A can do a work in 12 days and B can do the same work in 15 days. If they work on alternate days starting with A, then in how many days the work will be completed?


Question 3Answer

a.
None


b.
53/4 days


c.
55/9 days


d.
54/7 days

To solve this problem, let's first break down the work done by A and B individually, then determine how the work progresses when they alternate working days.

### Step 1: Calculate the work done by A and B individually per day.
- A can complete the work in 12 days. Therefore, A's work rate is:

\[
\text{A's work rate} = \frac{1}{12} \text{ of the work per day}.
\]

- B can complete the work in 15 days. Therefore, B's work rate is:

\[
\text{B's work rate} = \frac{1}{15} \text{ of the work per day}.
\]

### Step 2: Analyze the work done when A and B alternate days.
- On day 1, A works and completes \(\frac{1}{12}\) of the work.
- On day 2, B works and completes \(\frac{1}{15}\) of the work.

In 2 days, the total work completed is:

\[
\text{Work done in 2 days} = \frac{1}{12} + \frac{1}{15}.
\]

To add these fractions, find the least common denominator (LCD), which is 60:

\[
\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}.
\]

So, the total work done in 2 days is:

\[
\frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \text{ of the work}.
\]

### Step 3: Determine how many cycles of 2 days are needed to complete the work.
Let \( x \) be the number of complete 2-day cycles. The work completed in \( x \) cycles is:

\[
x \times \frac{3}{20} \text{ of the work}.
\]

To complete the entire work:

\[
x \times \frac{3}{20} = 1.
\]

Solving for \( x \):

\[
x = \frac{20}{3} \approx 6.67 \text{ cycles}.
\]

This means that after 6 complete cycles (12 days), 90% of the work is done.

### Step 4: Calculate the remaining work and the additional days required.
After 6 complete cycles, the work completed is:

\[
6 \times \frac{3}{20} = \frac{18}{20} = \frac{9}{10}.
\]

The remaining work is:

\[
1 - \frac{9}{10} = \frac{1}{10}.
\]

On day 13 (A's turn again), A completes \(\frac{1}{12}\) of the work. Since \(\frac{1}{12}\) is greater than the remaining \(\frac{1}{10}\), A will finish the work on this day.

Let's determine exactly when during the 13th day the work is completed. We want to find the fraction of the day needed for A to complete the remaining work:

\[
\text{Fraction of the day} = \frac{\frac{1}{10}}{\frac{1}{12}} = \frac{12}{10} = \frac{6}{5} \text{ days}.
\]

This is equivalent to 1.2 days.

### Final Calculation:
The total number of days is:

\[
12 + \frac{5}{9} = \

Understand how to solve a work completion problem where A and B alternate working days. Learn about their work rates, fractional work completion, and determine the total days required to complete the work. Suitable for high school students.

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