P can complete a work in 72 days. Q is 33.33% more efficient
than P. In how many days both complete the work if they
work on alternate days starting with P?

Options

A. 56 days
B. 60 days
C. 62 days
D. none of these

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Determine the work done by P and Q
P can complete the work in 72 days. Therefore, P's work rate is:
\[ \text{Work rate of P} = \frac{1}{72} \text{ (work per day)} \]

Q is 33.33% more efficient than P. This means Q can do more work in a day compared to P. To find Q's efficiency, we can calculate it as follows:
\[ \text{Efficiency of Q} = \text{Efficiency of P} \times \left(1 + \frac{33.33}{100}\right) = \frac{1}{72} \times \frac{4}{3} = \frac{4}{216} = \frac{1}{54} \]
So, Q can complete the work in 54 days.

### Step 2: Calculate the total work in terms of units
To find the total work, we can take the LCM of the days taken by P and Q:
- Days taken by P = 72
- Days taken by Q = 54

The LCM of 72 and 54 is 216. Therefore, the total work can be considered as 216 units.

### Step 3: Calculate the work done by P and Q in one cycle of two days
In one day:
- Work done by P = \(\frac{216}{72} = 3\) units
- Work done by Q = \(\frac{216}{54} = 4\) units

In two days (one day of P and one day of Q):
\[ \text{Total work done in 2 days} = 3 + 4 = 7 \text{ units} \]

### Step 4: Calculate how many complete cycles can be done
To find out how many complete cycles can be done in 216 units:
\[ \text{Number of complete cycles} = \frac{216}{7} \approx 30.857 \]
This means they can complete 30 full cycles (60 days) and then we will have some work left.

### Step 5: Calculate the work done in 60 days
In 60 days, the total work done is:
\[ \text{Total work done in 60 days} = 30 \times 7 = 210 \text{ units} \]

### Step 6: Calculate the remaining work
The remaining work after 60 days is:
\[ \text{Remaining work} = 216 - 210 = 6 \text{ units} \]

### Step 7: Determine who will do the remaining work
On the 61st day, P will work. P can do 3 units of work in one day. Since the remaining work is 6 units, P will do 3 units on the 61st day, leaving 3 units.

On the 62nd day, Q will work. Q can do 4 units of work in one day. Q will complete the remaining 3 units on the 62nd day.

### Step 8: Calculate the total days taken
Total days taken:
- 60 days for the complete cycles
- 1 day for P (61st day)
- 1 day for Q (62nd day)

Thus, the total days taken is:
\[ \text{Total days} = 60 + 1 + 1 = 62 \text{ days} \]

### Final Answer
Both P and Q can complete the work in **62 days**.

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