A can do a piece of work in 12 days and B in 20 days. If they together work on it for 5 days, and remaining work is completed by C in 3 days, then in how many days can C do the same work alone?

Options

A. 10 days
B. 9 days
C. 12 days
D. 15 days

Answer

To solve the problem step by step, we need to find out how many days C can complete the work alone after A and B have worked together for 5 days.

### Step 1: Calculate the work done by A and B in one day.

- A can complete the work in 12 days, so A's work rate is \( \frac{1}{12} \) of the work per day.
- B can complete the work in 20 days, so B's work rate is \( \frac{1}{20} \) of the work per day.

**Hint:** To find the work done in one day, use the formula: Work rate = Total work / Time taken.

### Step 2: Find the combined work rate of A and B.

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

\[ \text{Combined work rate} = \frac{1}{12} + \frac{1}{20} \]

To add these fractions, we need a common denominator. The least common multiple of 12 and 20 is 60.

\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{20} = \frac{3}{60} \]

So,

\[ \text{Combined work rate} = \frac{5}{60} + \frac{3}{60} = \frac{8}{60} = \frac{2}{15} \]

**Hint:** When adding fractions, find a common denominator to make the addition easier.

### Step 3: Calculate the total work done by A and B in 5 days.

Now, we can find out how much work A and B can complete together in 5 days:

\[ \text{Work done in 5 days} = 5 \times \frac{2}{15} = \frac{10}{15} = \frac{2}{3} \]

**Hint:** To find the total work done over multiple days, multiply the daily work rate by the number of days.

### Step 4: Determine the remaining work.

Since A and B completed \( \frac{2}{3} \) of the work, the remaining work is:

\[ \text{Remaining work} = 1 - \frac{2}{3} = \frac{1}{3} \]

**Hint:** To find the remaining work, subtract the completed work from the total work (which is considered as 1).

### Step 5: Calculate C's work rate.

C completes the remaining work in 3 days, so C's work rate is:

\[ \text{C's work rate} = \frac{1}{3} \text{ (work)} \div 3 \text{ (days)} = \frac{1}{9} \]

**Hint:** To find the work rate, divide the amount of work by the time taken.

### Step 6: Find out how many days C can do the work alone.

To find out how many days C can complete the entire work alone, we take the reciprocal of C's work rate:

\[ \text{Days taken by C} = \frac{1}{\frac{1}{9}} = 9 \text{ days} \]

**Hint:** To find the time taken to complete the work, take the reciprocal of the work rate.

### Final Answer:

C can complete the same work alone in **9 days**.