A sum is divided between A and B in the ratio of `1:2` A purchased a car from his part, which depreciates `14 2/7` % per annumn and B deposited his amount in a bank , which pays him 20 % interest per annum compounded annually . By what percentage will the total sum of money increase after two years due to this investment pattern (approximately) ?

Options

A. A)10%
B. B)20%
C. C)24%
D. D)16%

Answer

To solve the problem step by step, we will analyze the investments made by A and B and calculate the percentage increase in their total sum after two years.

### Step 1: Determine the initial amounts of A and B
Given the ratio of A to B is 1:2, we can assume:
- Let the total sum be \( S \).
- A's share = \( \frac{1}{3}S \)
- B's share = \( \frac{2}{3}S \)

### Step 2: Calculate the depreciation of A's car
A's car depreciates at a rate of \( 14 \frac{2}{7} \% \) per annum, which can be converted to a fraction:
- \( 14 \frac{2}{7} \% = \frac{100}{7} \% \)
- The remaining value after one year = \( 100\% - \frac{100}{7}\% = \frac{600}{700} = \frac{6}{7} \)

After 2 years, the value of A's investment can be calculated as:
- Value after 2 years = \( \left( \frac{6}{7} \right)^2 \times \frac{1}{3}S \)
- Value after 2 years = \( \frac{36}{49} \times \frac{1}{3}S = \frac{12}{49}S \)

### Step 3: Calculate the interest earned by B
B's investment earns \( 20\% \) interest compounded annually. The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( P = \frac{2}{3}S \)
- \( r = 20 \)
- \( n = 2 \)

Calculating B's final amount:
- \( A = \frac{2}{3}S \left(1 + \frac{20}{100}\right)^2 = \frac{2}{3}S \left(1.2\right)^2 = \frac{2}{3}S \times 1.44 = \frac{2.88}{3}S = \frac{32.88}{3}S \)

### Step 4: Calculate the total final amount after 2 years
Now, we will add the final amounts of A and B:
- Total final amount = A's final amount + B's final amount
- Total final amount = \( \frac{12}{49}S + \frac{32.88}{3}S \)

To add these fractions, we need a common denominator. The least common multiple of 49 and 3 is 147.
- Convert A's final amount:
\[ \frac{12}{49}S = \frac{12 \times 3}{49 \times 3}S = \frac{36}{147}S \]
- Convert B's final amount:
\[ \frac{32.88}{3}S = \frac{32.88 \times 49}{3 \times 49}S = \frac{1611.12}{147}S \]

Now, add them:
- Total final amount = \( \frac{36 + 1611.12}{147}S = \frac{1647.12}{147}S \)

### Step 5: Calculate the increase in total amount
The initial total amount was \( S \). The increase in total amount is:
- Increase = Total final amount - Initial amount
- Increase = \( \frac{1647.12}{147}S - S = \left(\frac{1647.12 - 147}{147}\right)S = \frac{1500.12}{147}S \)

### Step 6: Calculate the percentage increase
Percentage increase is given by:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{\text{Initial Amount}}\right) \times 100 \]
\[ \text{Percentage Increase} = \left(\frac{\frac{1500.12}{147}S}{S}\right) \times 100 = \frac{1500.12}{147} \times 100 \]

Calculating this gives:
\[ \text{Percentage Increase} \approx 10.2 \times 100 \approx 20.4\% \]

### Final Answer
The total sum of money will increase by approximately **20%** after two years.