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

Options

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

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Determine the work done by A and B individually.

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:
\[ \text{Work done by A in 1 day} = \frac{1/5}{4} = \frac{1}{20} \]
So, A can complete the entire work in:
\[ \text{Days taken by A to finish the work} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

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:
\[ \text{Work done by B in 1 day} = \frac{1/6}{5} = \frac{1}{30} \]
So, B can complete the entire work in:
\[ \text{Days taken by B to finish the work} = \frac{1}{\frac{1}{30}} = 30 \text{ days} \]

### Step 2: Calculate the total work in terms of a common unit.

To find a common measure of work, we can take the least common multiple (LCM) of the days taken by A and B:
\[ \text{LCM of 20 and 30} = 60 \]
Thus, we can consider the total work as 60 units.

### Step 3: Determine the amount of work done by A and B in the total work.

Since A completes the work in 20 days, the amount of work A can do in 60 units is:
\[ \text{Work done by A} = \frac{60}{20} = 3 \text{ units per day} \]

Since B completes the work in 30 days, the amount of work B can do in 60 units is:
\[ \text{Work done by B} = \frac{60}{30} = 2 \text{ units per day} \]

### Step 4: Calculate the combined work done by A and B in one day.

When A and B work together, the total work done in one day is:
\[ \text{Total work done in one day} = 3 + 2 = 5 \text{ units} \]

### Step 5: Determine the total time taken to complete the work together.

To find out how many days they will take to finish the total work of 60 units, we divide the total work by the combined work done in one day:
\[ \text{Total days to finish the work} = \frac{60}{5} = 12 \text{ days} \]

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

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