Understanding the Simple and Compound Interest Problem

The question asks us to first find the rate of interest from a simple interest scenario and then use that same rate to calculate the compound interest on a different principal amount over a specific time period. Let's break this down into steps.

Step 1: Finding the Simple Interest Rate

We are given that an amount increases by 75 percent in 12.5 years at simple interest. This 75 percent increase represents the total simple interest earned on the original principal amount.

• Let the principal amount be $\text{P}$.
• The simple interest ($\text{SI}$) earned is 75% of the principal, which is $0.75 \times \text{P}$.
• The time period ($\text{T}$) is 12.5 years.
• We need to find the simple interest rate ($\text{R}$) per annum.

The formula for simple interest is:

$\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}$

Substitute the given values into the formula:

$0.75 \times \text{P} = \frac{\text{P} \times \text{R} \times 12.5}{100}$

We can cancel $\text{P}$ from both sides of the equation (assuming $\text{P} \neq 0$):

$0.75 = \frac{\text{R} \times 12.5}{100}$

Now, solve for $\text{R}$:

$0.75 \times 100 = \text{R} \times 12.5$

$75 = \text{R} \times 12.5$}

$\text{R} = \frac{75}{12.5}$}

To make the division easier, we can multiply the numerator and denominator by 10:

$\text{R} = \frac{750}{125}$}

Dividing 750 by 125:

$\text{R} = 6$}

So, the simple interest rate is 6% per annum. This is the rate we will use for the compound interest calculation.

Step 2: Calculating the Compound Interest

Now, we need to find the compound interest on Rs. 20000 after 3 years at the rate of 6% per annum.

• Principal amount ($\text{P}'$) = Rs. 20000
• Time period ($\text{n}$) = 3 years
• Rate of interest ($\text{R}$) = 6% per annum (compounded annually, as is typical unless stated otherwise)

The formula for the amount ($\text{A}$) after $\text{n}$ years at compound interest is:

$\text{A} = \text{P}'\left(1 + \frac{\text{R}}{100}\right)^\text{n}$

Substitute the values:

$\text{A} = 20000\left(1 + \frac{6}{100}\right)^3$

$\text{A} = 20000\left(1 + 0.06\right)^3$

$\text{A} = 20000\left(1.06\right)^3$

Calculate $(1.06)^3$:

$(1.06)^3 = 1.06 \times 1.06 \times 1.06 = 1.1236 \times 1.06 = 1.191016$

Now, calculate the amount $\text{A}$:

$\text{A} = 20000 \times 1.191016$

$\text{A} = 23820.32$

The amount after 3 years is Rs. 23820.32.

The compound interest ($\text{CI}$) is the difference between the amount and the principal:

$\text{CI} = \text{A} - \text{P}'$

$\text{CI} = 23820.32 - 20000$

$\text{CI} = 3820.32$

The compound interest is Rs. 3820.32.

This calculation shows that the compound interest on Rs. 20000 after 3 years at 6% per annum is Rs. 3820.32.

Revision Table: Key Interest Formulas

Additional Information: Simple vs. Compound Interest

Understanding the difference between simple and compound interest is fundamental in financial mathematics.

• Simple Interest: In simple interest, the interest is calculated only on the original principal amount. The interest earned does not get added back to the principal for calculating interest in the subsequent periods. It's a straightforward calculation often used for short-term loans or deposits.
• Compound Interest: In compound interest, the interest for each period is calculated on the principal amount plus any accumulated interest from previous periods. This means that the interest itself earns interest. This leads to exponential growth over time and is commonly used for long-term investments and loans like mortgages. The frequency of compounding (annually, semi-annually, quarterly, etc.) affects the total interest earned or paid. In this problem, compounding is assumed to be annual as it's not specified otherwise.
• For the same principal, rate, and time period (greater than 1 year), compound interest is always greater than simple interest because of the interest-on-interest effect.