An interest rate is the percentage of the principal amount (the initial sum of money) that a borrower must pay to a lender in exchange for borrowing money or the return earned by an investor on their investments. It represents the cost of borrowing or the profit from lending money over a specified period of time, such as daily, monthly, quarterly, half-yearly, or annually.
Interest rates are commonly expressed as an annual percentage rate (APR) and can be applied in two main ways: simple interest and compound interest.
The interest rate formula is a mathematical equation which establishes a relation between the interest rate amount, principal amount, percent rate of interest and duration for which the amount is borrowed. Interest Rate Formula helps us to find out the amount of money that has to be returned to the lender after the specified time period.
Table of Content
Simple Interest Rate
Simple Interestexponentialney and the total amount of money that has to be returned to the lender after a specified time in the ,case of simple interest can be calculated using the following formula:
T_{s} = P+ I_{s}= P\left( 1+ \frac{ r\times t}{100}\right)
I_{s} = \frac{P\times r\times t}{100} Where,
- Ts represents Total amount to be paid
- Is stands for Simple Interest
- P represents principal amount
- r represents interest rate per annum (in percentage)
- t represent the duration of time (in years) the amount is borrowed
It might also be required to calculate the simple interest rate on a monthly basis, quarterly basis, etc. To do so, multiply the above formula with n/N where 'n' is the number of time periods in the chosen units and 'N' is the total number of time periods in a year for that unit. The values of N for some commonly used units are:
Unit | N |
|---|---|
Annually | 1 |
Half-yearly | 2 |
Quarterly | 4 |
Monthly | 12 |
Compound Interest Rate
A Compound interest rate is used when the interest is depended on the interest accumulated till then along with the borrowed amount. The amount of money to be paid as interest for the current year is calculated on the sum of the principal amount and interest charged till the previous year. As the principal amount at which the interest rate is applied keeps on increasing, it is said to be compounded every year.
Compound Interest Rate Formula
The interest money and the total amount of money that has to be returned to the lender after a specified time in case of compound interest can be calculated using the following formula:
T_{c} = P+ I_{c}= P\left( 1+ \frac{ r}{n \times100}\right)^{nt}
I_{c}= P\left( 1+ \frac{ r}{n\times100}\right)^{nt}-P where
- Tc represents Total amount to be paid
- Ic stands for Simple Interest
- P represents principal amount
- r represents compound interest rate per annum (in percentage)
- n is the number of times the interest is compounded each year
- t represent the duration of time (in years) the amount is borrowed
The value of 'n' is 1 if the amount is compounded annually. If not, 'n' changes with the variations in frequency in which the interest is compounded in a year. If 'r' is the rate per annum, then the values of 'n' can be seen in the following table:
Frequency | Value of n | Total amount |
|---|---|---|
Annually | 1 | |
Half-yearly | 2 | |
Quarterly | 4 | |
Monthly | 12 |
Simple Interest Rate vs Compound Interest Rate
We have already seen the formula for calculating simple interest and compound interest. The major difference between them is on the growth of the interest amount over time.
Simple Interest Formula: Linear Growth
The simple interest formula represents a linear relation. For a particular time period, the interest amount is only dependent on the principal amount. The interest amount to be paid remains constant throughout the entire duration if the interest rate is kept constant. It makes the calculation much simpler and so is used frequently for short-term loan agreements.
Compound Interest Formula: Exponential Growth
The compound interest formula establishes an exponential relation to calculate the interest amount. As the interest is calculated on the interest accumulated in the previous time periods along with the principal amount, it is said to be exponentially growing. Basically, the lender is earning interest over the interest money also. That's why compound interest is applied for long-term money lending or investments to gain maximum profits.
Read More: Difference Between Simple and Compound Interest
Also, Check
Solved Examples on Interest Rate Formula
Example 1: Alice has lent ₹10,000 to Bob for 3 years. Calculate the amount that Bob has to pay back at the end of 3 years, if
- The annual simple interest rate is 5%.
- The amount is compounded twice a year at the rate of 5%.
Solution:
Given,
Principal amount (P) = ₹10,000
Duration (t) = 3 years1. Simple interest rate (r) = 5%
Total amount to be paid (Ts) = P + Is
⇒ Ts = P + (P × r × t)/100
⇒ Ts = 10,000 + (10,000 × 5 × 3)/100
⇒ Ts = ₹ 11,500.002. Compound interest rate (r) = 5%
Total amount to be paid (Tc)
⇒ Tc = P + Ic
⇒ Tc = P( 1 + r/ (n × 100))n×t
⇒ Tc = 10,000 ( 1 + 5/ (2 × 100))2×3
⇒ Tc = ₹ 11,596.93
Example 2: Calculate the amount of time it takes to earn Rs. 81 as interest at 4.5% per annum for an initial amount of Rs. 450.
Solution:
Is = Rs. 81 , r = 4.5% , P = Rs. 450
⇒ t = (Is × 100)/(r × P) = (81 × 100)/(4.5 × 450)
⇒ t = 4 years
Example 3: It is given that the difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 5% per annum is Rs. 2. Find out the principal amount.
Solution:
Let the principal amount be Rs. P
Ic = P(1 + 5/100)2 - P = 0.1025P
Is= (P × 5 × 2)/100 = 0.1PGiven, Ic - Is = 2
⇒ 0.1025P - 0.1P = 2
⇒ 0.0025 × P = 2
⇒ P = Rs. 800.00
Example 4: A sum of money is put out at 10% interest compounded annually. What is the minimum number of years in which the total amount to be repaid is thrice the amount deposited?
Solution:
Let the amount deposited initially is Rs. P
P (1 + 10/100)t > 3P
⇒ (11/10)t > 3
⇒ t > 11.526Hence, it will take 12 years to triple the deposited amount.
Practice Problems on Interest Rate Formula
Question 1: What is the ratio of interest generated while a certain sum of money is subjected to simple interest at the same rate for 3 and 5 years respectively?in
Question 2: What is the difference in interest amount generated when a sum of Rs. 1000 is subjected to 10% compound interest rate, compounded monthly and annually?
Question 3: What is the ratio of interest amount generated when a sum of Rs. 8000 is compounded at 7% rate and a 7% simple interest rate separately for 5 years?
Question 4: Compound interest on Rs. 5000 for 3 years at 6% per annum is double the simple interest on some amount of money for 3 years at 8% per annum. What is the sum placed on simple Does?
Answer Key
Answer 1: The ratio of interest generated for 3 and 5 years will be 3 : 5.
Answer 2: The difference in interest is Rs. 4.71.
Answer 3: The ratio of Compound Interest to Simple Interest is: 1.15 : 1
Answer 4: The sum placed on simple interest is approximately Rs. 1981.42.