Introduction to Simple and Compound Interest
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.
FAQs on Simple Interest and Compound Interest
1. What is the fundamental definition of Simple Interest (SI) and how is it calculated?
Simple Interest is a fixed percentage of the original amount of money (principal) that is charged for a specific period. The key characteristic is that it is calculated only on the initial principal throughout the loan or investment term. The formula to calculate it is: SI = (P × R × T) / 100, where 'P' is the Principal, 'R' is the annual Rate of interest, and 'T' is the Time period in years.
2. How does Compound Interest (CI) work and what is its main difference from Simple Interest?
Compound Interest is the interest calculated on the initial principal and also on the accumulated interest from previous periods. It's often called “interest on interest.” The primary difference is:
- In Simple Interest, the principal amount never changes.
- In Compound Interest, the interest earned is added back to the principal at the end of each compounding period, creating a new, larger principal for the next period. This leads to faster growth of money over time.
3. What are the standard formulas used to find Simple and Compound Interest?
The standard formulas to determine Simple and Compound Interest are as follows:
- Simple Interest (SI) Formula: SI = (P × R × T) / 100
- Compound Interest (CI) Formula: First, you calculate the total Amount (A) using A = P(1 + R/100)T. Then, the Compound Interest is found by subtracting the principal from this amount: CI = A - P.
4. Can you provide a real-world example of how Simple Interest is applied?
A common real-world example of Simple Interest is a short-term personal loan or an auto loan. When you borrow money to buy a car, the interest is typically calculated on the original loan amount. Even though you make monthly payments, the interest calculation basis remains the same, making the total interest predictable and generally lower for the borrower compared to compound interest schemes.
5. Why does the principal stay the same in Simple Interest calculations but grow in Compound Interest?
The principal remains constant in Simple Interest because, by its very definition, interest is only calculated on the initial sum of money. The interest earned is treated separately and is not added back to the principal for subsequent calculations. In contrast, Compound Interest is designed to reinvest earnings. At the end of each period (e.g., a year), the interest is added to the principal, creating a new, higher principal for the next period to earn interest on.
6. From a financial perspective, why would a borrower prefer Simple Interest while a lender would prefer Compound Interest?
The preference depends on whether you are paying or receiving interest:
- A borrower prefers Simple Interest because the total interest paid over the loan's duration is lower and more predictable, making the loan less expensive.
- A lender or investor prefers Compound Interest because it maximises their returns. They earn interest not just on their initial capital but also on the interest that has already been earned, leading to exponential growth and greater wealth accumulation over time.
7. How can Compound Interest be understood as a form of “repeated Simple Interest”?
Thinking of Compound Interest as “repeated Simple Interest” is a great way to understand the concept without the formula. For a 3-year period:
- For Year 1: You calculate the Simple Interest on the original principal.
- For Year 2: You add the Year 1 interest to the original principal to get a new, larger principal. You then calculate Simple Interest for one year on this new amount.
- For Year 3: You repeat the process, adding the Year 2 interest to the Year 2 principal to get an even larger principal for the final calculation.
8. What is the effect of compounding interest more frequently, for instance, half-yearly instead of annually?
When interest is compounded more frequently (e.g., half-yearly or quarterly), the total amount of interest earned over the same time period will be higher. This is because the interest is added back to the principal more often, allowing the “interest on interest” effect to kick in sooner and more frequently. For calculations, the annual rate is divided by the number of compounding periods per year, and the time is multiplied by that same number.
