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.