A can do `(1)/(4)` of a work in 10 days B can do `(1)/(3)` of the work in 20 days. In how many days can both A and B together do the work ?

Options

A. 30 days
B. 32 days
C. 24 days
D. 25 days

Answer

To solve the problem step by step, we need to determine how long it will take for A and B to complete the work together.

### Step 1: Determine the work rate of A
A can do \( \frac{1}{4} \) of the work in 10 days. Therefore, to find out how long A takes to complete the entire work (1 unit), we can set up the following equation:

\[ \text{Time taken by A to complete 1 unit} = \frac{1}{\frac{1}{4}} \times 10 = 4 \times 10 = 40 \text{ days} \]

### Step 2: Determine the work rate of B
B can do \( \frac{1}{3} \) of the work in 20 days. Similarly, to find out how long B takes to complete the entire work (1 unit), we can set up the following equation:

\[ \text{Time taken by B to complete 1 unit} = \frac{1}{\frac{1}{3}} \times 20 = 3 \times 20 = 60 \text{ days} \]

### Step 3: Calculate the work rates of A and B
Now we can find the work rates (work done per day) for A and B:

- Work rate of A:
\[ \text{Work rate of A} = \frac{1 \text{ unit}}{40 \text{ days}} = \frac{1}{40} \text{ units per day} \]

- Work rate of B:
\[ \text{Work rate of B} = \frac{1 \text{ unit}}{60 \text{ days}} = \frac{1}{60} \text{ units per day} \]

### Step 4: Combine the work rates of A and B
To find the combined work rate of A and B, we add their individual work rates:

\[ \text{Combined work rate} = \frac{1}{40} + \frac{1}{60} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 40 and 60 is 120.

Converting the fractions:
\[ \frac{1}{40} = \frac{3}{120} \quad \text{and} \quad \frac{1}{60} = \frac{2}{120} \]

Now we can add them:
\[ \text{Combined work rate} = \frac{3}{120} + \frac{2}{120} = \frac{5}{120} = \frac{1}{24} \text{ units per day} \]

### Step 5: Calculate the time taken by A and B together to complete the work
If A and B together can complete \( \frac{1}{24} \) units of work in one day, then the time taken to complete 1 unit of work is the reciprocal of their combined work rate:

\[ \text{Time taken} = \frac{1 \text{ unit}}{\frac{1}{24} \text{ units per day}} = 24 \text{ days} \]

### Final Answer
A and B together can complete the work in **24 days**.

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