To learn Simple and Compound Interest quickly and correctly, begin with one distinction: simple interest is calculated on the original principal each period, while compound interest is calculated on the growing balance when interest is added and earns further interest. They can begin at the same rate but produce different amounts over time. These eduKateSG Mathematics Quick Notes explain the formulas as ordinary percentage calculations, show how to compare them and help students check their answers without confusing the principal, the interest earned and the final amount.
- Principal P: the original sum used in the calculation.
- Rate r: convert a percentage into a decimal, such as 5% = 0.05.
- Simple interest: I = Prt when r is annual and t is measured in years.
- Compound annual amount: A = P(1 + r)ⁿ for a fixed rate compounded once each year.
- Interest earned: subtract the principal from the final amount.
- Check frequency: the rate per compounding period must match the number of periods.
- Check direction: positive growth should increase the balance, while depreciation should decrease it.
First, understand simple interest as a repeated amount. Suppose you have a principal of $1,000 and the question specifies simple interest of 5% per year for three years. Each year adds 5% of the original $1,000, which is $50. Over three years, the total interest is 3 × $50 = $150, giving a final amount of $1,150. The key is that the $50 interest is not itself included in the principal for future simple-interest calculations. For a textbook example with annual rate r and time t in years, I = Prt.
Second, understand compound interest as repeated percentage growth. Begin with the same $1,000 at 5% compounded annually. After the first year, the amount is $1,000 × 1.05 = $1,050. After the second, multiply the new balance by 1.05 to get $1,102.50. After the third, multiply again to obtain $1,157.625, or $1,157.63 if reported to the nearest cent under the usual school rounding rule. The interest now earns interest, which is why the three-year compound amount is slightly larger than the simple-interest amount.
Third, see the repeated percentage change as a multiplier. An increase of 5% means keeping 100% of the original and adding 5%, so the new balance is 105% = 1.05 times the old balance. Apply the same multiplier n times to obtain A = P(1.05)ⁿ for this example. In general, A = P(1 + r)ⁿ when the same rate r applies each period and there are no added deposits, withdrawals or fees. This is the same structure as a geometric sequence. Compounding is easier when you understand the repeated multiplier rather than simply memorising a long formula.
Worked example: simple interest
A textbook account contains $1,500 and earns 4% simple interest per year for 2 years. Principal P = 1,500, annual rate r = 0.04 and time t = 2. Interest = Prt = 1,500 × 0.04 × 2 = $120. Final amount = P + I = 1,500 + 120 = $1,620. Check mentally: 4% of $1,500 is $60, and two years at $60 per year gives $120. Always distinguish the interest amount from the total balance.
Worked example: compound interest
A sum of $1,000 earns 10% annually, compounded once each year, for 2 years. The multiplier is 1 + 0.10 = 1.10. Final amount A = 1,000 × (1.10)² = $1,210. The total interest earned is 1,210 − 1,000 = $210. The first year adds $100; the second adds $110 because 10% is applied to $1,100, not only the original principal. For simple interest at the same rate and duration, the interest would be $200.
Compare the two methods over three years
Take $2,000 at 5% per year for 3 years. With simple interest, I = 2,000 × 0.05 × 3 = $300, so final amount = $2,300. With annual compounding, A = 2,000 × 1.05³ = $2,315.25. The compound amount exceeds the simple amount by $15.25. In this example both use the same annual percentage, time and principal, and the only difference is the method by which later interest is calculated. That makes it a fair comparison.
Why the compounding period matters
The rate and period must match. If a question explicitly states a nominal annual rate of 12% compounded monthly, the monthly rate under that convention is 12% ÷ 12 = 1% and there are 12 monthly periods in a year. A principal of $1,200 would become $1,200 × (1.01)¹² ≈ $1,352.19 after a year. This is different from multiplying by 1.12 once. Do not automatically divide every advertised annual rate by 12, however: some rates are described as effective annual rates and must be handled differently. Use the rate convention stated in the question.
A general repeated-compounding formula
For a fixed periodic rate i and n compounding periods, with no further money added or removed, the general growth formula is A = P(1 + i)ⁿ. The symbol i is the rate per period, not necessarily per year. If the question specifies a nominal annual rate r compounded m equal times per year, then i = r/m and the number of periods over t years is mt, giving A = P(1 + r/m)^(mt), assuming the stated nominal-rate convention. Keep the time unit and rate convention consistent before substituting.
