Step-by-step solution for a fraction addition and division problem in a math app.

Let's carefully go through the problem shown in the image and verify the solution step by step.

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Problem:

$$
2\frac{2}{3} + 3\frac{1}{5} \div 4\frac{4}{7}
$$

This is a mixed number expression involving addition and division. We follow the order of operations (PEMDAS), so we do the division first, then the addition.

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Step 1: Convert all mixed numbers to improper fractions

- $ 2\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} $
- $ 3\frac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{15 + 1}{5} = \frac{16}{5} $
- $ 4\frac{4}{7} = \frac{4 \times 7 + 4}{7} = \frac{28 + 4}{7} = \frac{32}{7} $

So the expression becomes:

$$
\frac{8}{3} + \frac{16}{5} \div \frac{32}{7}
$$

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Step 2: Perform the division

Division of fractions: multiply by the reciprocal.

$$
\frac{16}{5} \div \frac{32}{7} = \frac{16}{5} \times \frac{7}{32}
$$

Now simplify before multiplying:

- $ \frac{16}{32} = \frac{1}{2} $, so:
- $ \frac{16 \times 7}{5 \times 32} = \frac{1 \times 7}{5 \times 2} = \frac{7}{10} $

✔ So the division part simplifies to $ \frac{7}{10} $

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Step 3: Now add

$$
\frac{8}{3} + \frac{7}{10}
$$

We need a common denominator. The least common multiple (LCM) of 3 and 10 is 30.

Convert both fractions:

- $ \frac{8}{3} = \frac{8 \times 10}{3 \times 10} = \frac{80}{30} $
- $ \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30} $

Now add:

$$
\frac{80}{30} + \frac{21}{30} = \frac{101}{30}
$$

Convert to a mixed number:

$$
\frac{101}{30} = 3\frac{11}{30}
$$

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✔ Final Answer:

$$
\boxed{3\frac{11}{30}}
$$

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Reviewing the Image's Work:

The image shows:

$$
2\frac{2}{3} + 3\frac{1}{5} \div 4\frac{4}{7} = \frac{8}{3} + \frac{16}{5} \div \frac{32}{7}
$$

Then:

$$
= \frac{8}{3} + \frac{16}{5} \times \frac{7}{32} = \frac{8}{3} + \frac{112}{160}
$$

Wait — here’s a potential error!

- $ \frac{16}{5} \times \frac{7}{32} = \frac{112}{160} $ → this is correct numerically.
- But $ \frac{112}{160} $ can be simplified: divide numerator and denominator by 16 → $ \frac{7}{10} $

So they correctly write it as $ \frac{8}{3} + \frac{7}{10} $ later.

Then:

- LCM(3,10) = 30 → correct
- $ \frac{8 \times 10}{30} = \frac{80}{30} $, $ \frac{7 \times 3}{30} = \frac{21}{30} $
- $ \frac{80 + 21}{30} = \frac{101}{30} = 3\frac{11}{30} $

✔ All steps are correct.

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✔ Conclusion:

The solution shown in the image is correct.

Final Answer:
$$
\boxed{3\frac{11}{30}}
$$