Manoj invested a sum at x% per annum at C.I. If first year and second year C.1 on that sum is Rs. 845 and Rs. 910 Find Amount invested?

Options

A. Rs. 10985
B. Rs. 10000
C. Rs. 13000
D. Rs. 10900

Answer

To solve the problem step by step, we need to find the amount invested by Manoj based on the compound interest he received in the first and second years.

### Step 1: Understand the given information
We know that:
- The compound interest for the first year (CI1) = Rs. 845
- The compound interest for the second year (CI2) = Rs. 910

### Step 2: Calculate the interest earned in the second year
The interest earned in the second year can be calculated by subtracting the first year's interest from the second year's interest:
\[ \text{Interest for the second year} = CI2 - CI1 = 910 - 845 = 65 \text{ Rs.} \]

### Step 3: Relate the interest earned in the second year to the principal and rate of interest
The interest earned in the second year is also calculated based on the principal amount (P) and the rate of interest (R). The formula for the interest earned in the second year is:
\[ \text{Interest for the second year} = P \times R \times \frac{1}{100} \]
Where R is the rate of interest per annum.

### Step 4: Set up the equation for the interest earned in the second year
From the previous step, we can write:
\[ 65 = P \times R \times \frac{1}{100} \]
This implies:
\[ P \times R = 6500 \quad \text{(Equation 1)} \]

### Step 5: Relate the first year's interest to the principal and rate of interest
The interest earned in the first year is given by:
\[ CI1 = P \times \frac{R}{100} \]
Substituting the value of CI1:
\[ 845 = P \times \frac{R}{100} \]
This implies:
\[ P \times R = 84500 \quad \text{(Equation 2)} \]

### Step 6: Solve the two equations
Now we have two equations:
1. \( P \times R = 6500 \) (from step 4)
2. \( P \times R = 84500 \) (from step 5)

From Equation 1 and Equation 2, we can express R in terms of P:
\[ R = \frac{6500}{P} \]
Substituting this into Equation 2:
\[ P \times \frac{6500}{P} = 84500 \]
This simplifies to:
\[ 6500 = 84500 \]
This is incorrect, indicating we need to derive R from the interest values directly.

### Step 7: Calculate the rate of interest
From the interest earned in the second year:
\[ R = \frac{65 \times 100}{845} \]
Calculating this gives:
\[ R = \frac{6500}{845} \approx 7.69\% \]

### Step 8: Substitute R back to find P
Now we can substitute R back into either equation to find P. Using Equation 1:
\[ P \times 7.69 = 6500 \]
\[ P = \frac{6500}{7.69} \approx 845.00 \]

### Step 9: Calculate the principal amount
Using the derived rate of interest, we can find the principal amount:
Using the first year interest:
\[ 845 = P \times \frac{7.69}{100} \]
Solving for P:
\[ P = \frac{845 \times 100}{7.69} \approx 10985.00 \]

### Conclusion
Thus, the amount invested by Manoj is approximately Rs. 10,985.