P can complete a task in 15 days Q is 50% more efficient
then P. Both P and Q started working together on the
task and after few day Q lift task and P finished the
remaining 1/3of the given work. For how many days P and Q worled together.

Options

A. 3
B. 5
C. 4
D. 6

Explanation

To solve the problem step by step, we will analyze the work done by P and Q and determine how many days they worked together.

### Step 1: Determine the work rate of P
P can complete the task in 15 days. Therefore, the work rate of P is:
\[ \text{Work rate of P} = \frac{1}{15} \text{ (work per day)} \]

### Step 2: Determine the work rate of Q
Q is 50% more efficient than P. This means Q can do 1.5 times the work that P can do in a day. Thus, the work rate of Q is:
\[ \text{Work rate of Q} = 1.5 \times \frac{1}{15} = \frac{1.5}{15} = \frac{1}{10} \text{ (work per day)} \]

### Step 3: Calculate the combined work rate of P and Q
When P and Q work together, their combined work rate is:
\[ \text{Combined work rate} = \text{Work rate of P} + \text{Work rate of Q} = \frac{1}{15} + \frac{1}{10} \]
To add these fractions, we need a common denominator, which is 30:
\[ \text{Combined work rate} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \text{ (work per day)} \]

### Step 4: Determine the total work done
Since P finished the remaining \( \frac{1}{3} \) of the work alone, it means that together they completed \( 1 - \frac{1}{3} = \frac{2}{3} \) of the work.

### Step 5: Calculate the time taken by P and Q to complete \( \frac{2}{3} \) of the work
Let \( x \) be the number of days P and Q worked together. The amount of work they completed together in \( x \) days is:
\[ \text{Work done together} = x \times \frac{1}{6} \]
Setting this equal to \( \frac{2}{3} \):
\[ x \times \frac{1}{6} = \frac{2}{3} \]

### Step 6: Solve for \( x \)
To solve for \( x \), we can multiply both sides by 6:
\[ x = \frac{2}{3} \times 6 = 4 \]

### Conclusion
P and Q worked together for **4 days**.

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