Q is 50% more efficient than P, who completed a task in 45 days and R can do the same task in 7.5 days less than Q did. If P and Q start task together and after X days they left the task and R completed the remaining task in (X + 9) day. If Ratio between task done by (P + Q) together to task done by R alone is 1 : 2, Then find the value of X?

Options

A. 4.5 days
B. 7 days
C. 6 days
D. 5 days

Answer

To solve the problem step by step, we will follow the information provided and derive the necessary equations.

### Step 1: Determine the efficiencies of P, Q, and R
- Let the efficiency of P be \( E_P \).
- Since Q is 50% more efficient than P, we have:
\[ E_Q = E_P + 0.5E_P = 1.5E_P \]
- Given that P takes 45 days to complete the task, the total work \( W \) can be expressed as:
\[ W = E_P \times 45 \]

### Step 2: Calculate the efficiency of Q
- From the efficiency relation:
\[ E_Q = 1.5E_P \]
- The time taken by Q to complete the same work is:
\[ T_Q = \frac{W}{E_Q} = \frac{E_P \times 45}{1.5E_P} = 30 \text{ days} \]

### Step 3: Calculate the efficiency of R
- R can complete the task in 7.5 days less than Q:
\[ T_R = T_Q - 7.5 = 30 - 7.5 = 22.5 \text{ days} \]
- Therefore, the efficiency of R is:
\[ E_R = \frac{W}{T_R} = \frac{E_P \times 45}{22.5} = 2E_P \]

### Step 4: Set up the equation for work done
- Let \( X \) be the number of days P and Q work together.
- The work done by P and Q in \( X \) days is:
\[ \text{Work}_{PQ} = (E_P + E_Q) \times X = (E_P + 1.5E_P) \times X = 2.5E_P \times X \]
- The remaining work is:
\[ \text{Remaining Work} = W - \text{Work}_{PQ} = E_P \times 45 - 2.5E_P \times X = E_P(45 - 2.5X) \]

### Step 5: Work done by R
- R completes the remaining work in \( X + 9 \) days:
\[ \text{Work}_{R} = E_R \times (X + 9) = 2E_P \times (X + 9) \]

### Step 6: Set up the ratio equation
- According to the problem, the ratio of work done by P and Q together to the work done by R is 1:2:
\[ \frac{\text{Work}_{PQ}}{\text{Work}_{R}} = \frac{1}{2} \]
- Substituting the expressions we derived:
\[ \frac{2.5E_P \times X}{2E_P \times (X + 9)} = \frac{1}{2} \]
- Simplifying this gives:
\[ \frac{2.5X}{2(X + 9)} = \frac{1}{2} \]
- Cross-multiplying:
\[ 2.5X \cdot 2 = 2(X + 9) \]
\[ 5X = 2X + 18 \]
- Rearranging gives:
\[ 5X - 2X = 18 \implies 3X = 18 \implies X = 6 \]

### Final Answer
Thus, the value of \( X \) is \( 6 \) days.