Question: (7n+8)(2m-5)
(b) Express $\frac{5}{7}m - \frac{1}{3}(5 - 9n)$ as a single fraction in its simplest form.
1. **State the problem:**
We need to expand the expression $(7n+8)(2m-5)$ and then simplify the expression $\frac{5}{7}m - \frac{1}{3}(5 - 9n)$ as a single fraction.
---
2. **Expand $(7n+8)(2m-5)$ using distributive property:**
$$
(7n+8)(2m-5) = 7n \times 2m + 7n \times (-5) + 8 \times 2m + 8 \times (-5)
$$
$$
= 14nm - 35n + 16m - 40
$$
---
3. **Simplify $\frac{5}{7}m - \frac{1}{3}(5 - 9n)$:**
First, distribute $-\frac{1}{3}$ inside the parentheses:
$$
\frac{5}{7}m - \left(\frac{1}{3} \times 5 - \frac{1}{3} \times 9n\right) = \frac{5}{7}m - \frac{5}{3} + 3n
$$
Rewrite the expression:
$$
\frac{5}{7}m + 3n - \frac{5}{3}
$$
---
4. **Express as a single fraction:**
Find the least common denominator (LCD) for the terms. The denominators are 7 and 3, so LCD is 21.
Rewrite each term with denominator 21:
$$
\frac{5}{7}m = \frac{5 \times 3}{7 \times 3}m = \frac{15}{21}m
$$
$$
3n = \frac{3 \times 21}{21}n = \frac{63}{21}n
$$
$$
-\frac{5}{3} = -\frac{5 \times 7}{3 \times 7} = -\frac{35}{21}
$$
Combine all terms:
$$
\frac{15}{21}m + \frac{63}{21}n - \frac{35}{21} = \frac{15m + 63n - 35}{21}
$$
---
5. **Simplify numerator if possible:**
Factor out 7:
$$
\frac{7(3m + 9n - 5)}{21} = \frac{\cancel{7}(3m + 9n - 5)}{\cancel{7}3} = \frac{3m + 9n - 5}{3}
$$
---
**Final answers:**
- Expanded form of $(7n+8)(2m-5)$ is:
$$
14nm - 35n + 16m - 40
$$
- Simplified single fraction form of $\frac{5}{7}m - \frac{1}{3}(5 - 9n)$ is:
$$
\frac{3m + 9n - 5}{3}
$$