P and Q invest some amount under SI and CI, repectively but for the same period at `6%` per annum. Each gets a total amount of ₹ 65,000 at the end of 6 years Which of the following is definitely true?
Q’s initial principal is less than that of P
Q' s initial principal is equal to that of P
P's percentage earning is less than that of Q

Options

A. (i) only
B. (ii) only
C. (iii) only
D. (i) and (iii) only

Answer

To solve the problem, we need to analyze the investments made by P and Q under Simple Interest (SI) and Compound Interest (CI) respectively.

### Step-by-Step Solution:

1. **Understanding the Interest Types**:
- P invests under Simple Interest (SI).
- Q invests under Compound Interest (CI).

2. **Given Information**:
- Both P and Q receive a total amount of ₹65,000 after 6 years.
- The rate of interest for both is 6% per annum.

3. **Formula for Simple Interest**:
- The formula for the total amount (A) under Simple Interest is:
\[ A = P + SI \]
where \( SI = \frac{P \times r \times t}{100} \).
- Here, \( P \) is the principal, \( r \) is the rate of interest, and \( t \) is the time in years.

4. **Calculating P's Principal**:
- Let \( P \) be the principal amount invested by P.
- The total amount after 6 years for P is:
\[ 65000 = P + \frac{P \times 6 \times 6}{100} \]
- Simplifying the equation:
\[ 65000 = P + \frac{36P}{100} \]
\[ 65000 = P + 0.36P \]
\[ 65000 = 1.36P \]
- Solving for \( P \):
\[ P = \frac{65000}{1.36} \approx 47868.24 \]

5. **Formula for Compound Interest**:
- The formula for the total amount (A) under Compound Interest is:
\[ A = P(1 + \frac{r}{100})^t \]
- Let \( Q \) be the principal amount invested by Q.
- The total amount after 6 years for Q is:
\[ 65000 = Q(1 + \frac{6}{100})^6 \]
- Simplifying:
\[ 65000 = Q(1.06)^6 \]
- Calculating \( (1.06)^6 \):
\[ (1.06)^6 \approx 1.4185 \]
- Therefore:
\[ 65000 = Q \times 1.4185 \]
- Solving for \( Q \):
\[ Q = \frac{65000}{1.4185} \approx 45763.67 \]

6. **Comparison of Principals**:
- From our calculations:
- P's principal \( \approx 47868.24 \)
- Q's principal \( \approx 45763.67 \)
- Since \( Q < P \), we conclude that Q's initial principal is less than that of P.

7. **Percentage Earnings**:
- P earns simple interest, while Q earns compound interest. Given the same amount of time and interest rate, Q will always earn more than P.
- Therefore, P's percentage earning is less than that of Q.

### Conclusion:
- The statement that is definitely true is: **Q’s initial principal is less than that of P**.