominator**: To compare these fractions accurately, we can convert them to have a common denominator. The least common multiple of the denominators 6, 4, 2, and 5 is 60.

- $\frac{5}{6} = \frac{5 \times 10}{6 \times 10} = \frac{50}{60}$
   - $\frac{3}{4} = \frac{3 \times 15}{4 \times 15} = \frac{45}{60}$
   - $\frac{1}{2} = \frac{1 \times 30}{2 \times 30} = \frac{30}{60}$
   - $\frac{3}{5} = \frac{3 \times 12}{5 \times 12} = \frac{36}{60}$

2. **Compare the fractions**: Now that all fractions have the same denominator, we can compare their numerators directly:

- $\frac{50}{60}$ (from $\frac{5}{6}$)
   - $\frac{45}{60}$ (from $\frac{3}{4}$)
   - $\frac{36}{60}$ (from $\frac{3}{5}$)
   - $\frac{30}{60}$ (from $\frac{1}{2}$)

From the numerators, it is clear that $\frac{50}{60}$ is the largest, and $\frac{45}{60}$ is the second largest.

3. **Conclusion**: Therefore, the largest fraction is $\frac{5}{6}$ and the second largest is $\frac{3}{4}$.

**Explanation of Options:**

- **Option 1: $\frac{5}{6}$ and $\frac{3}{4}$** is correct because, as shown above, $\frac{5}{6}$ is the largest fraction and $\frac{3}{4}$ is the second largest.

- **Option 2: $\frac{5}{6}$ and $\frac{3}{5}$** is incorrect because, although $\frac{5}{6}$ is the largest, $\frac{3}{5}$ is not the second largest; $\frac{3}{4}$ is larger than $\frac{3}{5}$.

- **Option 3: $\frac{3}{4}$ and $\frac{3}{5}$** is incorrect because $\frac{3}{4}$ is not the largest fraction; $\frac{5}{6}$ is larger than $\frac{3}{4}$.

- **Option 4: $\frac{3}{4}$ and $\frac{1}{2}$** is incorrect because neither $\frac{3}{4}$ nor $\frac{1}{2}$ is the largest fraction; $\frac{5}{6}$ is larger than both.

Thus, Option 1 is the correct answer, as it accurately identifies the largest and second largest fractions among the given options.