Simple Interest vs. Compound Interest

Both formulas use principal, rate, and time, but the base changes. Simple interest uses the original principal each period. Compound interest adds earned interest to the balance, so later interest can be earned on earlier interest.

Simple interest formula

For simple interest, I = Prt, where P is principal, r is the rate per time period, and t is the number of matching periods. If $1,000 earns 5% simple interest per year for 3 years, interest is $1,000 × 0.05 × 3 = $150. The total is $1,150, assuming no other charges or payments.

Compound interest formula

For annual compounding, the accumulated amount is A = P(1 + r)^t. For m compounding periods each year, use A = P(1 + r/m)^(mt), with consistent units. Each period’s credited interest becomes part of the balance used for later calculations.

Read the units carefully

A rate must match the compounding period. If a nominal annual rate compounds monthly, convert it to the periodic rate by dividing by 12, and multiply years by 12 to count periods. If the problem gives an effective annual rate, do not divide it by the number of periods as though it were nominal.

When to use each formula

• Use simple interest when the problem states interest is calculated on original principal only.
• Use compound interest when interest is periodically added to the balance.
• Convert a percent to decimal form before substituting.
• Check whether the question asks for interest earned or the ending amount.

Compare balances period by period

With $1,000 at 5 percent for three years, simple interest adds $50 each year, for $150 total interest and a $1,150 balance. If it compounds annually, the balance is $1,000(1.05)³ = $1,157.625 before rounding. The first year earns $50; the second earns interest on $1,050; the third earns interest on $1,102.50. The difference comes from interest being added to principal between periods.

Separate rate, time, and compounding periods

A nominal annual rate of 12 percent compounded monthly uses a monthly rate of 0.12/12 = 0.01 and 12 periods per year. For two years, the exponent is 24: A = P(1.01)²⁴. Do not put 0.12 into the periodic-rate slot and then also compound monthly; that would apply the annual rate each month. If the problem states an effective annual rate, it already reflects within-year compounding and is used differently.

Find interest versus ending amount

The compound formula A = P(1 + r/m)^(mt) gives the ending amount. Interest earned is A − P if there are no additional deposits, withdrawals, or fees. For simple interest, I = Prt is the interest amount and A = P + I is the ending amount. Read the wording carefully so you do not report the total balance when asked for interest alone.

Understand what the formulas assume

These formulas assume a fixed stated rate and regular compounding as described. Real accounts may have variable rates, fees, taxes, deposits, or withdrawals that change the result. A periodic contribution requires a different calculation from a one-time principal unless the problem provides a separate contribution rule. In an exam question, use the stated assumptions rather than importing a product's real-world conditions.

Check growth and reasonableness

With a positive rate and no withdrawals, compound interest should not produce less interest than simple interest over multiple compounding periods on the same principal. The two methods match for one period when both apply the same rate to the same principal. Convert percentages to decimals, align the rate and time units, and estimate the balance before accepting a calculator result.

Simple interest grows from the original principal

Simple interest is calculated only on the original principal: I = Prt, where P is principal, r is the annual rate as a decimal, and t is time in years. If $1,000 earns 5% simple interest for 3 years, I = 1000(0.05)(3) = $150, so the total is $1,150. The interest earned each year stays $50 because the principal used in the calculation does not change.

Make units consistent. If the rate is annual and time is given in months, convert months to years or use a rate for the corresponding period. At 6% per year for 9 months, simple interest on $800 is 800(0.06)(9/12) = $36. A percent must be written as a decimal in the formula.

Compound interest earns interest on prior interest

With compounding, interest is added to the balance at each compounding period. For annual compounding, A = P(1 + r)ᵗ. For m compounding periods per year over t years, A = P(1 + r/m)ᵐᵗ. If $1,000 earns 5% compounded annually for 3 years, the balance is 1000(1.05)³ = $1,157.63, approximately. It exceeds the simple-interest total because interest itself earns interest.

More frequent compounding generally yields a larger balance for the same nominal annual rate and time, assuming no fees or withdrawals. The effective annual rate can be found from (1 + r/m)ᵐ − 1. Do not confuse the nominal annual rate with a periodic rate; divide by m for the rate per compounding period.

Choose the formula from the wording

If a problem says simple interest, use Prt for interest and add it to principal only if the question asks for the total amount. If it says compounded monthly or quarterly, use the compound formula with the appropriate m. If the question gives a balance after interest, distinguish that amount from the interest earned: interest = final amount − principal.

Check that the answer is greater than the original principal when a positive rate is applied for positive time. A negative time or a rate entered as 5 rather than 0.05 can produce an implausible result. Round currency to the nearest cent only at the end unless instructed otherwise.

• Simple interest: I = Prt; interest is based on original principal.
• Compound interest: multiply by a periodic growth factor each period.
• Convert annual rates and time to matching units.
• Distinguish interest earned from total balance.
• Keep full precision until the final currency rounding.

Exam takeaway

Simple: interest on principal only. Compound: interest on the changing balance. Match the rate and time units, then check whether the answer is interest or total accumulated value.