Percentages (II) · Ages 13–14
Compounding periods
Rate a period = yearly rate ÷ periods a year; periods = years × periods a year.
How to do it
Method
Rate a period = yearly rate ÷ periods a year; periods = years × periods a year
10000 × 1.044 = 11698.5856
Another way
Period by period
10000 → 10400 → 10816 → 11248.64 → 11698.5856
In full
A yearly rate is the rate for a whole year. Compounded half-yearly, each period's rate is the yearly rate ÷ 2, with 2 periods a year; quarterly, ÷ 4, with 4 periods a year; monthly, ÷ 12, with 12. So A = P × (1 + r%/k)^(k × n), with k periods a year for n years. $10000 at 8% a year, compounded half-yearly, for 2 years: each period earns 8% ÷ 2 = 4%, and there are 2 × 2 = 4 periods. One way, the formula: 10000 × 1.044 = 11698.5856, about $11699. Another, period by period: 10000 → 10400 → 10816 → 11248.64 → 11698.5856. Compounded yearly it makes only 10000 × 1.082 = $11664: at the same yearly rate, the more often it compounds, the more interest. The usual slips: 8% for every period, or 2 periods instead of 4.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
$36,000 at 10% a year, compounded every 6 months, for 2 years. The amount = $ (to the nearest dollar)
Show the answer and working
Answer: 43,758
Each period earns 10% ÷ 2 = 5%, over 2 × 2 = 4 periods: 36000 × 1.054, about 43758.
Example 2
$5,000 at 4% a year for 4 years. Which way of paying the interest gives the most at the end?
- A. Compounded yearly
- B. Simple interest
- C. Compounded every 6 months
- D. Compounded every 3 months
Choose the right answer
Show the answer and working
Answer: D. Compounded every 3 months
At the same yearly rate, the more often it compounds, the sooner interest joins the principal and earns more: every 3 months gives the most.
Example 3
$4,000 at 10% a year, compounded every 6 months: the amount after 6 months is $, after a year $, after a year and a half $.
Fill in each blank
Show the answer and working
Answer: 4,200, 4,410, 4,630.5
Each period earns 10% ÷ 2 = 5%, times 1.05 every 6 months: 4200, 4410, 4630.5. Or in one step: 4000 × 1.053 = 4630.5.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.At 10% a year compounded every 3 months, each period's rate is 10%.
True or false?
Answer:
2.$21,000 at 4% a year, compounded every 6 months, for 3 years. The amount = $ (to the nearest dollar)
Answer:
3.$20,000 at 12% a year for 3 years. Which way of paying the interest gives the most at the end?
- A. Simple interest
- B. Compounded every 3 months
- C. Compounded yearly
- D. Compounded every 6 months
Choose the right answer
Answer:
4.2% a year compounded every 6 months is the same as % a year compounded yearly
Answer:
5.$2,000 at 4% a year, compounded every 6 months: the amount after 6 months is $, after a year $, after a year and a half $.
Fill in each blank
Answer:
All the answers
- 1. False
- 2. 23,649
- 3. B. Compounded every 3 months
- 4. 2.01
- 5. 2,040, 2,080.8, 2,122.416
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake