To solve the expression \( \frac{4}{5} \div \frac{5}{6} \cdot \frac{7}{8} + \frac{9}{10} \cdot \frac{5}{12} \), we will follow the order of operations, dealing with division and multiplication first, and then addition.

### Step 1: Perform the division
\[
\frac{4}{5} \div \frac{5}{6} = \frac{4}{5} \times \frac{6}{5} = \frac{4 \times 6}{5 \times 5} = \frac{24}{25}
\]

### Step 2: Perform the first multiplication
\[
\frac{24}{25} \cdot \frac{7}{8} = \frac{24 \times 7}{25 \times 8} = \frac{168}{200} = \frac{42}{50} = \frac{21}{25}
\]

### Step 3: Perform the second multiplication
\[
\frac{9}{10} \cdot \frac{5}{12} = \frac{9 \times 5}{10 \times 12} = \frac{45}{120} = \frac{3}{8}
\]

### Step 4: Add the two fractions
\[
\frac{21}{25} + \frac{3}{8}
\]

To add these fractions, we need a common denominator. The least common denominator (LCD) of 25 and 8 is 200. Convert each fraction:

\[
\frac{21}{25} = \frac{21 \times 8}{25 \times 8} = \frac{168}{200}
\]
\[
\frac{3}{8} = \frac{3 \times 25}{8 \times 25} = \frac{75}{200}
\]

Now add the fractions:
\[
\frac{168}{200} + \frac{75}{200} = \frac{243}{200}
\]

So, the final answer is:
\[
\frac{243}{200} = 1 \frac{43}{200}
\]

Would you like more details on any part of this solution? Here are some related questions:

1. How do you simplify fractions after performing operations?
2. Why is finding a common denominator important when adding fractions?
3. What is the difference between dividing and multiplying fractions?
4. How do you convert an improper fraction to a mixed number?
5. Can the result \( \frac{243}{200} \) be simplified further?

**Tip:** Always check if fractions can be simplified before performing operations—it can make calculations easier!

Learn how to solve the expression 4/5 ÷ 5/6 ∙ 7/8 + 9/10 ∙ 5/12 step-by-step. This problem involves fractions, division, multiplication, and addition of fractions. Suitable for students in Grades 6-8.

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