A sum of money at the simple interest amounts to ₹5852 in 3 years and ₹7788 in 7 years. What is the rate of interest per annum?

Options

A. a) 8
B. b) 9
C. c) 1
D. d) 11

Explanation

To find the rate of interest per annum based on the given amounts at simple interest, we can follow these steps:

### Step 1: Understand the Given Information
We have two amounts:
- Amount after 3 years (A1) = ₹5852
- Amount after 7 years (A2) = ₹7788

### Step 2: Calculate the Simple Interest for the Given Periods
The simple interest (SI) can be calculated using the formula:
\[ SI = A - P \]
where \( A \) is the amount and \( P \) is the principal.

Let \( P \) be the principal amount and \( r \) be the rate of interest per annum.

For the first amount after 3 years:
\[ SI_1 = A_1 - P = 5852 - P \]

For the second amount after 7 years:
\[ SI_2 = A_2 - P = 7788 - P \]

### Step 3: Express the Simple Interest in Terms of Principal and Rate
The simple interest can also be expressed as:
\[ SI = \frac{P \times r \times t}{100} \]
where \( t \) is the time in years.

For the first amount after 3 years:
\[ SI_1 = \frac{P \times r \times 3}{100} \]

For the second amount after 7 years:
\[ SI_2 = \frac{P \times r \times 7}{100} \]

### Step 4: Set Up the Equations
From the above expressions, we can set up the following equations:
1. \( 5852 - P = \frac{P \times r \times 3}{100} \) (Equation 1)
2. \( 7788 - P = \frac{P \times r \times 7}{100} \) (Equation 2)

### Step 5: Subtract the Two Equations
Subtract Equation 1 from Equation 2 to eliminate \( P \):
\[ (7788 - P) - (5852 - P) = \frac{P \times r \times 7}{100} - \frac{P \times r \times 3}{100} \]
This simplifies to:
\[ 7788 - 5852 = \frac{P \times r \times (7 - 3)}{100} \]
\[ 1936 = \frac{P \times r \times 4}{100} \]

### Step 6: Rearrange to Find \( P \times r \)
Rearranging gives:
\[ P \times r = \frac{1936 \times 100}{4} \]
\[ P \times r = 48400 \]

### Step 7: Substitute Back to Find \( P \)
Now, substitute \( P \) back into one of the original equations to find \( r \). We can use Equation 1:
\[ 5852 - P = \frac{P \times r \times 3}{100} \]
Substituting \( P \times r = 48400 \):
\[ 5852 - P = \frac{48400 \times 3}{100P} \]
This gives us a quadratic equation in terms of \( P \). However, we can also find \( r \) directly from the difference in amounts.

### Step 8: Calculate the Rate of Interest
The difference in amounts over the difference in time gives:
\[ 7788 - 5852 = 1936 \quad \text{(this is the total interest for 4 years)} \]
Now, the simple interest for 4 years is:
\[ SI = \frac{P \times r \times 4}{100} \]
Thus,
\[ 1936 = \frac{P \times r \times 4}{100} \]
We already found \( P \times r = 48400 \), so:
\[ r = \frac{1936 \times 100}{4P} \]
Now, we can find \( r \) by substituting \( P \) back or using the total interest over the total time.

### Step 9: Final Calculation
Using the total interest:
\[ r = \frac{1936 \times 100}{4 \times P} \]
To find \( r \), we can also use the total interest per annum:
\[ r = \frac{1936}{4} = 484 \]
Thus, the rate of interest per annum is:
\[ r = 11\% \]

### Conclusion
The rate of interest per annum is **11%**.