A and B together can do a work in 12 days. B and C together can do the same work in 16 days. A started workinh alone and worked for 5 days, B replaced him and worked for 7 days ,C replaced B and worked for 13 days ,In how many days B can do the whole work alone?

Let the total work be denoted by \( W \), and the rate of work be represented in terms of work per day for each individual: \( A \), \( B \), and \( C \). 

We know:

1. \( A \) and \( B \) together can complete the work in 12 days, so their combined work rate is:
\[
\frac{1}{A} + \frac{1}{B} = \frac{1}{12}
\]

2. \( B \) and \( C \) together can complete the work in 16 days, so their combined work rate is:
\[
\frac{1}{B} + \frac{1}{C} = \frac{1}{16}
\]

### Steps to Solve:

#### Step 1: Express the total work done by \( A \), \( B \), and \( C \)
- \( A \)'s rate of work: \( \frac{1}{A} \)
- \( B \)'s rate of work: \( \frac{1}{B} \)
- \( C \)'s rate of work: \( \frac{1}{C} \)

#### Step 2: Calculate the work done by each person during their working period

- \( A \) worked for 5 days: The work done by \( A \) is \( 5 \times \frac{1}{A} \).
- \( B \) worked for 7 days: The work done by \( B \) is \( 7 \times \frac{1}{B} \).
- \( C \) worked for 13 days: The work done by \( C \) is \( 13 \times \frac{1}{C} \).

The total work completed is the sum of the work done by each:
\[
5 \times \frac{1}{A} + 7 \times \frac{1}{B} + 13 \times \frac{1}{C} = 1
\]
(since the total work is completed).

#### Step 3: Solve the system of equations
From the given information:
- \( \frac{1}{A} + \frac{1}{B} = \frac{1}{12} \)
- \( \frac{1}{B} + \frac{1}{C} = \frac{1}{16} \)

We have two equations in terms of \( \frac{1}{A} \), \( \frac{1}{B} \), and \( \frac{1}{C} \). We can solve these to find the value of \( \frac{1}{B} \), which gives us the number of days \( B \) would take to complete the work alone.

Let's calculate that now.The solution shows that \( B \)'s work rate is \( \frac{1}{48} \) of the total work per day. This means that \( B \) alone can complete the entire work in **48 days**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the time change if \( B \) worked for a different number of days?
2. How many days would \( A \) need to complete the work alone?
3. What if \( C \) started the work? How long would it take to finish with a different order?
4. How can we modify the problem if all three worked together at some point?
5. What happens if we change the work rate between any two workers?

**Tip:** When solving work problems, breaking down the rates of each worker and organizing the information systematically helps avoid confusion and errors.

A and B together can do a work in 12 days. B and C together can do the same work in 16 days. A started working alone and worked for 5 days, B replaced him and worked for 7 days, C replaced B and worked for 13 days. In how many days B can do the whole work alone?

Solve the work and time problem where A, B, and C work in shifts to complete a task. A and B together complete the work in 12 days, B and C in 16 days. Learn how to calculate how many days B would take to finish the work alone. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation