Comparison between Simple Interest and Compound Interest for the same principal amount.

Interest is of two kinds – Simple Interest and Compound Interest.

In the problems of interest, if the type of interest is not mentioned, we will consider it as simple interest.

If total interest on principal P for t years at r% per annum be I, then I = \(\frac{P × R × T}{100}\).

At r% per annum compound interest, if amount on Principal P for n years be A, then A = P\(\left ( 1 + \frac{r}{100} \right )^{n}\)

Banks and Post office generally calculate interest in different manner.

Simple interest for 1 year is calculated, and then they find the amount. This amount becomes the principal for the next year. This calculation is repeated every year for which the principal amount is kept as deposit. The difference between the final amount and the original amount is the compound interest (CI).

In case of simple interest the principal remains the same for the whole period of loan but in case of compound interest, the principal changes every year.

**1.** Find the difference between compound interest and simple interest for a principal amount of $10000 for 2 years at 5% rate of interest.

**Solution:**

Given, simple interest for 2 years = \(\frac{10000 × 5 × 2}{100}\)

= $1000

Interest for the first year = \(\frac{10000 × 5 × 1}{100}\)

= $500

Amount at the end of first year = $10000 + $500

= $10500

Interest for the second year = \(\frac{10500 × 5 × 1}{100}\)

= $525

Amount at the end of second year = $10500 + $525

= $11025

Therefore, compound interest = A – P

= final amount – original principal

= $11025 - $10000

= $1025

Therefore, difference between compound interest and simple interest = $1025 - $1000

= $25

**2. **Jason lends $ 10,000 to David at the simple interest rate of 10% for 2 years and $ 10,000 to James at the compound interest rate of 10% for 2 years. Find the sum of money that David and James will return to Jason after 2 years to repay the loan. Who will pay more and by how much?

**Solution:**

**For David:**

Principal (P) = $ 10000

Rate of Interest (R) = 10%

Time (T) = 2 Years

Therefore, interest = I = \(\frac{P × R × T}{100}\)

= \(\frac{10000 × 10 × 2}{100}\)

= $ 2000.

Therefore, amount A = P + I = $ 10000 + $ 2000 = $ 12000

Therefore David will repay $ 12,000 to Jason after 2 years.

**For James:**

Principal (P) = $ 10000

Rate of Interest (R) = 10%

Time (n) = 2 Years

From A = P \(\left ( 1 + \frac{r}{100} \right )^{n}\), we get

A = $ 10000 × \(\left ( 1 + \frac{10}{100} \right )^{2}\)

= $ 10000 × \(\left (\frac{110}{100} \right )^{2}\)

= $ 10000 × \(\left (\frac{11}{10} \right )^{2}\)

= $ 100 × 121

= $ 12100

Therefore, James will repay $ 12,100.

Now, $ 12100 > $ 12000, so, James will pay more. He will pay $ 12100 - $ 12000, i.e., $ 100 more than David.

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