A can complete of a work in 8 days. B can complete`3/5` of the same work in 12 days and C can complete `4/9` of the same work in 8 days. A and B worked together for 5 days. How much time(in days) will C alone take to complete the remaining work?

Options

A. `6`
B. `\2 (1)/(4)`
C. ` 8(1)/(2)`
D. `9`

Answer

To solve the problem step by step, let's first determine the work rates of A, B, and C.

### Step 1: Calculate the work rates of A, B, and C.

1. **A's work rate**: A can complete the entire work in 8 days.
\[ \text{Work rate of A} = \frac{1}{8} \text{ (work per day)} \]

2. **B's work rate**: B can complete \( \frac{3}{5} \) of the work in 12 days.
To find B's work rate, we first determine how much work B can do in one day:
\[ \text{Work done by B in one day} = \frac{3/5}{12} = \frac{3}{60} = \frac{1}{20} \]
Thus, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{20} \text{ (work per day)} \]

3. **C's work rate**: C can complete \( \frac{4}{9} \) of the work in 8 days.
To find C's work rate, we calculate:
\[ \text{Work done by C in one day} = \frac{4/9}{8} = \frac{4}{72} = \frac{1}{18} \]
Therefore, C's work rate is:
\[ \text{Work rate of C} = \frac{1}{18} \text{ (work per day)} \]

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

Now, we can add the work rates of A and B to find their combined work rate:
\[ \text{Combined work rate of A and B} = \frac{1}{8} + \frac{1}{20} \]

To add these fractions, we need a common denominator. The least common multiple of 8 and 20 is 40.
\[ \frac{1}{8} = \frac{5}{40}, \quad \frac{1}{20} = \frac{2}{40} \]
Thus,
\[ \text{Combined work rate of A and B} = \frac{5}{40} + \frac{2}{40} = \frac{7}{40} \]

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

Now we calculate how much work A and B can complete together in 5 days:
\[ \text{Work done by A and B in 5 days} = 5 \times \frac{7}{40} = \frac{35}{40} = \frac{7}{8} \]

### Step 4: Determine the remaining work.

Since A and B completed \( \frac{7}{8} \) of the work, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{7}{8} = \frac{1}{8} \]

### Step 5: Calculate the time taken by C to complete the remaining work.

Now we need to find out how long it will take C to complete the remaining \( \frac{1}{8} \) of the work. Since C's work rate is \( \frac{1}{18} \):
\[ \text{Time taken by C} = \frac{\text{Remaining work}}{\text{Work rate of C}} = \frac{1/8}{1/18} = \frac{1}{8} \times \frac{18}{1} = \frac{18}{8} = 2.25 \text{ days} \]

### Final Answer:
C will take **2.25 days** to complete the remaining work.

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