Hari Lal and Hari Prasad have equal amounts. Hari Lal invested all his amount at `10%` compounded annually for 2 years and Hari Prasad invested `1//4` at `10%` compound interest (annually) and rest at `r%` per annum at simple interest for the same 2 years period. The amount received by both at the end of 2 year is same. What is the value of r?

Options

A. `14%`
B. `12.5%`
C. `10.5%`
D. `11%`

Answer

To solve the problem, we need to find the value of \( r \) such that the total amounts received by Hari Lal and Hari Prasad after 2 years are equal.

### Step 1: Define the initial amount
Let the equal amount that both Hari Lal and Hari Prasad have be \( P \).

### Step 2: Calculate the amount received by Hari Lal
Hari Lal invests his entire amount \( P \) at \( 10\% \) compounded annually for 2 years. The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = final amount
- \( P \) = principal amount
- \( r \) = rate of interest
- \( n \) = number of years

For Hari Lal:
\[ A_{HL} = P \left(1 + \frac{10}{100}\right)^2 = P \left(1.1\right)^2 = P \times 1.21 = 1.21P \]

### Step 3: Calculate the amount received by Hari Prasad
Hari Prasad invests \( \frac{1}{4}P \) at \( 10\% \) compound interest and the remaining \( \frac{3}{4}P \) at \( r\% \) simple interest for 2 years.

For the amount invested at \( 10\% \):
\[ A_{HP1} = \frac{1}{4}P \left(1 + \frac{10}{100}\right)^2 = \frac{1}{4}P \times 1.21 = \frac{1.21P}{4} \]

For the amount invested at \( r\% \):
The formula for simple interest is:
\[ A = P + SI = P + \left(P \times \frac{r}{100} \times n\right) \]
For the remaining amount:
\[ A_{HP2} = \frac{3}{4}P + \left(\frac{3}{4}P \times \frac{r}{100} \times 2\right) = \frac{3}{4}P + \frac{3rP}{200} \]

### Step 4: Combine amounts for Hari Prasad
Now, combine both amounts for Hari Prasad:
\[ A_{HP} = A_{HP1} + A_{HP2} = \frac{1.21P}{4} + \left(\frac{3}{4}P + \frac{3rP}{200}\right) \]
\[ A_{HP} = \frac{1.21P}{4} + \frac{3P}{4} + \frac{3rP}{200} \]
To combine these, convert \( \frac{3P}{4} \) to a fraction with a denominator of 4:
\[ A_{HP} = \frac{1.21P}{4} + \frac{3P}{4} = \frac{1.21P + 3P}{4} = \frac{(1.21 + 3)P}{4} = \frac{4.21P}{4} = 1.0525P \]
Now add the simple interest part:
\[ A_{HP} = 1.0525P + \frac{3rP}{200} \]

### Step 5: Set the amounts equal
Since both amounts are equal:
\[ 1.21P = 1.0525P + \frac{3rP}{200} \]

### Step 6: Simplify the equation
Divide through by \( P \) (assuming \( P \neq 0 \)):
\[ 1.21 = 1.0525 + \frac{3r}{200} \]
Subtract \( 1.0525 \) from both sides:
\[ 1.21 - 1.0525 = \frac{3r}{200} \]
\[ 0.1575 = \frac{3r}{200} \]

### Step 7: Solve for \( r \)
Multiply both sides by 200:
\[ 200 \times 0.1575 = 3r \]
\[ 31.5 = 3r \]
Divide by 3:
\[ r = \frac{31.5}{3} = 10.5 \]

### Final Answer
The value of \( r \) is \( 10.5\% \).