A can do `(1)/(2)` of a piece of work in 5 days B can do `(3)/(5)` of the same work in 9 days and C can do `(2)/(3)` of that work in 8 days. In how many days can three of them to gether do the work ?

Options

A. 3 days
B. 5 days
C. `4(1)/(2)` days
D. 4 days

Answer

To solve the problem step by step, we will first determine how long it takes for each person (A, B, and C) to complete the entire work individually, and then find out how long they will take to complete the work together.

### Step 1: Determine the work rate of A
A can do \( \frac{1}{2} \) of the work in 5 days. Therefore, the time taken by A to complete the entire work is:
\[ \text{Time taken by A} = 5 \text{ days} \times 2 = 10 \text{ days} \]
So, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{10} \text{ work per day} \]

### Step 2: Determine the work rate of B
B can do \( \frac{3}{5} \) of the work in 9 days. Therefore, the time taken by B to complete the entire work is:
\[ \text{Time taken by B} = 9 \text{ days} \times \frac{5}{3} = 15 \text{ days} \]
So, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{15} \text{ work per day} \]

### Step 3: Determine the work rate of C
C can do \( \frac{2}{3} \) of the work in 8 days. Therefore, the time taken by C to complete the entire work is:
\[ \text{Time taken by C} = 8 \text{ days} \times \frac{3}{2} = 12 \text{ days} \]
So, C's work rate is:
\[ \text{Work rate of C} = \frac{1}{12} \text{ work per day} \]

### Step 4: Calculate the combined work rate of A, B, and C
Now we can add their work rates together to find the combined work rate:
\[ \text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} + \text{Work rate of C} \]
\[ = \frac{1}{10} + \frac{1}{15} + \frac{1}{12} \]

### Step 5: Find a common denominator
The least common multiple (LCM) of 10, 15, and 12 is 60. We will convert each fraction:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
Now, adding these fractions:
\[ \text{Combined work rate} = \frac{6}{60} + \frac{4}{60} + \frac{5}{60} = \frac{15}{60} = \frac{1}{4} \text{ work per day} \]

### Step 6: Calculate the time taken by A, B, and C together to complete the work
If A, B, and C together can complete \( \frac{1}{4} \) of the work in one day, then the time taken to complete the entire work is:
\[ \text{Time taken} = \frac{1 \text{ work}}{\frac{1}{4} \text{ work per day}} = 4 \text{ days} \]

### Final Answer
Thus, A, B, and C together can complete the work in **4 days**.

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