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.

- Work done by C in one day = 1/20
- Work done by D in one day = 1/30

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

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

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

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

Now, adding them together:

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

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

### Step 3: Relate the efficiency of A and B
We know that A is twice as efficient as B. Let the efficiency of B be \( x \). Then the efficiency of A will be \( 2x \).

### Step 4: Calculate the combined work rate of A and B
The combined work rate of A and B can be expressed as:

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

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

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

### Step 6: Solve for x
To find \( x \), we rearrange the equation:

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

This means B can complete the work in 36 days.

### Step 7: Find the efficiency of A
Since A is twice as efficient as B:

\[ \text{Efficiency of A} = 2x = 2 \cdot \frac{1}{36} = \frac{1}{18} \]

### Step 8: Calculate the time taken by A to complete the work
If A's work rate is \( \frac{1}{18} \), then the time taken by A to complete the work alone is the reciprocal of the work rate:

\[ \text{Time taken by A} = \frac{1}{\frac{1}{18}} = 18 \text{ days} \]

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

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