High School Maths · Year 2 · Unit 14 · Lesson 1 of 5
Compound Interest: Why Your Savings Grow Faster Every Year
Lina and Theo each put $1,000 into an account for 2 years at 10% a year. Lina gets simple interest. Theo gets compound interest.
Who has more money at the end, and how much more?
Theo, by $10. Lina earns $100 each year, so she ends with $1,200. Theo earns $100 in year 1, giving $1,100. In year 2 he earns 10% of $1,100 = $110, giving $1,210. With compound interest you also earn interest on your interest. In Year 1 you used simple interest, I = PRT ÷ 100. This lesson is about compound interest.
| Year | Lina (simple) | Theo (compound) |
|---|---|---|
| 0 | $1,000 | $1,000 |
| 1 | $1,100 | $1,100 |
| 2 | $1,200 | $1,210 |
In year 2 Theo also earns interest on his first $100 of interest.
Master the essentials
Compound interest
The interest is added to the account at the end of each year. Next year, interest is paid on the new, bigger amount.
$500 at 6%: 500 × 1.06 = $530, then 530 × 1.06 = $561.80
Multiplier
A rise of r% multiplies the amount by (1 + r ÷ 100).
7% → × 1.07 and 2.5% → × 1.025
The formula
Amount after n years: A = P(1 + r/100)ⁿ. P is the principal: the money put in at the start.
A = 800 × 1.05³ = $926.10
Interest earned
Interest = A − P. Read the question: does it ask for the amount or the interest?
926.10 − 800 = $126.10
Simple or compound?
Simple interest is the same every year. Compound interest gets bigger every year.
$800 at 5% for 3 years: simple $120, compound $126.10
Half-yearly
If interest is added every 6 months, use half the yearly rate and twice as many periods.
8% a year for 3 years, half-yearly: × 1.04⁶
Calculator
Type the whole calculation in one go. Round only the final answer, to the nearest cent.
1,250 × 1.04⁵ = 1,520.816128 → $1,520.82
Changing rates
If the rate changes, multiply by each year’s multiplier in turn.
4% then 3%: P × 1.04 × 1.03
See it done
1. $3,600 is invested at 4.5% compound interest per year. Find the amount after 3 years. Give your answer to the nearest cent.
- The multiplier for 4.5% is 1 + 0.045 = 1.045.
- A = 3,600 × 1.045³.
- = 4,108.19805 on the calculator.
- To the nearest cent: $4,108.20.
$4,108.20
2. $2,400 is invested for 4 years at 6% per year. How much more interest is earned with compound interest than with simple interest?
- Simple: I = 2,400 × 6 × 4 ÷ 100 = $576.
- Compound: A = 2,400 × 1.06⁴ = 3,029.944704, so the interest is $629.94.
- Difference: $629.94 − $576 = $53.94.
$53.94 more with compound interest
3. $5,000 is invested for 2 years at 8% per year, with the interest added every 6 months. Find the interest earned.
- Every 6 months the rate is 8% ÷ 2 = 4%.
- In 2 years there are 2 × 2 = 4 half-years.
- A = 5,000 × 1.04⁴ = 5,849.2928, which is $5,849.29.
- Interest: $5,849.29 − $5,000 = $849.29. (Added once a year it would be only $832.)
$849.29
Common mistake: Multiplying one year’s interest by the number of years. That is simple interest. For compound interest the interest changes every year, so use a power: 3 years at 5% is × 1.05³, not × 1.15.
10 questions with instant feedback
Choose the correct answer. A calculator is allowed. Write the multiplier first, use a power for the years, and round to the nearest cent only at the end. Questions 1–3 are warm-ups, 4–7 are at exam standard and 8–10 are challenge questions. After you answer, the feedback shows the full solution.
Warm-up (The basic skill: one or two steps.)
1.Which calculation gives the amount after 5 years when $700 is invested at 3% compound interest per year?
Not quite. This adds 5 × 3% = 15% once. That is simple interest.
Solution
- A 3% rise multiplies by 1.03.
- This happens 5 times, once each year.
- Amount = 700 × 1.03⁵.
Answer: D. 700 × 1.03⁵.
Not quite. 0.03 is only the interest part. Add 1 to keep the $700: 1.03.
Solution
- A 3% rise multiplies by 1.03.
- This happens 5 times, once each year.
- Amount = 700 × 1.03⁵.
Answer: D. 700 × 1.03⁵.
