The sum of the largest and smallest fractions among $\frac{6}{7}, 2\frac{3}{5}, \frac{3}{8}$ and $2\frac{8}{15}$ is:

Question

Accepted Answer

To determine the sum of the largest and smallest fractions among the given options, we need to first convert all the mixed numbers into improper fractions and then compare them.

The fractions given are:
1. $\frac{6}{7}$
2. $2\frac{3}{5}$
3. $\frac{3}{8}$
4. $2\frac{8}{15}$

Let's convert the mixed numbers into improper fractions:

- $2\frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}$
- $2\frac{8}{15} = \frac{2 \times 15 + 8}{15} = \frac{30 + 8}{15} = \frac{38}{15}$

Now, we have the fractions:
1. $\frac{6}{7}$
2. $\frac{13}{5}$
3. $\frac{3}{8}$
4. $\frac{38}{15}$

To compare these fractions, we need to find a common denominator or convert them to decimal form:

- $\frac{6}{7} \approx 0.857$
- $\frac{13}{5} = 2.6$
- $\frac{3}{8} = 0.375$
- $\frac{38}{15} \approx 2.533$

From these values, we can see that the smallest fraction is $\frac{3}{8}$ and the largest fraction is $\frac{13}{5}$.

Now, we calculate the sum of the largest and smallest fractions:

$$
\frac{13}{5} + \frac{3}{8}
$$

To add these fractions, we need a common denominator. The least common multiple of 5 and 8 is 40.

Convert each fraction to have a denominator of 40:

- $\frac{13}{5} = \frac{13 \times 8}{5 \times 8} = \frac{104}{40}$
- $\frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40}$

Now add the fractions:

$$
\frac{104}{40} + \frac{15}{40} = \frac{119}{40}
$$

Convert $\frac{119}{40}$ to a mixed number:

$$
\frac{119}{40} = 2\frac{39}{40}
$$

Thus, the sum of the largest and smallest fractions is $2\frac{39}{40}$, which corresponds to Option 4.

**Explanation for why other options are incorrect:**

1. **Option 1: 2$\frac{9}{35}$** - This option is incorrect because the sum $\frac{119}{40}$ does not simplify to 2$\frac{9}{35}$.

2. **Option 2: 2$\frac{37}{40}$** - This option is incorrect because the sum $\frac{119}{40}$ is not equal to 2$\frac{37}{40}$.

3. **Option 3: 2$\frac{7}{35}$** - This option is incorrect because the sum $\frac{119}{40}$ does not simplify to 2$\frac{7}{35}$.

Therefore, the correct answer is indeed Option 4: 2$\frac{39}{40}$.