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 annum 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. `20%`
B. `26.66%`
C. `30%`
D. `25%`

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the amounts for A and B
Given that the sum is divided between A and B in the ratio of 1:2, we can assume:
- Let the amount for A be \( x \).
- Then the amount for B will be \( 2x \).

### Step 2: Calculate the depreciation of A's car
A's car depreciates at a rate of \( 14 \frac{2}{7}\% \) per annum. First, we convert this percentage into a decimal:
\[ 14 \frac{2}{7}\% = 14.2857\% = \frac{100}{7} \approx 0.142857 \]
The depreciation formula for the value after \( n \) years is:
\[ \text{Value after } n \text{ years} = \text{Initial Value} \times (1 - \text{Depreciation Rate})^n \]
For A:
\[ \text{Value after 2 years} = x \times (1 - 0.142857)^2 \]
Calculating this:
\[ = x \times (0.857143)^2 \approx x \times 0.734694 \approx \frac{734694x}{1000000} \]

### Step 3: Calculate the amount for B after 2 years with interest
B deposits his amount in a bank with an interest rate of 20% compounded annually. The formula for compound interest is:
\[ A = P(1 + r)^n \]
Where:
- \( P = 2x \) (initial amount)
- \( r = 0.20 \) (interest rate)
- \( n = 2 \) (number of years)

Calculating this:
\[ A = 2x(1 + 0.20)^2 = 2x(1.2)^2 = 2x \times 1.44 = 2.88x \]

### Step 4: Calculate the total value after 2 years
Now we can find the total value after 2 years:
\[ \text{Total Value} = \text{Value of A's car} + \text{Value of B's investment} \]
\[ = \frac{734694x}{1000000} + 2.88x \]
To combine these, we convert \( 2.88x \) into a fraction:
\[ 2.88x = \frac{2880000x}{1000000} \]
Now, adding these:
\[ \text{Total Value} = \frac{734694x + 2880000x}{1000000} = \frac{3614694x}{1000000} \]

### Step 5: Calculate the increase in total value
The initial total amount was:
\[ \text{Initial Total} = x + 2x = 3x \]
The increase in total value is:
\[ \text{Increase} = \text{Total Value} - \text{Initial Total} = \frac{3614694x}{1000000} - 3x \]
Converting \( 3x \) into a fraction:
\[ 3x = \frac{3000000x}{1000000} \]
Now, calculating the increase:
\[ \text{Increase} = \frac{3614694x - 3000000x}{1000000} = \frac{614694x}{1000000} \]

### Step 6: Calculate the percentage increase
The percentage increase is given by:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Initial Total}} \right) \times 100 \]
Substituting the values:
\[ = \left( \frac{\frac{614694x}{1000000}}{3x} \right) \times 100 = \left( \frac{614694}{3000000} \right) \times 100 \]
Calculating this gives:
\[ = \frac{614694 \times 100}{3000000} \approx 20.49\% \]

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