Simple interest uses I = PRT/100: the interest is a fixed amount each year, calculated only on the original principal P. Compound interest uses A = P(1 + r/100)ⁿ: each period's interest is added to the amount, so the next period earns interest on interest too. The two formulas answer different questions (total interest for simple, final amount for compound), and mixing them up is the single most common error in this topic.
Both topics use the words "principal", "rate" and "interest", so it is easy to reach for the wrong formula under time pressure, or to confuse either with a profit and loss question, which uses cost price and selling price instead. The real difference is what happens to the interest once it is earned. With simple interest, each year's interest is set aside: it never itself earns more interest, so the yearly interest amount stays the same throughout. With compound interest, each period's interest is folded back into the principal, so the amount it earns interest on grows every period.
This is also why compound interest always grows faster than simple interest over the same rate and time, once there is more than one compounding period. In the first period the two are identical. From the second period onward, compound interest is calculated on a slightly larger amount, so it pulls ahead, and the gap widens with every extra period.
Simple interest
I = PRT / 100
A = P + I
Compound interest
A = P(1 + r/100)ⁿ
I = A − P
P, R, r
Principal, and the rate per compounding period (%)
T, n
Time in years (simple), number of compounding periods (compound)
The part students lose marks on is reading the compounding period correctly. "Per annum compounded half-yearly" does not mean plug the annual rate straight into the formula. It means the interest is actually added twice a year, so you must convert both the rate and the number of periods before using A = P(1 + r/100)ⁿ:
Converting r and n for half-yearly compounding
r = (annual rate) ÷ 2.n = (number of years) × 2.Kai Wen invests $4000 at a simple interest rate of R% per annum. After 3 years, the investment earns $540 in interest. Find R.
Solution
I = PRT/100 and substitute the known values:540 = 4000 × R × 3 / 1004000 × 3 / 100 = 120, so the equation becomes 540 = 120R.R = 540 ÷ 120 = 4.5.Quick check: put R back in. I = 4000 × 4.5 × 3 / 100 = 4000 × 13.5 / 100 = 54000/100 = 540. Matches the question, so R = 4.5 is correct.
Mei Ling deposits $5000 in an account paying 6% per annum, compounded half-yearly, for 2 years. Find the final amount, and the total interest earned.
Solution
r = 6 ÷ 2 = 3 per half year, and n = 2 × 2 = 4 compounding periods.A = P(1 + r/100)ⁿ:A = 5000 × (1 + 3/100)⁴ = 5000 × (1.03)⁴A = 5000 × 1.12550881 = 5627.54405, so A = $5627.54 (2 decimal places).I = A − P = 5627.54 − 5000 = $627.54.The classic mistake: using the simple interest formula on a compound question
If a student wrongly applied I = PRT/100 to worked example 2 above, using the annual rate 6 and T = 2, they would get I = 5000 × 6 × 2 / 100 = 600, giving a final amount of $5600. The correct compound answer is $627.54 interest and a final amount of $5627.54, a difference of $27.54. The word "compounded" in the question is the signal: it means the formula must be A = P(1 + r/100)ⁿ, never I = PRT/100, and the rate and periods must match the stated compounding frequency, not the annual figures as given.
Read the question twice before choosing a formula: "simple interest" or a fixed yearly amount signals I = PRT/100. Any mention of "compounded" (annually, half-yearly, quarterly, monthly) signals A = P(1 + r/100)ⁿ with r and n adjusted to the compounding period. Also check exactly what is being asked: total interest (I) or final amount (A) are different numbers and both appear as possible answers on the mark scheme.
Why does compound interest grow faster than simple interest?
Because compound interest adds each period's interest back into the amount, so the next period earns interest on a larger sum. Simple interest always earns interest on the same original principal, so the yearly interest amount never changes. Over enough periods the compound total pulls further and further ahead.
Does "per annum" always mean I use T or n in years?
"Per annum" describes the rate, not automatically the time unit in the formula. For simple interest, T is in years to match the annual rate directly. For compound interest, if the compounding is annual, n is in years too, but if it is compounded half-yearly or quarterly, you must convert both r and n to match the actual compounding period before substituting.
How do I know if a question wants total interest or the final amount?
Read the exact wording. "Find the interest earned" or "how much interest" asks for I. "Find the amount in the account" or "the value of the investment" asks for A. If you compute one when the question asks for the other, you get zero marks even with correct working, so underline the exact quantity being asked before you start.
Can compound interest questions use a formula other than A = P(1 + r/100)ⁿ?
For O-Level E-Maths, this is the formula to use. Some questions dress it up as depreciation, where the value decreases each period instead of increasing, using A = P(1 − r/100)ⁿ with a minus sign. The structure and the r/n conversion rules are identical, only the sign changes. If a question instead gives you the final amount and asks for the original principal, you are working a reverse percentage problem and the method is different again.
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