Compound Interest
Simple interest pays on the principal only — the same amount every year. Compound interest pays on the growing balance, so each year's interest also earns interest the next year. That small difference is why compound accounts pull ahead over time.
The formula is , and the key reading skill is knowing what means: it is the total balance in the account after years, not just the interest. Get that straight and compound problems are two clean steps — a power, then a multiplication.
The formula: A = P(1 + r)^t
Here is the principal, is the rate as a decimal, and is the time in years — the same three ingredients as simple interest. The new piece is the exponent: is one year's growth factor, and raising it to the power applies that growth times in a row.
For example, at 10% the growth factor is . Two years of compounding means multiplying by twice: . The output is the whole balance. If a question asks for the interest earned, subtract the principal at the end: .
Why compound beats simple
Put $1,000 at 10% in each kind of account. Simple interest pays every single year: the balance goes $1,100, $1,200, $1,300. Compound interest pays 10% of whatever the balance is now: $1,100 after year one, then 10% of $1,100 is $110, so $1,210 after year two, then $1,331.
Both accounts match after the first year. From then on, compound wins — by $10 after two years, $31 after three, and the gap keeps widening because every year's interest joins the balance and starts earning too. The table below tracks the two balances side by side.
Spotting compound growth in a table
Many problems show two account balances year by year and ask which one earns compound interest. Check how much each account grows from one year to the next. Simple interest adds the same amount every year. Compound interest adds a little more each year, because the balance it pays on keeps growing.
Worked examples
Example 1: find the balance
You deposit $1,000 at 10% compound interest for 2 years. Find the balance.
Answer: A balance of $1,210
Example 2: a bigger principal
You deposit $2,000 at 10% compound interest for 2 years. Find the balance.
Answer: A balance of $2,420
Example 3: find the interest, not the balance
You deposit $500 at 10% compound interest for 2 years. How much interest do you earn?
Answer: $105 of interest
Try one yourself
Common questions
What is the difference between A and I?
in the compound formula is the total balance after years — principal and interest together. in the simple interest formula is the interest alone. To get compound interest by itself, compute and then subtract the principal.
Why is compound interest bigger than simple interest?
Simple interest is always calculated on the original principal. Compound interest is calculated on the current balance, which includes every earlier year's interest — so the amount added grows each year.
What does the (1 + r) part mean?
It is one year's growth factor. Multiplying by keeps 100% of what you had and adds more. At 10%, multiplying a balance by does both steps — the old balance plus its interest — in one move.
How can I tell from a table whether interest is compound?
Subtract each year's balance from the next. If the account grows by the same amount every year, it is simple interest. If it grows by a little more each year, it is compound.
Want the video version?
Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.