A can do a piece of work in 12 days and B in 24 days. If they work togeher, in how many days will they finish the work?

Options

A. 15 days
B. 20 days
C. 12days
D. 8 days

Answer

To solve the problem of how many days A and B will take to finish the work together, we can follow these steps:

### Step 1: Determine the work done by A and B individually.
- A can complete the work in 12 days. Therefore, A's work rate (efficiency) is:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{12 \text{ days}} = \frac{1}{12} \text{ work per day} \]

- B can complete the work in 24 days. Therefore, B's work rate (efficiency) is:
\[ \text{Efficiency of B} = \frac{1 \text{ work}}{24 \text{ days}} = \frac{1}{24} \text{ work per day} \]

### Step 2: Calculate the combined work rate of A and B.
- To find their combined efficiency, we add the individual efficiencies:
\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{12} + \frac{1}{24} \]

- To add these fractions, we need a common denominator. The least common multiple of 12 and 24 is 24. Thus:
\[ \frac{1}{12} = \frac{2}{24} \]
\[ \text{Combined Efficiency} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \text{ work per day} \]

### Step 3: Calculate the total time taken to complete the work together.
- Since their combined efficiency is \(\frac{1}{8}\) work per day, it means together they can complete the work in:
\[ \text{Time taken} = \frac{1 \text{ work}}{\text{Combined Efficiency}} = \frac{1}{\frac{1}{8}} = 8 \text{ days} \]

### Final Answer:
A and B will finish the work together in **8 days**.

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