A can complete `33(1)/(3)`% of a work in 5 days and B can complete 40% of the same work in 10 days. They work together for 5 days and then B left the work. A alone will complete the remaining work in:

Options

A. `7(1)/(2)` days
B. 5 days
C. 7 days
D. `5(1)/(2)` days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the total work
A can complete \(33 \frac{1}{3}\%\) of the work in 5 days.

First, convert \(33 \frac{1}{3}\%\) to a fraction:
\[ 33 \frac{1}{3}\% = \frac{100}{3}\% \]
This means A completes \(\frac{100}{3} \div 100 = \frac{1}{3}\) of the work in 5 days.

To find the total work, we can set up the equation:
\[ \text{Total Work} = 5 \text{ days} \times 3 = 15 \text{ days} \]
Thus, the total work can be considered as 15 units.

### Step 2: Determine B's work rate
B can complete \(40\%\) of the work in 10 days.

Convert \(40\%\) to a fraction:
\[ 40\% = \frac{40}{100} = \frac{2}{5} \]
This means B completes \(\frac{2}{5}\) of the work in 10 days.

To find the total work, we can set up the equation:
\[ \text{Total Work} = 10 \text{ days} \times \frac{5}{2} = 25 \text{ days} \]
Thus, the total work can be considered as 25 units.

### Step 3: Find the LCM of work rates
Now, we need to find the least common multiple (LCM) of the days taken by A and B to complete the work:
- A takes 15 days to complete the work.
- B takes 25 days to complete the work.

The LCM of 15 and 25 is 75 days.

### Step 4: Calculate work done by A and B in one day
Now, let's find the work done by A and B in one day:
- A's work rate = \(\frac{1}{15}\) of the work per day.
- B's work rate = \(\frac{1}{25}\) of the work per day.

### Step 5: Work done by A and B together in one day
The combined work done by A and B in one day:
\[ \text{Work done by A + Work done by B} = \frac{1}{15} + \frac{1}{25} \]
To add these fractions, find a common denominator (which is 75):
\[ \frac{1}{15} = \frac{5}{75}, \quad \frac{1}{25} = \frac{3}{75} \]
Thus,
\[ \text{Combined work} = \frac{5}{75} + \frac{3}{75} = \frac{8}{75} \]

### Step 6: Work done in 5 days
In 5 days, the total work done by A and B together:
\[ \text{Total work in 5 days} = 5 \times \frac{8}{75} = \frac{40}{75} = \frac{8}{15} \]

### Step 7: Remaining work
Now, we need to find the remaining work after 5 days:
\[ \text{Remaining work} = 1 - \frac{8}{15} = \frac{15 - 8}{15} = \frac{7}{15} \]

### Step 8: Time taken by A to complete remaining work
Now, we need to find out how long A will take to complete the remaining work:
A's work rate is \(\frac{1}{15}\) of the work per day. To find the time taken to complete \(\frac{7}{15}\) of the work:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{A's work rate}} = \frac{\frac{7}{15}}{\frac{1}{15}} = 7 \text{ days} \]

### Final Answer
A alone will complete the remaining work in **7 days**.

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