There is 50% increase in an amount in 5 years at simple interest. The compound interest on ₹10,000 after 2 years at the same rate will be:

We are given the following details:

1. **Increase at Simple Interest:** 50% increase in 5 years.
2. **Principal Amount for Compound Interest:** ₹10,000
3. **Time for Compound Interest Calculation:** 2 years

### Step 1: Determine the Rate of Interest
At simple interest, the amount increases by 50% in 5 years. This means:
- The amount becomes 150% of the principal in 5 years.
- In simple interest, this is expressed as:  
  \[
  A = P + P \cdot \frac{R \cdot T}{100}
  \]
  where \(P\) is the principal, \(R\) is the rate of interest, and \(T\) is the time in years.

So,
\[
P + P \cdot \frac{R \cdot 5}{100} = P + 0.5P
\]
\[
P \cdot \left(1 + \frac{5R}{100}\right) = 1.5P
\]
Solving for \(R\):
\[
1 + \frac{5R}{100} = 1.5
\]
\[
\frac{5R}{100} = 0.5
\]
\[
R = \frac{0.5 \times 100}{5} = 10\%
\]

Thus, the rate of interest is \(10\%\) per annum.

### Step 2: Calculate Compound Interest
We now calculate the compound interest on ₹10,000 for 2 years at a rate of \(10\%\).

The formula for compound interest is:
\[
A = P \left(1 + \frac{R}{100}\right)^T
\]
Where:
- \(A\) is the amount after time \(T\),
- \(P\) is the principal,
- \(R\) is the rate of interest,
- \(T\) is the time period.

Substituting the values:
\[
A = 10,000 \left(1 + \frac{10}{100}\right)^2 = 10,000 \times \left(1.1\right)^2
\]
\[
A = 10,000 \times 1.21 = 12,100
\]

The total amount after 2 years is ₹12,100. The compound interest is the difference between the amount and the principal:
\[
CI = 12,100 - 10,000 = 2,100
\]

### Final Answer:
The compound interest on ₹10,000 after 2 years at the same rate is ₹2,100.

Would you like any further details or have questions? Here are some related questions for deeper understanding:

1. How does the formula for simple interest differ from compound interest?
2. What happens if the interest is compounded quarterly instead of annually?
3. How would the compound interest change if the time period were extended to 3 years?
4. Can you calculate the total amount for a 10-year period using the same rate and compound interest?
5. What is the difference between nominal interest rate and effective interest rate?

**Tip:** Always verify whether the interest rate is compounded annually, semi-annually, or quarterly to avoid miscalculations.

Find out how to calculate the compound interest on ₹10,000 after 2 years at a 10% interest rate. Learn the difference between simple and compound interest, including key formulas and concepts for high school-level mathematics.

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