Two students A and B can finish solving some set of questions in 10 minutes and 15 minutes respectively. In what time will the task be finished if B had started solving 3 minutes after A ?

Options

A. 6 minutes 12 seconds
B. 7 minutes 12 seconds
C. 8 minutes 12 seconds
D. 9 minutes 12 seconds

Explanation

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the rate of work for each student.
- Student A can complete the task in 10 minutes, so A's rate of work is:
\[ \text{Rate of A} = \frac{1 \text{ task}}{10 \text{ minutes}} = \frac{1}{10} \text{ tasks per minute} \]
- Student B can complete the task in 15 minutes, so B's rate of work is:
\[ \text{Rate of B} = \frac{1 \text{ task}}{15 \text{ minutes}} = \frac{1}{15} \text{ tasks per minute} \]

### Step 2: Calculate how much work A does in the first 3 minutes.
- In 3 minutes, A will solve:
\[ \text{Work done by A in 3 minutes} = 3 \times \frac{1}{10} = \frac{3}{10} \text{ of the task} \]

### Step 3: Determine the remaining work after A's initial contribution.
- The total work is 1 task, so the remaining work after A has worked for 3 minutes is:
\[ \text{Remaining work} = 1 - \frac{3}{10} = \frac{7}{10} \text{ of the task} \]

### Step 4: Calculate the combined rate of work when A and B work together.
- The combined rate of A and B working together is:
\[ \text{Combined rate} = \frac{1}{10} + \frac{1}{15} \]
- To add these fractions, find a common denominator (which is 30):
\[ \text{Combined rate} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ tasks per minute} \]

### Step 5: Calculate the time taken to complete the remaining work.
- To find the time taken to finish the remaining \(\frac{7}{10}\) of the task at the combined rate of \(\frac{1}{6}\) tasks per minute:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Combined rate}} = \frac{\frac{7}{10}}{\frac{1}{6}} = \frac{7}{10} \times 6 = \frac{42}{10} = 4.2 \text{ minutes} \]

### Step 6: Convert 4.2 minutes to minutes and seconds.
- \(4.2\) minutes can be expressed as:
\[ 4 \text{ minutes and } 0.2 \text{ minutes} \]
- To convert \(0.2\) minutes to seconds:
\[ 0.2 \times 60 = 12 \text{ seconds} \]
- Thus, \(4.2\) minutes is equal to \(4\) minutes and \(12\) seconds.

### Step 7: Add the initial 3 minutes that A worked alone.
- The total time taken to complete the task is:
\[ \text{Total time} = 4 \text{ minutes } 12 \text{ seconds} + 3 \text{ minutes} = 7 \text{ minutes } 12 \text{ seconds} \]

### Final Answer:
The total time taken to finish the task is **7 minutes 12 seconds**.

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