C alone can complete a work in 20 days and D alone can complete the same workin 30 days. In how many days C and D together can complete the same work?

Options

A. 6 days
B. 12 days
C. 15 days
D. 8 days

Answer

To solve the problem, we need to find out how many days C and D together can complete the work. Let's break it down step by step.

### Step 1: Determine the work done by C and D individually.
- C can complete the work in 20 days.
- Therefore, the work done by C in one day is:
\[ \text{Work done by C in one day} = \frac{1}{20} \]

- D can complete the work in 30 days.
- Therefore, the work done by D in one day is:
\[ \text{Work done by D in one day} = \frac{1}{30} \]

### Step 2: Combine the work done by C and D.
- To find the total work done by C and D together in one day, we add their individual work rates:
\[ \text{Total work done in one day} = \frac{1}{20} + \frac{1}{30} \]

### Step 3: Find the common denominator and add the fractions.
- The least common multiple (LCM) of 20 and 30 is 60.
- We convert the fractions:
\[ \frac{1}{20} = \frac{3}{60} \quad \text{and} \quad \frac{1}{30} = \frac{2}{60} \]
- Now we can add these fractions:
\[ \frac{3}{60} + \frac{2}{60} = \frac{5}{60} \]

### Step 4: Simplify the total work done.
- The total work done by C and D together in one day is:
\[ \frac{5}{60} = \frac{1}{12} \]

### Step 5: Calculate the total time taken to complete the work together.
- If C and D together can complete \(\frac{1}{12}\) of the work in one day, then the total time taken to complete the entire work is the reciprocal of \(\frac{1}{12}\):
\[ \text{Total days} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

Thus, C and D together can complete the work in **12 days**.

### Final Answer:
C and D together can complete the work in **12 days**.

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