Understanding Simple and Compound Interest in Maths

Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time. In simple interest, the principal amount is always the same, unlike compound interest where we add the interest of the previous year's principal to calculate the interest of the next year. Compound interest is an interest accumulated on the principal and interest together over a given time period. The interest accumulated on a principal over a period of time is also accounted under the principal. In this article you will learn about compound interest, simple interest formulas and some solved examples of compound interest as repeated simple interest.

Interest Formulas for SI and CI

Simple interest and compound interest are both included in the interest formula. Interest is the cost a borrower pays a lender for a loan. This additional sum, or the interest, must be paid in addition to the loan itself. The compound interest formula and the simple interest formula are both discussed in the interest formula.

The Interest formulas are given as,

Simple Interest, S.I. $= \dfrac{\mathrm{P} \times \mathrm{R} \times \mathrm{T}}{100}$

Where,

P – Principal amount

R – Rate of interest

T – Time period (Number of years)

Compound Interest = Amount – Principal

Where

Amount, $A=P\left(1+\dfrac{r}{n}\right)^{n t}$

Here, P = principal

r = rate of interest

t = time in years

n = number of times interest is compounded per year

Difference Between Simple and Compound Interest

Following are the difference between simple and compound interest:

Compound Interest as Repeated Simple Interest

Let's learn how to calculate compound interest as repeated simple interest.

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

Clearly, the compound interest on a principal P for 1 year =simple interest on a principal for 1 year, when the interest is calculated yearly.

The compound interest on a principal for 2 years > the simple interest on the same principal for 2 years.

Remember, if the principal = P, amount at the end of the period = A and compound interest = CI, CI = A - P

Example: Find the compound interest on $\$ 14000$ at the rate of interest $5 \%$ per annum.

Ans:

$\text { Interest for the first year }=\dfrac{14000 \times 5 \times 1}{100}$ = $\$700$

Amount at the end of the first year $=\$ 14000+\$ 700$

$=\$ 14700$

Principal for the second year $=\$ 14700$

Interest for the second year $=\dfrac{14700 \times 5 \times 1}{100}$

$=\$ 735$

Amount at the end of the second year $=\$ 14700+\$ 735$

$=\$ 15435$

Therefore, compound interest $=\mathrm{A}-\mathrm{P}$

$=$ final amount - original principal

$=\$ 15435-\$ 14000$

$=\$ 1435$

Solved Examples of Simple Interest and Compound Interest

Let's look into some of the solved examples on how to calculate simple interest and compound interest.

Q 1. Namita borrowed Rs 50,000 for 3 years at the rate of 3.5% per annum. Find the interest accumulated at the end of 3 years.​

Ans: Given that:-

Principal $(\mathrm{P})=R s .50,000$

Rate $(\mathrm{R})=3.5 \%$

Time $(T)=3$ years

To find:- Simple Interest $=$ ?

The formula to be used:-

$S I=\dfrac{P \times R \times T}{100}$

Now, plug all the given values:-

$S I =\dfrac{50,000 \times 3.5 \times 3}{100}$

$=\mathrm{RS} .5250$

Hence, the value of SI is Rs. $\mathbf{5 2 5 0}$

Q2. Find the amount if Rs. 10,000 is invested at $10 \%$ p.a. for 2 years when compounded annually?

Ans: We know $A=P(1+\dfrac{R}{100})^n$

From given data $P=10,000$

$R=10 \%$

$n=2 \text { years }$

Substituting the input values we have the equation as under

$A=10,000(1+ \dfrac{10}{100})^2$

$=10,000(1+0.1)^2$

$=10,000(1.1)^2$

$=10,000(1.21)$

$=\text { Rs. } 12,100$

Solved Examples of Compound Interest as Repeated Simple Interest

Let's look into some solved examples of compound interest as repeated simple interest.

Q 1. Find the compound interest on $\$ 30000$ for 3 years at the rate of interest $4 \%$ per annum.

Ans:

$\text { Interest for the first year }=\dfrac{30000 \times 4 \times 1}{100}$

$\quad=\$ 1200$

Amount at the end of first year $=\$ 30000+\$ 1200$

$=\$ 31200$

Principal for the second year $=\$ 31200$

Interest for the second year $=\dfrac{31200 \times 4 \times 1}{100}$

$=\$ 1248$

Amount at the end of second year $=\$ 31200+\$ 1248$

$=\$ 32448$

Principal for the third year $=\$ 32448$

Interest for the third year $=\dfrac{32448 \times 4 \times 1}{100}$

$=\$ 1297.92$

Amount at the end of third year $=\$ 32448+\$ 1297.92$

$=\$ 33745.92$

Therefore, compound interest $=\mathrm{A}-\mathrm{P}$

$=$ final amount $-$ original principal

$=\$ 33745.92-\$ 30000$

$=\$ 3745.92$

Q2. Calculate the amount and compound interest on $\$ 10000$ for 3 years at $9 \%$ p.a.

Ans: Interest for the first year $=\dfrac{10000 \times 9 \times 1}{100}$

$=\$ 900$

Amount at the end of first year $=\$ 10000+\$ 900$

$=\$ 10900$

Principal for the second year $=\$ 10900$

Interest for the second year $=\dfrac{10900 \times 9 \times 1}{100}$

$=\$ 981$

Amount at the end of second year $=\$ 10900+\$ 981$

$=\$ 11881$

Principal for the third year $=\$ 11881$

Interest for the third year $=\dfrac{11881 \times 9 \times 1}{100}$

$=\$ 1069.29$

Amount at the end of third year $=\$ 11881+\$ 1069.29$

$=\$ 12950.29$

Therefore, the required amount $=\$ 12950.29$

Therefore, compound interest $=\mathrm{A}-\mathrm{P}$

= final amount - original principal

$=\$ 12950.29-\$ 10000$

$=\$ 2950.29$

Practice Questions

Q1. The simple interest on a sum of money for 2 years at 8% per annum is Rs.2400. What will be the compound interest on that sum at the same rate and for the same period?

Ans: Rs. 2496

Q2. A machine is purchased for Rs. 625000. Its value depreciates at the rate of 8% per annum. What will be its value after 2 years? Find the compound interest.

Ans: Rs. 529000

Summary

We learned in this article that simple interest paid or received over a specific period is a fixed percentage of the principal amount borrowed or lent. Borrowers must pay interest on interest as well as principal because compound interest accrues and is added to the accumulated interest from previous periods. Simple interest is preferable as a borrower because you are not paying interest on interest. Simple interest is easier to repay, whereas compound interest can help you build wealth over time because your earnings also earn money.