A can do a piece of work in 30 days and B can do the same piece of work in 20 days. They start working together and work for 4 days and then both leave the work. C alone finishes the remaining work in 18 days. In how many days will C alone finish the whole work?

Options

A. 30
B. 27
C. 25
D. 24

Answer

To solve the problem step by step, we will first determine the work rates of A, B, and C, and then calculate how long it would take C to finish the entire work alone.

### Step 1: Determine the work rates of A and B
- A can complete the work in 30 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{30} \text{ (work per day)} \]
- B can complete the work in 20 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{20} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
When A and B work together, their combined work rate is:
\[ \text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} = \frac{1}{30} + \frac{1}{20} \]
To add these fractions, we need a common denominator. The least common multiple of 30 and 20 is 60.
\[ \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Thus,
\[ \text{Combined work rate} = \frac{2}{60} + \frac{3}{60} = \frac{5}{60} = \frac{1}{12} \text{ (work per day)} \]

### Step 3: Calculate the work done by A and B in 4 days
In 4 days, the amount of work done by A and B together is:
\[ \text{Work done in 4 days} = \text{Combined work rate} \times 4 = \frac{1}{12} \times 4 = \frac{4}{12} = \frac{1}{3} \]

### Step 4: Determine the remaining work
Since A and B completed \(\frac{1}{3}\) of the work, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 5: Calculate the time taken by C to finish the remaining work
It is given that C finishes the remaining \(\frac{2}{3}\) of the work in 18 days. Therefore, we can find C's work rate:
\[ \text{Work rate of C} = \frac{\text{Remaining work}}{\text{Time taken}} = \frac{\frac{2}{3}}{18} = \frac{2}{3 \times 18} = \frac{2}{54} = \frac{1}{27} \text{ (work per day)} \]

### Step 6: Calculate the total time taken by C to finish the whole work
If C can do \(\frac{1}{27}\) of the work in a day, then the total time C would take to finish the entire work is:
\[ \text{Total time for C} = \frac{1}{\text{Work rate of C}} = \frac{1}{\frac{1}{27}} = 27 \text{ days} \]

### Final Answer
C alone will finish the whole work in **27 days**.

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