Not quite. 3% = 0.03, so the multiplier is 1.03, not 1.3.
Solution
- A 3% rise multiplies by 1.03.
- This happens 5 times, once each year.
- Amount = 700 × 1.03⁵.
Answer: D. 700 × 1.03⁵.
Correct.
Solution
- A 3% rise multiplies by 1.03.
- This happens 5 times, once each year.
- Amount = 700 × 1.03⁵.
2.$1,500 is invested at 4% compound interest per year. Find the amount after 2 years.
Not quite. This adds $60 each year. That is simple interest.
Solution
- Multiplier: 1.04.
- After 1 year: 1,500 × 1.04 = $1,560.
- After 2 years: 1,560 × 1.04 = $1,622.40.
Answer: B. $1,622.40.
Correct.
Solution
- Multiplier: 1.04.
- After 1 year: 1,500 × 1.04 = $1,560.
- After 2 years: 1,560 × 1.04 = $1,622.40.
Not quite. This is the interest only. The question asks for the amount.
Solution
- Multiplier: 1.04.
- After 1 year: 1,500 × 1.04 = $1,560.
- After 2 years: 1,560 × 1.04 = $1,622.40.
Answer: B. $1,622.40.
Not quite. This is the amount after 1 year only.
Solution
- Multiplier: 1.04.
- After 1 year: 1,500 × 1.04 = $1,560.
- After 2 years: 1,560 × 1.04 = $1,622.40.
Answer: B. $1,622.40.
3.$6,000 is invested at 2.5% compound interest per year for 3 years. How much interest is earned? Give your answer to the nearest cent.
Correct.
Solution
- A = 6,000 × 1.025³.
- = 6,461.34375.
- Interest = 6,461.34375 − 6,000 = $461.34 to the nearest cent.
Not quite. This is the amount. Take away the $6,000 to find the interest.
Solution
- A = 6,000 × 1.025³.
- = 6,461.34375.
- Interest = 6,461.34375 − 6,000 = $461.34 to the nearest cent.
Answer: A. $461.34.
Not quite. This is simple interest: 6,000 × 2.5 × 3 ÷ 100.
Solution
- A = 6,000 × 1.025³.
- = 6,461.34375.
- Interest = 6,461.34375 − 6,000 = $461.34 to the nearest cent.
Answer: A. $461.34.
Not quite. 2.5% = 0.025, so the multiplier is 1.025, not 1.25.
Solution
- A = 6,000 × 1.025³.
- = 6,461.34375.
- Interest = 6,461.34375 − 6,000 = $461.34 to the nearest cent.
Answer: A. $461.34.
Exam standard (Several steps, like real exam questions.)
4.The table shows three savings accounts. Ravi has $4,200 to invest for 4 years. Which account leaves him with the most money, and how much does he have at the end?
| Account | Interest | Rate per year |
|---|---|---|
| Harbour | compound | 3.4% |
| Summit | simple | 3.6% |
| Meadow | compound | 3.5%, minus a $40 fee at the end |
Three savings accounts.
Not quite. Harbour gives $4,801, which is less than Summit’s $4,804.80. A higher simple rate can beat a lower compound rate.
Solution
- Harbour: 4,200 × 1.034⁴ = $4,801.
- Summit: 4,200 + 4,200 × 3.6 × 4 ÷ 100 = $4,804.80.
- Meadow: 4,200 × 1.035⁴ − 40 = $4,779.60.
- The most is Summit.
Answer: D. Summit: $4,804.80.
Not quite. This forgets the $40 fee. Meadow leaves him with $4,779.60.
Solution
- Harbour: 4,200 × 1.034⁴ = $4,801.
- Summit: 4,200 + 4,200 × 3.6 × 4 ÷ 100 = $4,804.80.
- Meadow: 4,200 × 1.035⁴ − 40 = $4,779.60.
- The most is Summit.
Answer: D. Summit: $4,804.80.
Not quite. Summit pays simple interest, so it adds the same $151.20 each year.
Solution
- Harbour: 4,200 × 1.034⁴ = $4,801.
- Summit: 4,200 + 4,200 × 3.6 × 4 ÷ 100 = $4,804.80.
- Meadow: 4,200 × 1.035⁴ − 40 = $4,779.60.
- The most is Summit.
Answer: D. Summit: $4,804.80.
Correct.
Solution
- Harbour: 4,200 × 1.034⁴ = $4,801.
- Summit: 4,200 + 4,200 × 3.6 × 4 ÷ 100 = $4,804.80.
