What is the difference in the share of profit between P and Q in joint business the end of one year
(i) P invested Rs. 80,000 and withdrew Rs. 20,000 after 6 months.
(ii) Q joined four months after the start of business.
(iii) Q's amount was `80%` of P's amount during the last six months

Options

A. (i) and (ii) only
B. (ii) and (iii) only
C. The sum earned simple interest in 10 years.
D. Even not with all (i), (ii) and (iii) together

Answer

To solve the problem of finding the difference in the share of profit between P and Q in a joint business at the end of one year, we will follow these steps:

### Step 1: Calculate P's Investment Duration and Effective Capital
P invested Rs. 80,000 and withdrew Rs. 20,000 after 6 months.
- For the first 6 months, P's investment was Rs. 80,000.
- For the next 6 months, P's investment was Rs. 60,000 (after withdrawal).

**Effective Capital Calculation for P:**
- For the first 6 months:
\[ \text{Investment} = 80,000 \times 6 = 480,000 \]
- For the next 6 months:
\[ \text{Investment} = 60,000 \times 6 = 360,000 \]
- Total effective capital for P:
\[ \text{Total} = 480,000 + 360,000 = 840,000 \]

### Step 2: Calculate Q's Investment Duration and Effective Capital
Q joined 4 months after the start of the business, meaning Q was in the business for 8 months.
- During the last 6 months, Q's amount was 80% of P's amount.

**P's amount during the last 6 months:**
- P had Rs. 60,000 for the last 6 months.
- Q's amount:
\[ Q's \text{ amount} = 80\% \text{ of } 60,000 = 0.8 \times 60,000 = 48,000 \]

**Effective Capital Calculation for Q:**
- For the last 8 months:
\[ \text{Investment} = 48,000 \times 8 = 384,000 \]

### Step 3: Calculate Total Effective Capital for Both
- Total effective capital for P = Rs. 840,000
- Total effective capital for Q = Rs. 384,000

### Step 4: Calculate the Profit Share Ratio
To find the profit share ratio, we need to divide the total effective capital of each partner by the total effective capital combined.

**Total Effective Capital:**
\[ \text{Total} = 840,000 + 384,000 = 1,224,000 \]

**Profit Share Ratio:**
- P's share:
\[ \text{P's share} = \frac{840,000}{1,224,000} = \frac{840}{1224} = \frac{70}{102} = \frac{35}{51} \]
- Q's share:
\[ \text{Q's share} = \frac{384,000}{1,224,000} = \frac{384}{1224} = \frac{32}{102} = \frac{16}{51} \]

### Step 5: Calculate the Difference in Profit Shares
To find the difference in profit shares, we can calculate:
\[ \text{Difference} = P's \text{ share} - Q's \text{ share} = \frac{35}{51} - \frac{16}{51} = \frac{19}{51} \]

### Final Answer
The difference in the share of profit between P and Q at the end of one year is:
\[ \text{Difference} = \frac{19}{51} \text{ of the total profit} \]