To solve these problems, we’ll use the concept of proportional relationships. The work done by painters is proportional to the number of houses they can paint in a given time, so we set up proportions based on the given values.

Let's solve each blank:

Problem 1

Given:

• Painters: $8$
• Houses: $17$
• Weeks: $22$

This is our base ratio, which we can use to solve the other scenarios. We calculate the rate as:

$\text{Rate of Work} = \frac{\text{Houses}}{\text{Painters} \times \text{Weeks}} = \frac{17}{8 \times 22} = \frac{17}{176} \approx 0.0966$

Problem 2

Given:

• Painters: $25$
• Houses: (unknown)
• Weeks: $42$

Let $H$ represent the number of houses. Using the same rate:

$\frac{H}{25 \times 42} = 0.0966$

Solving for $H$:

$H = 0.0966 \times (25 \times 42) = 0.0966 \times 1050 = 101.43$

Rounding to the nearest 0.01:

$H \approx 101.43$

Answer for Problem 2: $H = 101.43$

Problem 3

Given:

• Painters: (unknown)
• Houses: $100$
• Weeks: $26$

Let $P$ represent the number of painters. Using the same rate:

$\frac{100}{P \times 26} = 0.0966$

Solving for $P$:

$P = \frac{100}{0.0966 \times 26} = \frac{100}{2.5116} \approx 39.81$

Rounding to the nearest 0.01:

$P \approx 39.81$

Answer for Problem 3: $P = 39.81$

Problem 4

Given:

• Painters: $50$
• Houses: $250$
• Weeks: (unknown)

Let $W$ represent the number of weeks. Using the same rate:

$\frac{250}{50 \times W} = 0.0966$

Solving for $W$:

$W = \frac{250}{0.0966 \times 50} = \frac{250}{4.83} \approx 51.76$

Rounding to the nearest 0.01:

$W \approx 51.76$

Answer for Problem 4: $W = 51.76$

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Summary of Answers:

1. Problem 2: $H = 101.43$
2. Problem 3: $P = 39.81$
3. Problem 4: $W = 51.76$

Would you like a detailed explanation of any step or have further questions?

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Related Questions:

1. How would the answer change if the number of painters doubled in each problem?
2. What would happen to the rate if the number of weeks was halved in each scenario?
3. How would you set up the proportion if the number of houses was unknown in all cases?
4. What would the answers be if each value was increased by 10%?
5. How could this method be applied to a real-world project timeline estimation?

Tip:

When working with proportions, cross-multiplying can simplify solving for unknowns in direct proportional relationships.