Understanding Individual Work Rates

To solve this problem, we first need to understand how much work each person completes per day. This is known as their individual work rate.

• Person A completes the entire work in 12 days. Therefore, A's work rate is $\frac{1}{12}$ of the work per day.
• Person B completes the entire work in 15 days. So, B's work rate is $\frac{1}{15}$ of the work per day.
• Person C completes the entire work in 20 days. Thus, C's work rate is $\frac{1}{20}$ of the work per day.

Calculating Combined Work Rate

Initially, A, B, and C started working together. We can find their combined work rate by adding their individual rates.

Combined rate of A, B, and C = Rate of A + Rate of B + Rate of C

Combined rate = $\frac{1}{12} + \frac{1}{15} + \frac{1}{20}$

To add these fractions, we find the least common multiple (LCM) of the denominators (12, 15, and 20). The LCM is 60.

Combined rate = $\frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{5+4+3}{60} = \frac{12}{60}$

Simplifying the fraction $\frac{12}{60}$, we get $\frac{1}{5}$. So, A, B, and C together complete $\frac{1}{5}$ of the work per day.

Work Done in the First Two Days

A, B, and C worked together for the first 2 days before C left.

Work done in 2 days = Combined rate $\times$ Number of days worked

Work done = $\frac{1}{5} \times 2 = \frac{2}{5}$

This means $\frac{2}{5}$ of the total work was completed in the first 2 days.

Calculating Remaining Work

After C left, there was a portion of the work that still needed to be completed.

Remaining work = Total work - Work done in the first 2 days

Remaining work = $1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}$

So, $\frac{3}{5}$ of the work is remaining.

Determining the Combined Rate of A and B

Now, A and B are responsible for finishing the remaining $\frac{3}{5}$ of the work. Let's calculate their combined work rate.

Combined rate of A and B = Rate of A + Rate of B

Combined rate (A+B) = $\frac{1}{12} + \frac{1}{15}$

The LCM of 12 and 15 is 60.

Combined rate (A+B) = $\frac{5}{60} + \frac{4}{60} = \frac{5+4}{60} = \frac{9}{60}$

Simplifying this fraction, the combined rate of A and B is $\frac{3}{20}$ of the work per day.

Calculating Time to Finish Remaining Work

To find out how many days A and B will take to complete the remaining work, we divide the remaining work by their combined rate.

Time = $\frac{\text{Remaining Work}}{\text{Combined Rate of A and B}}$

Time = $\frac{\frac{3}{5}}{\frac{3}{20}}$

Dividing fractions involves multiplying the first fraction by the reciprocal of the second fraction:

Time = $\frac{3}{5} \times \frac{20}{3}$

Time = $\frac{3 \times 20}{5 \times 3} = \frac{60}{15}$

Time = 4 days.

Thus, A and B will finish the remaining work in 4 days.