Question: Choose the best answer. If necessary, use the paper you were given.
Question
Carlos takes 20 minutes more to paint a rectangular wall measuring $25$ feet by $12$ feet than to paint a rectangular wall measuring $18$ feet by $10$ feet. If he paints the two walls at the same rate per square foot, how long does it take him to paint both walls?
1. **State the problem:** Carlos paints two rectangular walls at the same rate per square foot. The first wall measures $25$ feet by $12$ feet, and the second wall measures $18$ feet by $10$ feet. He takes $20$ minutes more to paint the first wall than the second. We need to find how long it takes him to paint both walls combined.
2. **Define variables and formulas:** Let $r$ be the rate of painting in minutes per square foot.
- Area of first wall: $$A_1 = 25 \times 12 = 300 \text{ square feet}$$
- Area of second wall: $$A_2 = 18 \times 10 = 180 \text{ square feet}$$
Time to paint first wall: $$t_1 = r \times 300$$
Time to paint second wall: $$t_2 = r \times 180$$
Given: $$t_1 = t_2 + 20$$
3. **Set up the equation:**
$$r \times 300 = r \times 180 + 20$$
4. **Solve for $r$:**
$$300r - 180r = 20$$
$$120r = 20$$
$$r = \frac{20}{120} = \frac{1}{6}$$
5. **Calculate time to paint both walls:**
Total area: $$A = 300 + 180 = 480$$
Total time: $$t = r \times A = \frac{1}{6} \times 480 = 80$$ minutes
6. **Answer:** It takes Carlos $80$ minutes to paint both walls.