The compound interest (compounded annually) on ₹9300 for 2 years@ R% pa is ₹4092. Had the rate of interest been (R-10)%, what would have been the interest on the same sum of money for the same time? (2 years)

Options

A. ₹ 2046
B. ₹ 1974
C. ₹ 2027
D. ₹ 1953

Answer

To solve the problem step by step, we will follow the outlined procedure in the video transcript.

### Step 1: Identify the given values
- Principal (P) = ₹9300
- Compound Interest (CI) for 2 years = ₹4092
- Time (T) = 2 years

### Step 2: Calculate the Amount (A)
The amount (A) after 2 years can be calculated using the formula:
\[ A = P + CI \]
Substituting the values:
\[ A = 9300 + 4092 = 13,392 \]

### Step 3: Use the Compound Interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the known values:
\[ 13,392 = 9300 \left(1 + \frac{R}{100}\right)^2 \]

### Step 4: Simplify the equation
To isolate \( R \), we first divide both sides by 9300:
\[ \frac{13,392}{9300} = \left(1 + \frac{R}{100}\right)^2 \]
Calculating the left side:
\[ \frac{13,392}{9300} = 1.44 \]
Thus, we have:
\[ 1.44 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: Take the square root
Taking the square root of both sides:
\[ \sqrt{1.44} = 1 + \frac{R}{100} \]
Calculating the square root:
\[ 1.2 = 1 + \frac{R}{100} \]

### Step 6: Solve for R
Subtracting 1 from both sides:
\[ 0.2 = \frac{R}{100} \]
Multiplying both sides by 100:
\[ R = 20 \]

### Step 7: Calculate the new rate of interest
The new rate of interest is:
\[ R - 10 = 20 - 10 = 10\% \]

### Step 8: Calculate the new compound interest for the new rate
Using the new rate (10%) for the same principal and time:
\[ A = P \left(1 + \frac{10}{100}\right)^2 \]
Substituting the values:
\[ A = 9300 \left(1 + 0.1\right)^2 \]
\[ A = 9300 \left(\frac{11}{10}\right)^2 \]
\[ A = 9300 \times \frac{121}{100} \]
Calculating:
\[ A = 9300 \times 1.21 = 11,253 \]

### Step 9: Calculate the new compound interest
The new compound interest (CI) is:
\[ CI = A - P \]
Substituting the values:
\[ CI = 11,253 - 9300 = 1,953 \]

### Final Answer
The compound interest at the new rate of 10% for 2 years is ₹1,953.

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