A painting job can be completed by Team A alone in 2.5 hours or by Team B alone in 75 minutes. On one occasion, after Team A had completed a fraction  of the job, Team B took over immediately. The whole painting job was completed in 1.5 hours. If m and n are positive integers with no common factors, find the value of m + n.

6

7

5

4

Let's break down the problem step by step.

### Step 1: Determine Work Rates for Team A and Team B

- **Team A** can complete the entire job in 2.5 hours. Therefore, their work rate is:
  \[
  \text{Rate of Team A} = \frac{1 \text{ job}}{2.5 \text{ hours}} = \frac{1}{2.5} = \frac{2}{5} \text{ jobs per hour.}
  \]

- **Team B** can complete the entire job in 75 minutes (which is 1.25 hours). Therefore, their work rate is:
  \[
  \text{Rate of Team B} = \frac{1 \text{ job}}{1.25 \text{ hours}} = \frac{1}{1.25} = \frac{4}{5} \text{ jobs per hour.}
  \]

### Step 2: Analyze the Situation

Let \( x \) be the fraction of the job that Team A completes before Team B takes over. Then Team A works for a time \( t_1 \) and Team B works for a time \( t_2 \), such that:
- \( t_1 \times \text{Rate of Team A} = x \)
- \( t_2 \times \text{Rate of Team B} = 1 - x \)
- \( t_1 + t_2 = 1.5 \text{ hours} \)

### Step 3: Express \( t_1 \) and \( t_2 \) in Terms of \( x \)

- From the equation \( t_1 \times \frac{2}{5} = x \), we get:
  \[
  t_1 = \frac{5x}{2}
  \]

- From the equation \( t_2 \times \frac{4}{5} = 1 - x \), we get:
  \[
  t_2 = \frac{5(1 - x)}{4}
  \]

### Step 4: Use the Total Time Equation

Substituting \( t_1 \) and \( t_2 \) into the equation \( t_1 + t_2 = 1.5 \):
\[
\frac{5x}{2} + \frac{5(1 - x)}{4} = 1.5
\]

### Step 5: Solve for \( x \)

First, find a common denominator to simplify:
\[
\frac{10x}{4} + \frac{5(1 - x)}{4} = \frac{3}{2}
\]
\[
\frac{10x + 5(1 - x)}{4} = \frac{3}{2}
\]
\[
\frac{10x + 5 - 5x}{4} = \frac{3}{2}
\]
\[
\frac{5x + 5}{4} = \frac{3}{2}
\]

Multiply both sides by 4:
\[
5x + 5 = 6
\]
\[
5x = 1
\]
\[
x = \frac{1}{5}
\]

### Step 6: Identify the Fraction \( \frac{m}{n} \) and Compute \( m + n \)

Here, \( x = \frac{1}{5} \), which implies \( m = 1 \) and \( n = 5 \). Therefore:
\[
m + n = 1 + 5 = 6
\]

So, the correct answer is **6**.

Would you like further details or have any other questions? Here are some related questions to consider:

1. How do you calculate the work done by two teams working at different rates?
2. What is the importance of finding a common denominator in these types of problems?
3. How can you use proportions to solve problems involving time and work?
4. Can this problem be solved using the concept of weighted averages?
5. How does the concept of rate relate to the time taken to complete a task?

**Tip**: When dealing with work problems, always remember to express the rates as jobs per unit time, and ensure that time units are consistent across all calculations.

Explore how to solve time and work problems using the rates of Team A and Team B. Understand how fractional work and different work rates contribute to completing tasks efficiently. Suitable for high school students learning about rates and time in mathematics.

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