Math Problem Statement

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:

Solution

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+PRT100A = P + P \cdot \frac{R \cdot T}{100} where PP is the principal, RR is the rate of interest, and TT is the time in years.

So, P+PR5100=P+0.5PP + P \cdot \frac{R \cdot 5}{100} = P + 0.5P P(1+5R100)=1.5PP \cdot \left(1 + \frac{5R}{100}\right) = 1.5P Solving for RR: 1+5R100=1.51 + \frac{5R}{100} = 1.5 5R100=0.5\frac{5R}{100} = 0.5 R=0.5×1005=10%R = \frac{0.5 \times 100}{5} = 10\%

Thus, the rate of interest is 10%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%10\%.

The formula for compound interest is: A=P(1+R100)TA = P \left(1 + \frac{R}{100}\right)^T Where:

  • AA is the amount after time TT,
  • PP is the principal,
  • RR is the rate of interest,
  • TT is the time period.

Substituting the values: A=10,000(1+10100)2=10,000×(1.1)2A = 10,000 \left(1 + \frac{10}{100}\right)^2 = 10,000 \times \left(1.1\right)^2 A=10,000×1.21=12,100A = 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=12,10010,000=2,100CI = 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.

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Math Problem Analysis

Mathematical Concepts

Simple Interest
Compound Interest
Percentage Increase
Rate of Interest

Formulas

Simple Interest: A = P + P × (R × T) / 100
Compound Interest: A = P × (1 + R/100)^T

Theorems

Interest Rate Calculation
Exponential Growth in Compound Interest

Suitable Grade Level

Grades 9-12