If the simple interest on certain sum of money be Rs. 40 for 2 years and the compound interest on the same sum at the same rate and for the same time be Rs. 45. What is the principal ?

Options

A. Rs.70
B. Rs.90
C. Rs.85
D. Rs.80

Answer

To find the principal amount given the simple interest and compound interest, we can follow these steps:

### Step 1: Understand the given information
- Simple Interest (SI) for 2 years = Rs. 40
- Compound Interest (CI) for 2 years = Rs. 45

### Step 2: Calculate the Simple Interest for 1 year
Since the simple interest is the same every year, we can find the interest for 1 year by dividing the total simple interest by the number of years.
\[ \text{SI for 1 year} = \frac{\text{SI for 2 years}}{2} = \frac{40}{2} = 20 \text{ Rs.} \]

### Step 3: Establish the relationship between SI and CI
The difference between the compound interest and the simple interest for the same time period can be used to find the rate of interest.
\[ \text{Difference} = \text{CI} - \text{SI} = 45 - 40 = 5 \text{ Rs.} \]

### Step 4: Relate the difference to the principal and rate
The difference of Rs. 5 represents the interest earned on the initial simple interest of Rs. 20 (for the first year) in the second year. This can be expressed as:
\[ \text{Difference} = \text{SI for 1 year} \times \frac{r}{100} \]
Where \( r \) is the rate of interest.

### Step 5: Set up the equation
Substituting the known values into the equation:
\[ 5 = 20 \times \frac{r}{100} \]

### Step 6: Solve for \( r \)
Rearranging the equation to find \( r \):
\[ \frac{r}{100} = \frac{5}{20} \implies r = \frac{5 \times 100}{20} = 25\% \]

### Step 7: Use the rate to find the principal
Now, we can use the simple interest formula to find the principal. The formula for simple interest is:
\[ \text{SI} = \frac{P \times r \times t}{100} \]
Where \( P \) is the principal, \( r \) is the rate, and \( t \) is the time in years. We know:
- \( \text{SI} = 40 \)
- \( r = 25\% \)
- \( t = 2 \)

Substituting these values into the formula:
\[ 40 = \frac{P \times 25 \times 2}{100} \]

### Step 8: Solve for \( P \)
Rearranging the equation to find \( P \):
\[ 40 = \frac{50P}{100} \implies 40 = \frac{P}{2} \implies P = 40 \times 2 = 80 \]

### Conclusion
The principal amount is Rs. 80.

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