A can do one-third of a piece of work in 4 days and B can do one-fourth of the work in 6 days. How long will they take to complete the work, working together ?

Options

A. 12 days
B. 6 days
C. 8 days
D. 10 days

Explanation

To solve the problem step by step, we will first determine the work rates of A and B, and then find out how long they will take to complete the work together.

### Step 1: Determine A's Work Rate
A can do one-third of the work in 4 days. To find A's work rate, we can use the formula:

\[ \text{Rate of A} = \frac{\text{Work}}{\text{Time}} = \frac{1/3}{4} = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12} \]

This means A can complete \(\frac{1}{12}\) of the work in one day.

### Step 2: Determine B's Work Rate
B can do one-fourth of the work in 6 days. Similarly, we find B's work rate:

\[ \text{Rate of B} = \frac{\text{Work}}{\text{Time}} = \frac{1/4}{6} = \frac{1}{4} \times \frac{1}{6} = \frac{1}{24} \]

This means B can complete \(\frac{1}{24}\) of the work in one day.

### Step 3: Combine A's and B's Work Rates
To find the combined work rate when A and B work together, we add their individual rates:

\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{12} + \frac{1}{24} \]

To add these fractions, we need a common denominator. The least common multiple of 12 and 24 is 24.

\[ \frac{1}{12} = \frac{2}{24} \]

Now we can add:

\[ \text{Combined Rate} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \]

### Step 4: Calculate the Time to Complete the Work Together
The combined rate of A and B is \(\frac{1}{8}\) of the work per day. To find out how many days it will take them to complete the entire work (1 unit of work), we use the formula:

\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Rate}} = \frac{1}{\frac{1}{8}} = 8 \text{ days} \]

### Final Answer
A and B will take 8 days to complete the work together.

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