A plumber works twice as fast as his apprentice. After the plumber has worked alone for 3 hours, his apprentice joins him and working together they complete the job 4 hours later. How many hours would it have taken the plumber to do the entire job by himself?

Options

A. `9`
B. `12`
C. `14`
D. `18`

Answer

To solve the problem step by step, we will define the work rates of the plumber and his apprentice, calculate the work done in different time intervals, and finally determine how long it would take the plumber to complete the job alone.

### Step 1: Define Work Rates
Let the work rate of the apprentice be \(1\) unit per hour. Since the plumber works twice as fast as his apprentice, his work rate will be:
\[ \text{Work rate of plumber} = 2 \times \text{Work rate of apprentice} = 2 \times 1 = 2 \text{ units per hour} \]

**Hint:** Define the work rates based on the relationship given in the problem.

### Step 2: Calculate Work Done by the Plumber Alone
The plumber works alone for \(3\) hours. The work done by the plumber in this time is:
\[ \text{Work done by plumber in 3 hours} = \text{Work rate of plumber} \times \text{Time} = 2 \text{ units/hour} \times 3 \text{ hours} = 6 \text{ units} \]

**Hint:** Use the formula for work done: Work = Rate × Time.

### Step 3: Calculate Work Done Together
After \(3\) hours, the apprentice joins the plumber, and they work together for \(4\) hours. The combined work rate when they work together is:
\[ \text{Combined work rate} = \text{Work rate of plumber} + \text{Work rate of apprentice} = 2 \text{ units/hour} + 1 \text{ unit/hour} = 3 \text{ units/hour} \]
The work done together in \(4\) hours is:
\[ \text{Work done together in 4 hours} = \text{Combined work rate} \times \text{Time} = 3 \text{ units/hour} \times 4 \text{ hours} = 12 \text{ units} \]

**Hint:** Add the work rates to find the combined rate when working together.

### Step 4: Calculate Total Work Done
Now, we can find the total work done:
\[ \text{Total work} = \text{Work done by plumber alone} + \text{Work done together} = 6 \text{ units} + 12 \text{ units} = 18 \text{ units} \]

**Hint:** Sum the work done in each phase to find the total work.

### Step 5: Calculate Time for Plumber to Complete the Job Alone
To find out how long it would take the plumber to complete the entire job by himself, we use the formula:
\[ \text{Time taken by plumber} = \frac{\text{Total work}}{\text{Work rate of plumber}} = \frac{18 \text{ units}}{2 \text{ units/hour}} = 9 \text{ hours} \]

**Hint:** Use the formula for time: Time = Work / Rate.

### Final Answer
The plumber would take **9 hours** to complete the entire job by himself.

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