A can do a work in 10 days and B can do the same in 20 days. If they start the work together and A leaves that work 5 days before completion, in how many days the work will be finished?

Options

A. 10
B. 8
C. 12
D. 9

Answer

To solve the problem step by step, we will first determine the work rates of A and B, then calculate how long they work together, and finally find out how many days it takes to complete the work after A leaves.

### Step 1: Determine the work rates of A and B
- A can complete the work in 10 days, so A's work rate is:
\[ \text{Rate of A} = \frac{1 \text{ work}}{10 \text{ days}} = \frac{1}{10} \text{ work/day} \]
- B can complete the work in 20 days, so B's work rate is:
\[ \text{Rate of B} = \frac{1 \text{ work}}{20 \text{ days}} = \frac{1}{20} \text{ work/day} \]

### Step 2: Calculate the combined work rate of A and B
- When A and B work together, their combined work rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{10} + \frac{1}{20} \]
- To add these fractions, we find a common denominator (which is 20):
\[ \text{Combined Rate} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \text{ work/day} \]

### Step 3: Determine how long they work together before A leaves
- Let \( x \) be the number of days they work together. Since A leaves 5 days before the work is completed, B will work alone for 5 days after A leaves.
- Therefore, the total time taken to complete the work can be expressed as:
\[ \text{Total time} = x + 5 \]

### Step 4: Calculate the total work done
- The total work done by A and B together for \( x \) days is:
\[ \text{Work done by A and B} = \text{Combined Rate} \times x = \frac{3}{20} \times x \]
- The work done by B alone for 5 days is:
\[ \text{Work done by B alone} = \text{Rate of B} \times 5 = \frac{1}{20} \times 5 = \frac{5}{20} = \frac{1}{4} \]

### Step 5: Set up the equation for total work
- The total work done by both A and B together and B alone must equal 1 (the whole work):
\[ \frac{3}{20}x + \frac{1}{4} = 1 \]

### Step 6: Solve for \( x \)
- Convert \(\frac{1}{4}\) to a fraction with a denominator of 20:
\[ \frac{1}{4} = \frac{5}{20} \]
- Substitute this back into the equation:
\[ \frac{3}{20}x + \frac{5}{20} = 1 \]
- Multiply through by 20 to eliminate the denominator:
\[ 3x + 5 = 20 \]
- Solve for \( x \):
\[ 3x = 20 - 5 \]
\[ 3x = 15 \]
\[ x = 5 \]

### Step 7: Calculate the total time to finish the work
- Since \( x = 5 \), the total time taken to finish the work is:
\[ \text{Total time} = x + 5 = 5 + 5 = 10 \text{ days} \]

### Final Answer
The work will be finished in **10 days**.

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