A can do `(1)/(5)th` of a work in 4 days and B can do `(1)/(6)`th of the same work in 5 days. In how many days they can finish the work, if they work together?

Options

A. 12
B. 20
C. 15
D. 30

Answer

To solve the problem step by step, we will calculate the efficiency of both A and B and then determine how long it will take for them to complete the work together.

### Step 1: Calculate A's Work Rate
A can do \( \frac{1}{5} \) of the work in 4 days.
To find out how much work A can do in one day, we calculate:

\[ \text{Work done by A in 1 day} = \frac{1/5}{4} = \frac{1}{20} \]

This means A can complete \( \frac{1}{20} \) of the work in one day.

### Step 2: Calculate B's Work Rate
B can do \( \frac{1}{6} \) of the work in 5 days.
To find out how much work B can do in one day, we calculate:

\[ \text{Work done by B in 1 day} = \frac{1/6}{5} = \frac{1}{30} \]

This means B can complete \( \frac{1}{30} \) of the work in one day.

### Step 3: Calculate Combined Work Rate
Now, we will find the combined work rate of A and B when they work together:

\[ \text{Combined work rate} = \text{Work done by A in 1 day} + \text{Work done by B in 1 day} \]

\[ \text{Combined work rate} = \frac{1}{20} + \frac{1}{30} \]

To add these fractions, we need a common denominator. The least common multiple of 20 and 30 is 60.

\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]

So,

\[ \text{Combined work rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]

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

### Step 4: Calculate Total Time to Complete the Work
To find out how many days it will take for A and B to finish the work together, we take the reciprocal of their combined work rate:

\[ \text{Total time} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

Thus, A and B can finish the work together in **12 days**.

### Final Answer
A and B can finish the work together in **12 days**.

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