Let’s break down and solve the two problems visible in the image.

Problem 26:

Beth is making a rope ladder. Each step of the ladder is $2 \frac{1}{3}$ feet wide. Beth has rope that is 24 feet long. How many steps can she make from the rope?

Solution:

• Convert $2 \frac{1}{3}$ feet into an improper fraction: $2 \frac{1}{3} = \frac{7}{3} \text{ feet}$

• Beth has 24 feet of rope. To find out how many steps she can make, divide the total length of rope by the length per step: $\frac{24}{\frac{7}{3}} = 24 \times \frac{3}{7} = \frac{72}{7} \approx 10.29$ Beth can make approximately 10 full steps from the rope.

Problem 27:

The area of a rectangle is $257 \frac{1}{4}$ square inches. One side is $10 \frac{1}{2}$ inches. Find the other side $w$.

Solution:

• Convert both mixed numbers to improper fractions:

  • $257 \frac{1}{4} = \frac{1029}{4} \text{ square inches}$
  • $10 \frac{1}{2} = \frac{21}{2} \text{ inches}$

• Use the formula for the area of a rectangle: $\text{Area} = \text{length} \times \text{width}$ Solving for $w$ (the unknown side): $w = \frac{\text{Area}}{\text{length}} = \frac{\frac{1029}{4}}{\frac{21}{2}} = \frac{1029}{4} \times \frac{2}{21} = \frac{2058}{84} = \frac{343}{14} = 24 \frac{7}{14} = 24 \frac{1}{2} \text{ inches}$ Therefore, the other side $w$ is 24.5 inches.

Would you like further details on these solutions, or explanations for any specific step?

Related Questions:

1. How would the solution change if Beth had only 20 feet of rope instead of 24 feet?
2. What if the width of each step in the ladder were increased to 3 feet?
3. How would you calculate the number of steps if the rope length was not a whole number (e.g., 22.5 feet)?
4. How would you approach the rectangle problem if the area was given in terms of meters instead of inches?
5. What other geometric shapes could be involved in area-related problems similar to problem 27?

Tip:

When dividing by a fraction, remember you can multiply by the reciprocal of that fraction to make the calculation simpler.