What if the value decreases each year?
Depreciation and repeated percentage decreases use the same multiplier idea with subtraction. If a product loses 20% of its value per year, each new value is 80% of the previous one, so the multiplier is 0.80. Starting from $5,000, after one year it is $4,000. After two years it is 4,000 × 0.80 = $3,200. The formula is A = P(1 − r)ⁿ for a fixed fractional reduction per period, when 0 ≤ r ≤ 1. Do not simply subtract 40% of the original over two years if the loss is compounded.
Finding the original principal
Sometimes a question gives the final amount and asks how much money was there at the start. Suppose an amount has grown to $1,210 after two years at 10% compounded annually. Since A = P × 1.1², divide by 1.21: P = 1,210/1.21 = $1,000. Check forward: 1,000 × 1.21 = 1,210. Reversing the multiplier is more reliable than subtracting 10% twice from the final amount, because each percentage would use a different base.
Read percentages precisely
- 5%: the decimal rate is 0.05.
- 5% increase: multiply by 1.05.
- 5% decrease: multiply by 0.95.
- Simple interest: the interest each period is based on the original principal.
- Compound interest: the interest each period is based on the current balance.
- Final amount: principal plus accumulated interest, where applicable.
- Interest earned: final amount minus original principal, assuming no other cash flows.
Real financial terms versus a school calculation
A textbook problem often assumes a fixed rate, no fees and a precise compounding schedule. Real savings products and loans may use different interest conventions, variable rates, minimum balances, additional deposits, withdrawals, fees, taxes, repayment schedules or rounding rules. So the formulas above explain the specified mathematical model; they are not a promise of what any particular bank or loan will pay or charge. When applying the mathematics outside school, read the exact rate definition, time period and terms.
Mistakes that make simple calculations wrong
- Using 5 instead of 0.05: convert percentages into decimal rates.
- Calling the interest the final amount: the final amount includes the principal.
- Using the compound formula in a simple-interest question: identify the calculation method first.
- Adding the same percentage to the original every year in a compound question: use a repeated multiplier on the new balance.
- Mixing monthly periods with an annual rate: match the periodic rate to the number of periods.
- Rounding after every step without instructions: retain precision until the final answer unless the specified model rounds each period.
- Assuming depreciation is linear: a repeated percentage decrease applies to the reduced value.
Mini-test: try these without looking back
- Find the simple interest and final amount on $2,500 at 3% per year for 4 years.
- Find the final amount of $800 at 10% compounded annually for 2 years.
- A product initially costs $5,000 and depreciates by 20% annually for 2 years. Find its value after 2 years.
- Find the final amount of $2,000 invested at 5% compounded annually for 3 years.
- An amount becomes $1,210 after 2 years at 10% compounded annually. Find the starting principal.
Answers with the correct method
1. Simple interest = 2,500 × 0.03 × 4 = $300; final amount = $2,800. 2. 800 × 1.1² = $968. 3. 5,000 × 0.8² = $3,200. 4. 2,000 × 1.05³ = $2,315.25. 5. 1,210 ÷ 1.1² = $1,000. Each answer can be checked by applying the stated percentage for the relevant periods.
Your twelve-minute interest study routine
- Minutes 1–2: define principal, rate, time and final amount.
- Minutes 3–5: calculate one simple-interest example and verify the yearly interest.
- Minutes 6–8: calculate one annually compounded example using the multiplier.
- Minutes 9–10: compare the two methods using the same principal, rate and period.
- Minutes 11–12: solve a repeated percentage decrease or a reverse-principal question, then check the amount.
The fastest correct way to learn interest is to ask what amount the percentage acts upon. If it acts on the original principal every year, think simple interest. If it acts on a changing balance, think repeated multiplication. That single distinction explains most of the formulas.
Continue the Mathematics Quick Notes learning pathway
For connected practice, read Percentages · Direct and Inverse Proportion · Number Patterns and Sequences · Mathematics Word Problems. Test the ideas using new numbers before you move to the next topic.
For the longer foundation guide, see Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The immutable reference has not been changed.