In the above (i.e.., previous) problem, what is the ratio of P’s final amount to that of Q, if P and Q invest the same amounts??

Options

A. `(1.06)^6`
B. `136/100xx(100/106)^6`
C. `100/136xx(106/100)^6`
D. none of these

Answer

To find the ratio of P's final amount to that of Q's when both invest the same amount, we can follow these steps:

### Step 1: Define the Investment Amount
Let the amount invested by both P and Q be \( x \).

### Step 2: Calculate P's Final Amount
P invests under Simple Interest (SI). The formula for the final amount \( A \) in SI is given by:
\[ A = P + SI \]
Where \( SI = \frac{P \cdot r \cdot t}{100} \).
Here, \( P = x \), \( r = 6\% \), and \( t = 6 \) years.

Calculating the Simple Interest for P:
\[ SI = \frac{x \cdot 6 \cdot 6}{100} = \frac{36x}{100} = \frac{36}{100}x \]
Thus, the final amount for P is:
\[ A_P = x + \frac{36}{100}x = x \left(1 + \frac{36}{100}\right) = x \cdot \frac{136}{100} \]

### Step 3: Calculate Q's Final Amount
Q invests under Compound Interest (CI). The formula for the final amount \( A \) in CI is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Using the same values as before:
\[ A_Q = x \left(1 + \frac{6}{100}\right)^6 = x \left(\frac{106}{100}\right)^6 \]

### Step 4: Find the Ratio of P's Final Amount to Q's Final Amount
Now we need to find the ratio \( \frac{A_P}{A_Q} \):
\[ \frac{A_P}{A_Q} = \frac{x \cdot \frac{136}{100}}{x \left(\frac{106}{100}\right)^6} \]
The \( x \) cancels out:
\[ \frac{A_P}{A_Q} = \frac{\frac{136}{100}}{\left(\frac{106}{100}\right)^6} \]

### Step 5: Simplify the Ratio
This can be simplified further:
\[ \frac{A_P}{A_Q} = \frac{136}{100} \cdot \left(\frac{100}{106}\right)^6 = \frac{136 \cdot 100^6}{100 \cdot 106^6} = \frac{136 \cdot 100^5}{106^6} \]

### Final Answer
Thus, the ratio of P's final amount to Q's final amount is:
\[ \frac{136 \cdot 100^5}{106^6} \]