Problem Analysis:
We are given the following information:
- Andy can complete a job alone in
40 minutes.
- Brenda can complete the same job alone in
50 minutes.
- We need to determine how long it would take for Andy and Brenda to complete the job if they work together.
Step-by-Step Solution:
#### Step 1: Determine the work rates of Andy and Brenda
The work rate is defined as the fraction of the job completed per minute.
- Andy's work rate:
Since Andy completes the job in 40 minutes, his work rate is:
$$
\text{Andy's work rate} = \frac{1}{40} \text{ (job per minute)}.
$$
- Brenda's work rate:
Since Brenda completes the job in 50 minutes, her work rate is:
$$
\text{Brenda's work rate} = \frac{1}{50} \text{ (job per minute)}.
$$
#### Step 2: Combine their work rates
When Andy and Brenda work together, their combined work rate is the sum of their individual work rates:
$$
\text{Combined work rate} = \text{Andy's work rate} + \text{Brenda's work rate}.
$$
Substituting the values:
$$
\text{Combined work rate} = \frac{1}{40} + \frac{1}{50}.
$$
To add these fractions, we need a common denominator. The least common multiple (LCM) of 40 and 50 is 200. Rewrite the fractions with a denominator of 200:
$$
\frac{1}{40} = \frac{5}{200}, \quad \frac{1}{50} = \frac{4}{200}.
$$
Now add the fractions:
$$
\frac{1}{40} + \frac{1}{50} = \frac{5}{200} + \frac{4}{200} = \frac{9}{200}.
$$
Thus, the combined work rate is:
$$
\text{Combined work rate} = \frac{9}{200} \text{ (job per minute)}.
$$
#### Step 3: Determine the time taken to complete the job together
The time taken to complete the job is the reciprocal of the combined work rate. If the combined work rate is $\frac{9}{200}$ jobs per minute, then the time $t$ (in minutes) to complete 1 job is:
$$
t = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{9}{200}} = \frac{200}{9}.
$$
#### Step 4: Simplify the result
The value $\frac{200}{9}$ is approximately:
$$
\frac{200}{9} \approx 22.22 \text{ minutes}.
$$
However, we will keep the exact fractional form for precision.
Final Answer:
The time it takes for Andy and Brenda to complete the job together is:
$$
\boxed{\frac{200}{9}}
$$ minutes.
Parent Tip: Review the logic above to help your child master the concept of work problem.
