A is 30% more efficient than B. How much time will they, working

together, take to complete a job which A

alone could have done in 23 days?

Let's denote:
- \( T_A \) as the time taken by A to complete the job alone.
- \( T_B \) as the time taken by B to complete the job alone.

Given:
- \( T_A = 23 \) days.
- A is 30% more efficient than B.

Efficiency is inversely proportional to the time taken to complete a job. Therefore, if A is 30% more efficient than B, then the time taken by B to complete the job is:
\[
T_B = \frac{T_A}{1 + 0.30} = \frac{23}{1.30} = \frac{23}{1.3} \approx 17.69 \text{ days}
\]

Let's denote the work rates (amount of job completed per day) of A and B as \( W_A \) and \( W_B \) respectively. These are given by:
\[
W_A = \frac{1}{T_A} = \frac{1}{23}
\]
\[
W_B = \frac{1}{T_B} = \frac{1}{\frac{23}{1.3}} = \frac{1.3}{23}
\]

When A and B work together, their combined work rate is:
\[
W_A + W_B = \frac{1}{23} + \frac{1.3}{23} = \frac{1 + 1.3}{23} = \frac{2.3}{23}
\]

The time \( T_{AB} \) taken for A and B working together to complete the job is the reciprocal of their combined work rate:
\[
T_{AB} = \frac{1}{W_A + W_B} = \frac{1}{\frac{2.3}{23}} = \frac{23}{2.3} \approx 10 \text{ days}
\]

Therefore, A and B, working together, will take approximately 10 days to complete the job.

Would you like any details or have any questions about this solution?

Here are some related questions for further exploration:
1. If B alone starts the job and works for 5 days, how many days will A need to complete the remaining job?
2. How long will it take for A and B to complete two identical jobs working together?
3. If A works for 15 days and then B joins, how many more days are needed to complete the job?
4. How much of the job can A and B complete together in 4 days?
5. What is the combined work rate of A and B in terms of job per day?
6. How long would it take for three workers, each as efficient as B, to complete the job?
7. If B becomes 50% more efficient, how much time will they take to complete the job together?
8. If A takes a break for 3 days, how much time will B need to complete the remaining job alone?

**Tip:** Understanding the relationship between efficiency and time can help solve work rate problems more easily, especially when dealing with multiple workers.

Learn how to solve a time and efficiency problem where A is 30% more efficient than B. Understand the concept of work rates and how to calculate combined work rates when A and B work together. Suitable for Grades 7-9.

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