On a certain sum and at the certain rate of interest, CI at the.end of two years is Rs 252 while CI at the end of four years is Rs 556.92. Find that amount?

Options

A. A)Rs 1000
B. B)Rs 1500
C. C)Rs 1600
D. D)Rs. 1200

Answer

To solve the problem step by step, we will use the information provided about the compound interest (CI) at the end of 2 years and 4 years.

### Step 1: Understand the given information
We know:
- CI at the end of 2 years = Rs 252
- CI at the end of 4 years = Rs 556.92

### Step 2: Calculate the interest earned in the 3rd and 4th years
The interest earned from the end of the 2nd year to the end of the 4th year can be calculated as:
\[ \text{Interest from Year 2 to Year 4} = \text{CI at end of 4 years} - \text{CI at end of 2 years} \]
\[ = 556.92 - 252 = 304.92 \]

### Step 3: Determine the interest for each year
Since the interest is compounded annually, the interest earned in the 3rd and 4th years is the same, and we can denote it as \( I \). Therefore:
\[ 2I = 304.92 \]
\[ I = \frac{304.92}{2} = 152.46 \]

### Step 4: Calculate the total amount at the end of 2 years
The total amount at the end of 2 years (A2) can be calculated as:
\[ A2 = P + \text{CI at end of 2 years} \]
Let \( P \) be the principal amount. Thus:
\[ A2 = P + 252 \]

### Step 5: Calculate the total amount at the end of 4 years
The total amount at the end of 4 years (A4) can be calculated as:
\[ A4 = P + \text{CI at end of 4 years} \]
Thus:
\[ A4 = P + 556.92 \]

### Step 6: Set up the equations
From the previous steps, we have two equations:
1. \( A2 = P + 252 \)
2. \( A4 = P + 556.92 \)

### Step 7: Relate the two amounts
Since the amount at the end of 4 years is also equal to the amount at the end of 2 years plus the interest for the next 2 years, we can write:
\[ A4 = A2 + 304.92 \]

### Step 8: Substitute A2 in the equation
Substituting the value of \( A2 \) into the equation:
\[ P + 556.92 = (P + 252) + 304.92 \]

### Step 9: Simplify the equation
Now, simplify the equation:
\[ P + 556.92 = P + 556.92 \]
This confirms our calculations are consistent.

### Step 10: Calculate the principal amount
Now we can find the principal amount using the interest earned in 2 years:
We know that:
\[ \text{CI for 2 years} = \frac{P \times r \times 2}{100} \]
Where \( r \) is the rate of interest. We have already calculated the interest earned in 2 years as Rs 252.

Using the relationship between the amounts:
1. \( A2 = P + 252 \)
2. \( A4 = P + 556.92 \)

We can find that:
\[ A2 = P + 252 \]

Now, we can find the principal:
Let’s assume the rate of interest is 10% (from the video transcript):
\[ P = 100 \text{ units} \]
Thus:
\[ A2 = 110 \text{ units} \]
\[ 110 = P + 252 \]
So:
\[ P = 110 - 252 = -142 \text{ (not possible)} \]

Instead, we can directly calculate the principal:
Using the difference in interest:
\[ 21 \text{ units} = 252 \]
So:
\[ 1 \text{ unit} = 12 \]
Thus:
\[ P = 100 \text{ units} = 100 \times 12 = 1200 \]

### Final Answer
The principal amount \( P \) is Rs 1200.