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Interest — Definition, Formula & Examples

Interest

The process by which an amount of money increases over time. With interest, a fixed percentage of the money is added at regular time intervals.

 

Types of interest: simple interest, compound interest, continuously compounded interest

Key Formula

I=P⋅r⋅tI = P \cdot r \cdot t
Where:
  • II = Total interest earned (in dollars or other currency)
  • PP = Principal — the original amount of money
  • rr = Interest rate per time period (as a decimal)
  • tt = Number of time periods

Worked Example

Problem: You deposit $1,000 in a savings account that earns 5% simple interest per year. How much interest do you earn after 3 years, and what is the total amount in the account?
Step 1: Identify the values: principal P = $1,000, rate r = 5% = 0.05, time t = 3 years.
P=1000,r=0.05,t=3P = 1000, \quad r = 0.05, \quad t = 3
Step 2: Apply the simple interest formula to find the interest earned.
I=P⋅r⋅t=1000×0.05×3=150I = P \cdot r \cdot t = 1000 \times 0.05 \times 3 = 150
Step 3: Add the interest to the principal to find the total amount.
A=P+I=1000+150=1150A = P + I = 1000 + 150 = 1150
Answer: You earn 150ininterest,andthetotalamountintheaccountafter3yearsis150 in interest, and the total amount in the account after 3 years is1,150.

Another Example

Problem: You deposit $1,000 in an account that earns 5% compound interest per year (compounded annually). How much is in the account after 3 years?
Step 1: Use the compound interest formula with P = $1,000, r = 0.05, n = 1 (compounded once per year), and t = 3.
A=P(1+rn)nt=1000(1+0.05)3A = P\left(1 + \frac{r}{n}\right)^{nt} = 1000\left(1 + 0.05\right)^{3}
Step 2: Compute the growth factor and the final amount.
A=1000×(1.05)3=1000×1.157625=1157.63A = 1000 \times (1.05)^3 = 1000 \times 1.157625 = 1157.63
Step 3: Find the total interest earned by subtracting the principal.
I=1157.63−1000=157.63I = 1157.63 - 1000 = 157.63
Answer: With compound interest, the account holds 1,157.63after3years—thatis1,157.63 after 3 years — that is7.63 more than simple interest would give. The extra comes from earning interest on previously earned interest.

Frequently Asked Questions

What is the difference between interest rate and interest?
The interest rate is the percentage used to calculate the charge (e.g., 5%), while interest is the actual dollar amount produced by applying that rate over time. For example, a 5% rate on 1,000foroneyearyields1,000 for one year yields50 of interest.
Why does compound interest grow faster than simple interest?
With simple interest, you only earn interest on the original principal. With compound interest, each period's interest is added to the balance, so in the next period you earn interest on a larger amount. Over time, this "interest on interest" effect causes the balance to grow exponentially rather than linearly.

Simple Interest vs. Compound Interest

Simple interest is calculated only on the original principal, so the interest earned each period stays constant. Compound interest is calculated on the principal plus all previously accumulated interest, so the amount earned each period increases. Over short time spans or low rates the difference is small, but over long periods compound interest produces significantly more growth. For example, 1,000at51,000 at 5% for 3 years yields150 with simple interest but $157.63 with annual compound interest.

Why It Matters

Interest is the foundation of personal finance: it determines how much you pay on loans, credit cards, and mortgages, and how much your savings and investments grow over time. Understanding interest helps you compare financial products — a small difference in rate or compounding frequency can mean thousands of dollars over a loan's lifetime. In mathematics, interest problems introduce exponential growth, one of the most important patterns in science and economics.

Common Mistakes

Mistake: Forgetting to convert the interest rate from a percentage to a decimal before calculating.
Correction: Always divide the percentage by 100 first. For instance, 5% becomes 0.05 in formulas. Using 5 instead of 0.05 will give an answer 100 times too large.
Mistake: Using the simple interest formula when interest is compounded.
Correction: Check whether the problem states simple or compound interest. If interest is compounded, use A=P(1+r/n)ntA = P(1 + r/n)^{nt}, not I=PrtI = Prt. Applying the wrong formula underestimates the final amount.

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