Jack takes thrice is much time as Peter and twice as much as Justin to finish a work working together they can finish the work in 15 days. The time (in days) Justin will take to finish the work alone is:

Options

A. 60
B. 85
C. 75
D. 45

Answer

To solve the problem, we need to establish the relationships between the times taken by Jack, Peter, and Justin to complete the work. Let's denote:

- The time taken by Peter to finish the work alone as \( P \) days.
- The time taken by Justin to finish the work alone as \( J \) days.
- The time taken by Jack to finish the work alone as \( K \) days.

From the problem, we know:

1. Jack takes thrice as much time as Peter:
\[ K = 3P \]

2. Jack takes twice as much time as Justin:
\[ K = 2J \]

3. When working together, they can finish the work in 15 days.

Now, we can express the work done by each person in terms of their rates:

- The work rate of Peter is \( \frac{1}{P} \) (work done per day).
- The work rate of Justin is \( \frac{1}{J} \).
- The work rate of Jack is \( \frac{1}{K} \).

When they work together, their combined work rate is:
\[ \frac{1}{P} + \frac{1}{J} + \frac{1}{K} = \frac{1}{15} \]

Substituting \( K \) from the first two equations into the combined work rate equation, we get:
\[ \frac{1}{P} + \frac{1}{J} + \frac{1}{3P} = \frac{1}{15} \]

To combine the fractions, we first find a common denominator, which is \( 3PJ \):
\[ \frac{3J}{3PJ} + \frac{3P}{3PJ} + \frac{J}{3PJ} = \frac{1}{15} \]

This simplifies to:
\[ \frac{3J + 3P + J}{3PJ} = \frac{1}{15} \]
\[ \frac{4J + 3P}{3PJ} = \frac{1}{15} \]

Cross-multiplying gives:
\[ 15(4J + 3P) = 3PJ \]

Expanding and rearranging:
\[ 60J + 45P = 3PJ \]

Now, we can express \( P \) in terms of \( J \) using \( K = 2J \):
From \( K = 3P \), we have:
\[ P = \frac{K}{3} = \frac{2J}{3} \]

Substituting \( P \) back into the equation:
\[ 60J + 45\left(\frac{2J}{3}\right) = 3J\left(\frac{2J}{3}\right) \]
\[ 60J + 30J = 2J^2 \]
\[ 90J = 2J^2 \]

Dividing by \( J \) (assuming \( J \neq 0 \)):
\[ 90 = 2J \]
\[ J = 45 \]

Thus, Justin will take **45 days** to finish the work alone.