- Meadow: 4,200 × 1.035⁴ − 40 = $4,779.60.
- The most is Summit.
5.Which working shows that $2,000 invested at 7% compound interest per year is worth $2,621.59 after 4 years, to the nearest cent?
Not quite. This adds 4 × 7% = 28% once: simple interest.
Solution
- Multiplier: 1.07, used once for each of the 4 years.
- 2,000 × 1.07⁴ = 2,621.59202.
- To the nearest cent: $2,621.59 ✓.
Answer: C. 2,000 × 1.07⁴ = 2,621.59202.
Not quite. This multiplies by 4. The 4 years make a power: 1.07⁴.
Solution
- Multiplier: 1.07, used once for each of the 4 years.
- 2,000 × 1.07⁴ = 2,621.59202.
- To the nearest cent: $2,621.59 ✓.
Answer: C. 2,000 × 1.07⁴ = 2,621.59202.
Correct.
Solution
- Multiplier: 1.07, used once for each of the 4 years.
- 2,000 × 1.07⁴ = 2,621.59202.
- To the nearest cent: $2,621.59 ✓.
Not quite. 7% = 0.07, so the multiplier is 1.07, not 1.7.
Solution
- Multiplier: 1.07, used once for each of the 4 years.
- 2,000 × 1.07⁴ = 2,621.59202.
- To the nearest cent: $2,621.59 ✓.
Answer: C. 2,000 × 1.07⁴ = 2,621.59202.
6.Money is invested at 5% compound interest per year. After 2 years it is worth $3,307.50. How much was invested?
Correct.
Solution
- Amount = P × 1.05².
- So P = 3,307.50 ÷ 1.05² = 3,307.50 ÷ 1.1025.
- P = $3,000.
Not quite. This takes 5% off twice. The $3,307.50 is 105% of 105% of the start, so divide by 1.05².
Solution
- Amount = P × 1.05².
- So P = 3,307.50 ÷ 1.05² = 3,307.50 ÷ 1.1025.
- P = $3,000.
Answer: A. $3,000.
Not quite. This divides by 1.1 (simple interest of 10%). Compound interest needs 1.05² = 1.1025.
Solution
- Amount = P × 1.05².
- So P = 3,307.50 ÷ 1.05² = 3,307.50 ÷ 1.1025.
- P = $3,000.
Answer: A. $3,000.
Not quite. This takes 10% off. Working backwards means dividing by the multiplier.
Solution
- Amount = P × 1.05².
- So P = 3,307.50 ÷ 1.05² = 3,307.50 ÷ 1.1025.
- P = $3,000.
Answer: A. $3,000.
7.$8,500 earns compound interest at 4% in the first year and 6% in each of the next 2 years. Find the amount after 3 years, to the nearest cent.
Correct.
Solution
- Multipliers: 1.04, then 1.06, then 1.06.
- A = 8,500 × 1.04 × 1.06² = 9,932.624.
- To the nearest cent: $9,932.62.
Not quite. This uses the average rate, 5%, for all 3 years. Use each year’s own multiplier.
Solution
- Multipliers: 1.04, then 1.06, then 1.06.
- A = 8,500 × 1.04 × 1.06² = 9,932.624.
- To the nearest cent: $9,932.62.
Answer: A. $9,932.62.
Not quite. This uses 6% for only one year. There are 2 years at 6%.
Solution
- Multipliers: 1.04, then 1.06, then 1.06.
- A = 8,500 × 1.04 × 1.06² = 9,932.624.
- To the nearest cent: $9,932.62.
Answer: A. $9,932.62.
Not quite. This adds 4% + 6% + 6% = 16%. Multiply the multipliers instead.
Solution
- Multipliers: 1.04, then 1.06, then 1.06.
- A = 8,500 × 1.04 × 1.06² = 9,932.624.
- To the nearest cent: $9,932.62.
Answer: A. $9,932.62.
Challenge (Harder problems that mix ideas or need reasoning.)
8.$2,800 is invested for 3 years at 5% per year, with the interest added every 6 months. Find the amount at the end, to the nearest cent.
Not quite. This adds interest once a year. Here it is added every 6 months.
Solution
- Each half-year the rate is 5% ÷ 2 = 2.5%.
- 3 years = 6 half-years.
- A = 2,800 × 1.025⁶ = 3,247.141…, which is $3,247.14.
Answer: B. $3,247.14.
Correct.
