Key takeaway: Compound interest is interest calculated on the original amount plus all the interest already earned, so a balance grows exponentially instead of by equal steps.
Grade 9 – Algebra 1F-IF.C.8bFunctions and Graphs
Definition of Compound Interest
Compound interest pays you interest on your interest. After each compounding period, the interest earned is added to the balance, and the next period’s interest is figured on that larger amount. Simple interest, by contrast, always uses the original principal.
How often interest is added matters. Quarterly compounding adds interest 4 times a year, monthly compounding 12 times . More frequent compounding gives slightly more money for the same annual rate. Since the balance is multiplied by the same factor every period, the model is an exponential function, and the same idea works for loans that grow when payments are missed.
Formula and notation
With principal \(P\), annual rate \(r\) as a decimal, \(n\) compounding periods per year and \(t\) years, the final amount is
\[A = P(1 + \frac{r}{n})^{nt}\]
The interest earned is \(A – P\). For annual compounding, \(n = 1\) and the formula becomes \(A = P(1 + r)^{t}\).
Key facts
- The rate per period is the annual rate divided by the number of periods per year.
- The exponent counts all the periods: n times t.
- Compound interest earns more than simple interest at the same rate and time.
- Round the final answer to the nearest cent, not the intermediate factors.
Compound Interest example
Marisol puts \$2,500 into an account that pays 6% annual interest compounded quarterly. How much is in the account after 3 years?
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Identify: \(P = 2500\), \(r = 0.06\), \(n = 4\), \(t = 3\).
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Rate per quarter: \(\frac{0.06}{4} = 0.015\). Number of quarters: \(4 \times 3 = 12\).
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Compute: \(A = 2500(1.015)^{12} \approx 2500 \times 1.195618 \approx 2989.05\).
After 3 years the account holds about \$2,989.05, which is \$489.05 of interest.
Common mistake
Watch out
Ravi uses the full 6% in every quarter and writes \(2500(1.06)^{12} \approx \$5{,}030.49\).
Why it is wrong
The 6% is a yearly rate. Each quarter earns only one fourth of it, 1.5%. Divide by \(n\) first: \(2500(1.015)^{12} \approx \$2{,}989.05\). A result that doubles the money in 3 years should look too good to be true for a 6% account.
Related terms and lessons
- Algebra 1 · Exponential Growth and DecayLessonWorksheetQuiz
- Algebra 2 · Exponential Functions and the Number eLessonWorksheetQuiz
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