Find the number of days in which Q can do a job if P can do the same job in 8 days.
A. Q is `60%` more efficient than P.
B. P and Q togetyher can do the job in 10/3 days.
C. P is `75//2%` less efficient than Q.

Options

A. only A is sufficient
B. Only B is sufficient
C. Either A or B is sufficient
D. Any of them

Explanation

To find the number of days in which Q can do a job if P can do the same job in 8 days, we will analyze the information provided in each case (A, B, and C) step by step.

### Step 1: Understanding the efficiency of P
P can complete the job in 8 days. Therefore, P's work rate is:
\[ \text{Work rate of P} = \frac{1 \text{ job}}{8 \text{ days}} = \frac{1}{8} \text{ jobs per day} \]

### Step 2: Analyzing Case A
**Case A:** Q is 60% more efficient than P.

If Q is 60% more efficient than P, we can calculate Q's work rate as follows:
\[ \text{Efficiency of Q} = \text{Efficiency of P} + 60\% \text{ of Efficiency of P} \]
\[ \text{Efficiency of Q} = \frac{1}{8} + 0.6 \times \frac{1}{8} = \frac{1}{8} + \frac{0.6}{8} = \frac{1 + 0.6}{8} = \frac{1.6}{8} = \frac{0.2}{1} = \frac{1}{5} \text{ jobs per day} \]

Now, to find the number of days Q takes to complete the job:
\[ \text{Days taken by Q} = \frac{1 \text{ job}}{\text{Work rate of Q}} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Step 3: Analyzing Case B
**Case B:** P and Q together can do the job in \( \frac{10}{3} \) days.

To find their combined work rate:
\[ \text{Combined work rate} = \frac{1 \text{ job}}{\frac{10}{3} \text{ days}} = \frac{3}{10} \text{ jobs per day} \]

Let Q's work rate be \( x \). Then:
\[ \text{Work rate of P} + \text{Work rate of Q} = \frac{1}{8} + x = \frac{3}{10} \]

Solving for \( x \):
\[ x = \frac{3}{10} - \frac{1}{8} \]
Finding a common denominator (40):
\[ x = \frac{12}{40} - \frac{5}{40} = \frac{7}{40} \text{ jobs per day} \]

Now, to find the number of days Q takes to complete the job:
\[ \text{Days taken by Q} = \frac{1 \text{ job}}{x} = \frac{1}{\frac{7}{40}} = \frac{40}{7} \text{ days} \approx 5.71 \text{ days} \]

### Step 4: Analyzing Case C
**Case C:** P is \( \frac{75}{2}\% \) less efficient than Q.

This means:
\[ \text{Efficiency of P} = \text{Efficiency of Q} - \frac{75}{2}\% \text{ of Efficiency of Q} \]
Let Q's efficiency be \( y \):
\[ \frac{1}{8} = y - \frac{75}{200}y = y \left(1 - \frac{75}{200}\right) = y \left(\frac{125}{200}\right) = \frac{5}{8}y \]

Now solving for \( y \):
\[ y = \frac{1/8}{5/8} = \frac{1}{5} \text{ jobs per day} \]

Now, to find the number of days Q takes to complete the job:
\[ \text{Days taken by Q} = \frac{1 \text{ job}}{y} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Conclusion
From the analysis of all three cases:
- **Case A:** Q takes 5 days.
- **Case B:** Q takes approximately 5.71 days.
- **Case C:** Q takes 5 days.

Thus, Cases A and C provide a definitive answer that Q can complete the job in 5 days.