Solution
- Each half-year the rate is 5% ÷ 2 = 2.5%.
- 3 years = 6 half-years.
- A = 2,800 × 1.025⁶ = 3,247.141…, which is $3,247.14.
Not quite. This uses 5% for each half-year. Each half-year gets half the rate: 2.5%.
Solution
- Each half-year the rate is 5% ÷ 2 = 2.5%.
- 3 years = 6 half-years.
- A = 2,800 × 1.025⁶ = 3,247.141…, which is $3,247.14.
Answer: B. $3,247.14.
Not quite. This uses 3 periods. There are 6 half-years in 3 years.
Solution
- Each half-year the rate is 5% ÷ 2 = 2.5%.
- 3 years = 6 half-years.
- A = 2,800 × 1.025⁶ = 3,247.141…, which is $3,247.14.
Answer: B. $3,247.14.
9.Sasha invests some money at 8% compound interest per year. After 2 years she has earned $1,164.80 in interest. How much did she invest?
Not quite. This treats $1,164.80 as the amount. It is only the interest: 0.1664 of the money invested.
Solution
- After 2 years the money is multiplied by 1.08² = 1.1664.
- So the interest is 0.1664 of the money invested.
- P = 1,164.80 ÷ 0.1664 = $7,000. Check: 7,000 × 1.1664 − 7,000 = 1,164.80 ✓.
Answer: C. $7,000.
Not quite. This uses simple interest: 2 × 8% = 16%. Compound interest gives 1.08² − 1 = 0.1664.
Solution
- After 2 years the money is multiplied by 1.08² = 1.1664.
- So the interest is 0.1664 of the money invested.
- P = 1,164.80 ÷ 0.1664 = $7,000. Check: 7,000 × 1.1664 − 7,000 = 1,164.80 ✓.
Answer: C. $7,000.
Correct.
Solution
- After 2 years the money is multiplied by 1.08² = 1.1664.
- So the interest is 0.1664 of the money invested.
- P = 1,164.80 ÷ 0.1664 = $7,000. Check: 7,000 × 1.1664 − 7,000 = 1,164.80 ✓.
Not quite. This uses one year of interest only.
Solution
- After 2 years the money is multiplied by 1.08² = 1.1664.
- So the interest is 0.1664 of the money invested.
- P = 1,164.80 ÷ 0.1664 = $7,000. Check: 7,000 × 1.1664 − 7,000 = 1,164.80 ✓.
Answer: C. $7,000.
10.$10,000 is invested for 2 years at 6% per year. Account X adds the interest once a year. Account Y adds it every 6 months. How much more does account Y pay, to the nearest cent?
Not quite. This uses 6% for each of the 4 half-years in Y. Each half-year gets 3%.
Solution
- X: 10,000 × 1.06² = $11,236.
- Y: 10,000 × 1.03⁴ = 11,255.0881.
- Difference: $19.09 more in account Y.
Answer: B. $19.09.
Correct.
Solution
- X: 10,000 × 1.06² = $11,236.
- Y: 10,000 × 1.03⁴ = 11,255.0881.
- Difference: $19.09 more in account Y.
Not quite. This uses only 2 half-years in Y. There are 4 half-years in 2 years.
Solution
- X: 10,000 × 1.06² = $11,236.
- Y: 10,000 × 1.03⁴ = 11,255.0881.
- Difference: $19.09 more in account Y.
Answer: B. $19.09.
Not quite. The yearly rate is the same, but Y adds interest sooner, so Y earns interest on more of its interest.
Solution
- X: 10,000 × 1.06² = $11,236.
- Y: 10,000 × 1.03⁴ = 11,255.0881.
- Difference: $19.09 more in account Y.
Answer: B. $19.09.
Saving for a course
Nadia puts $1,800 into an account that pays 3.8% compound interest per year. Give money answers to the nearest cent.
1. Find the amount in Nadia’s account after 3 years.
2. How much interest has she earned?
3. Her friend puts $1,800 into an account paying 4% simple interest per year. Who earns more interest in 3 years, and by how much?
- A = 1,800 × 1.038³ = 2,013.096…, so the amount is $2,013.10.
- Interest = $2,013.10 − $1,800 = $213.10.
- Friend: 1,800 × 4 × 3 ÷ 100 = $216. The friend earns $2.90 more: over a short time, a higher simple rate can beat a lower compound rate.
Next: the Lesson 1 exit quiz. Five questions, recorded on your dashboard. Aim for 4 out of 5.