How to Calculate the Compound Interest Rate for Money to Double in a Given Time Frame
Understanding compound interest is essential for anyone involved in finance or investment. This article will guide you through the process of calculating the compound interest rate when a specific amount of money grows to a target amount over a certain period. We will use various examples and formulas to illustrate the steps, ensuring clarity and practical applicability.
Compound Interest Formula Explained
The formula for compound interest is:
$$ A P left( 1 frac{r}{100} right)^n $$where:
A is the amount of money accumulated after n years, including interest. P is the principal amount (initial investment). r is the annual interest rate (in percentage). n is the number of years the money is invested or borrowed for.Example 1: Finding the Compound Interest Rate
Suppose you have ( P Rs. 1250 ) and you want to know the annual interest rate that would make this grow to ( A Rs. 1800 ) in 2 years. Here is the step-by-step process:
Step-by-Step Calculation
Given:
( A 1800 ) ( P 1250 ) ( n 2 )Using the compound interest formula:
$$ 1800 1250 left( 1 frac{r}{100} right)^2 $$Divide both sides by 1250:
$$ frac{1800}{1250} left(1 frac{r}{100} right)^2 $$Calculate the left side:
$$ 1.44 left(1 frac{r}{100} right)^2 $$Take the square root of both sides:
$$ sqrt{1.44} 1 frac{r}{100} $$Calculate the square root:
$$ 1.2 1 frac{r}{100} $$Subtract 1 from both sides:
$$ 0.2 frac{r}{100} $$Multiply both sides by 100:
$$ r 20 $$Therefore, the rate of compound interest is 20% per annum.
Additional Examples
Let's explore a few more examples to further solidify the concept:
Example 2: Finding the Compound Interest Rate for Rs. 1100 to Become Rs. 1400 in 2 Years
Given:
( A 1400 ) ( P 1100 ) ( n 2 )Using the formula:
$$ 1400 1100 left( 1 frac{r}{100} right)^2 $$Divide both sides by 1100:
$$ frac{1400}{1100} left(1 frac{r}{100} right)^2 $$Calculate the left side:
$$ 1.27 left(1 frac{r}{100} right)^2 $$Take the square root of both sides:
$$ sqrt{1.27} 1 frac{r}{100} $$Calculate the square root:
$$ 1.127 1 frac{r}{100} $$Subtract 1 from both sides:
$$ 0.127 frac{r}{100} $$Multiply both sides by 100:
$$ R 12.7 $%$Therefore, the rate of compound interest is 12.7% per annum.
Example 3: Another Example Using Different Numbers
Given:
( A 1440 ) ( P 1000 ) ( n 2 )Using the formula:
$$ 1440 1000 left( 1 frac{r}{100} right)^2 $$Divide both sides by 1000:
$$ frac{1440}{1000} left(1 frac{r}{100} right)^2 $$Calculate the left side:
$$ 1.44 left(1 frac{r}{100} right)^2 $$Take the square root of both sides:
$$ sqrt{1.44} 1 frac{r}{100} $$Calculate the square root:
$$ 1.2 1 frac{r}{100} $$Subtract 1 from both sides:
$$ 0.2 frac{r}{100} $$Multiply both sides by 100:
$$ R 20 $%$Therefore, the rate of compound interest is 20% per annum.
Verification Steps
To ensure that our calculations are correct, let's verify by substituting the derived rates back into the compound interest formula:
Verification of the first example:
$$ 1800 1250 left( 1 0.2 right)^2 $$Calculate the right side:
$$ 1800 1250 left( 1.2 right)^2 $$ $$ 1800 1250 (1.44) $$ $$ 1800 1800 $%$Verification confirms that the rate is correct.