A sum of money at simple interest amount to Rs. 1260 in 2 yr and Rs. 1350 in 5 yr, then the rate per cent per annum is

Options

A. `30%`
B. `10%`
C. `2.5%`
D. `5%`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Identify the given information
We know:
- Amount after 2 years (A1) = Rs. 1260
- Amount after 5 years (A2) = Rs. 1350

### Step 2: Calculate the difference in amounts
The difference in amounts over the period from 2 years to 5 years is:
\[ A2 - A1 = 1350 - 1260 = Rs. 90 \]

### Step 3: Determine the time difference
The time difference between the two amounts is:
\[ 5 \text{ years} - 2 \text{ years} = 3 \text{ years} \]

### Step 4: Calculate the interest for 3 years
The interest earned over these 3 years is Rs. 90, which means:
- Interest for 3 years = Rs. 90

### Step 5: Calculate the interest per year
To find the interest earned in one year, we divide the total interest by the number of years:
\[ \text{Interest per year} = \frac{90}{3} = Rs. 30 \]

### Step 6: Calculate the total interest for 2 years
Now, we can calculate the total interest earned in 2 years:
\[ \text{Total Interest for 2 years} = 30 \times 2 = Rs. 60 \]

### Step 7: Calculate the principal amount
We know that the amount after 2 years is the sum of the principal and the interest earned:
\[ A1 = P + \text{SI for 2 years} \]
Substituting the known values:
\[ 1260 = P + 60 \]
Thus, we can find the principal:
\[ P = 1260 - 60 = Rs. 1200 \]

### Step 8: Use the formula for Simple Interest to find the rate
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- SI = Interest earned
- P = Principal
- R = Rate of interest per annum
- T = Time in years

Substituting the known values:
\[ 60 = \frac{1200 \times R \times 2}{100} \]

### Step 9: Rearranging the equation to solve for R
Multiplying both sides by 100:
\[ 6000 = 1200 \times R \times 2 \]
Now simplify:
\[ 6000 = 2400R \]
Dividing both sides by 2400:
\[ R = \frac{6000}{2400} = \frac{25}{10} = 2.5 \]

### Step 10: Conclusion
Thus, the rate of interest per annum is:
\[ R = 2.5\% \]