A is twice efficient as B and together they do the same work in as much time as C and D together. If C and D can complete the work in 20 and 30 days respectively, working alone, then in how many days A can complete the work individually :

Options

A. 12 days
B. 18 days
C. 24 days
D. 30 days

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the work rates of C and D
C can complete the work in 20 days, and D can complete it in 30 days. We can find their work rates (work done per day) as follows:

- Work rate of C = 1/20 (work done per day)
- Work rate of D = 1/30 (work done per day)

### Step 2: Calculate the combined work rate of C and D
To find the combined work rate of C and D, we add their individual work rates:

\[ \text{Combined work rate of C and D} = \frac{1}{20} + \frac{1}{30} \]

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,

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

This means C and D together can complete the work in 12 days.

### Step 3: Relate the work rates of A and B
According to the problem, A is twice as efficient as B. If we let the work rate of B be \( x \), then the work rate of A will be \( 2x \).

### Step 4: Calculate the combined work rate of A and B
The combined work rate of A and B is:

\[ \text{Combined work rate of A and B} = 2x + x = 3x \]

### Step 5: Set the work rates equal
Since A and B together complete the same work in the same time as C and D together, we can set their combined work rates equal:

\[ 3x = \frac{1}{12} \]

### Step 6: Solve for x
To find \( x \):

\[ x = \frac{1}{12} \times \frac{1}{3} = \frac{1}{36} \]

This means B can complete the work in 36 days.

### Step 7: Find A's work rate and time to complete the work
Since A's work rate is \( 2x \):

\[ 2x = 2 \times \frac{1}{36} = \frac{2}{36} = \frac{1}{18} \]

This means A can complete the work in 18 days.

### Final Answer
A can complete the work individually in **18 days**.

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