A, B and C can doa piece of work in 10, 15 and 30 days, respectively. If B and C both assist A on every third day, then in how many days will 40% of the work be completed?

Options

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

Answer

To solve the problem, we need to determine how much work A, B, and C can complete together under the given conditions. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the work rate of A, B, and C
- A can complete the work in 10 days, so in one day, A completes:
\[ \text{Work done by A in 1 day} = \frac{1}{10} \text{ of the work} \]
- B can complete the work in 15 days, so in one day, B completes:
\[ \text{Work done by B in 1 day} = \frac{1}{15} \text{ of the work} \]
- C can complete the work in 30 days, so in one day, C completes:
\[ \text{Work done by C in 1 day} = \frac{1}{30} \text{ of the work} \]

### Step 2: Calculate the total work done by A, B, and C
To find a common measure, we can consider the total work as 30 units (the least common multiple of 10, 15, and 30).

- Work done by A in 1 day:
\[ \text{A's work} = \frac{30}{10} = 3 \text{ units} \]
- Work done by B in 1 day:
\[ \text{B's work} = \frac{30}{15} = 2 \text{ units} \]
- Work done by C in 1 day:
\[ \text{C's work} = \frac{30}{30} = 1 \text{ unit} \]

### Step 3: Calculate the work done in a 3-day cycle
- **Day 1**: A works alone:
\[ \text{Total work on Day 1} = 3 \text{ units} \]
- **Day 2**: A works alone:
\[ \text{Total work on Day 2} = 3 \text{ units} \]
- **Day 3**: A, B, and C work together:
\[ \text{Total work on Day 3} = 3 + 2 + 1 = 6 \text{ units} \]

### Step 4: Calculate total work done in 3 days
In 3 days, the total work done is:
\[ \text{Total work in 3 days} = 3 + 3 + 6 = 12 \text{ units} \]

### Step 5: Determine how many days to complete 40% of the work
40% of the total work (30 units) is:
\[ 0.4 \times 30 = 12 \text{ units} \]

Since we have calculated that 12 units of work can be completed in 3 days, we conclude that:
\[ \text{Total days to complete 40% of the work} = 3 \text{ days} \]

### Final Answer
Thus, A, B, and C will complete 40% of the work in **3 days**.

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