A sum of 1000 after 3 years at compound interest becomes a certain amount that is equal to the amount that is the result of a 3 year depreciation from 1728. Find the difference between the rates of C.I. and depreciation. (Given C.I. is 10% p.a.) (Approximately)

Options

A. `3.33%`
B. `1.66%`
C. `3%`
D. `2%`

Answer

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

### Step 1: Understand the Problem
We need to find the difference between the rates of compound interest (C.I.) and depreciation. We know that the principal amount is ₹1000, the C.I. rate is 10% per annum, and after 3 years, this amount becomes equal to the amount after 3 years of depreciation from ₹1728.

### Step 2: Calculate the Amount after 3 Years of C.I.
Using the formula for compound interest:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = Amount after time \(n\)
- \(P\) = Principal amount (₹1000)
- \(r\) = Rate of interest (10%)
- \(n\) = Number of years (3)

Substituting the values:
\[ A = 1000 \left(1 + \frac{10}{100}\right)^3 \]
\[ A = 1000 \left(1 + 0.1\right)^3 \]
\[ A = 1000 \left(1.1\right)^3 \]
Calculating \(1.1^3\):
\[ 1.1^3 = 1.331 \]
So,
\[ A = 1000 \times 1.331 = 1331 \]

### Step 3: Set Up the Depreciation Equation
Let the rate of depreciation be \(X\% \). The amount after 3 years of depreciation from ₹1728 can be calculated as:
\[ A = 1728 \left(1 - \frac{X}{100}\right)^3 \]
Since we established that the amount after 3 years of C.I. is equal to the depreciated amount:
\[ 1331 = 1728 \left(1 - \frac{X}{100}\right)^3 \]

### Step 4: Solve for \(X\)
Now, we can rearrange the equation:
\[ \left(1 - \frac{X}{100}\right)^3 = \frac{1331}{1728} \]
Calculating \(\frac{1331}{1728}\):
\[ \frac{1331}{1728} = \left(\frac{11}{12}\right)^3 \]
Taking the cube root on both sides:
\[ 1 - \frac{X}{100} = \frac{11}{12} \]
Now, solving for \(X\):
\[ \frac{X}{100} = 1 - \frac{11}{12} \]
\[ \frac{X}{100} = \frac{1}{12} \]
\[ X = \frac{100}{12} = 8.33 \]

### Step 5: Calculate the Difference Between C.I. and Depreciation
Now we find the difference between the C.I. rate and the depreciation rate:
\[ \text{Difference} = 10 - 8.33 = 1.67 \]

### Final Answer
The difference between the rates of C.I. and depreciation is approximately **1.67%**.

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