Growth & Decay flashcards
10 cards on Growth & Decay: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 10
Growth & Decay
Percentage multipliers
What do you multiply by to increase or decrease an amount by a percentage?
Answer it in your head, then turn the card over.
Percentage multipliers
Key points
- An increase of r%: multiply by 1 + . An 8% rise is × 1.08.
- A decrease of r%: multiply by 1 − . A 12% fall is × 0.88, because 88% is left.
- A multiplier above 1 means growth; below 1 means decay.
Example
Up 8%: 100% + 8% = 108% = 1.08
Down 12%: 100% − 12% = 88% = 0.88
Up 3.5%: 103.5% = 1.035
Watch out
Multiplying by 0.12 finds only the part lost: to find what is left after a 12% fall, use 0.88.
Growth & Decay
Changes one after another
How do you combine several percentage changes in a row, and why can't you add them?
Answer it in your head, then turn the card over.
Changes one after another
Key points
- Multiply the multipliers together to get one overall multiplier.
- Each new change is a percentage of the new amount, not the original.
- Read the overall change from the result: 1.21 is a 21% rise, 0.88 a 12% fall.
Example
10% rise, then another 10% rise:
1.1 × 1.1 = 1.21, a 21% rise (not 20%)
10% rise, then a 20% fall: 1.1 × 0.8 = 0.88, a 12% fall
Watch out
Percentages in a row never simply add: +10% then −20% is a 12% fall, not a 10% fall.
Growth & Decay
Compound interest
What is compound interest, and how do you work out what an investment is worth?
Answer it in your head, then turn the card over.
Compound interest
Key points
- Compound interest is paid on the interest already earned as well as on the original amount.
- So multiply by the multiplier once for each year: use a power.
- Round money to the nearest penny only at the end.
Formula
amount = P ×
£1500 at 4% a year for 3 years
1500 × 1.04³ = 1500 × 1.124864
= £1687.30 to the nearest penny
Watch out
Adding 4% of £1500 three times is simple interest: it gives £1680, too little for compound interest.
Growth & Decay
Interest earned, and simple interest
How is compound interest different from simple interest, and how do you find the interest alone?
Answer it in your head, then turn the card over.
Interest earned, and simple interest
Key points
- Simple interest pays the same amount every year, worked out on the original amount.
- Compound interest grows each year, because each year's total earns interest too.
- If the question asks for the interest, take the starting amount away at the end.
£800 at 5% a year for 2 years
Compound: 800 × 1.05² = 800 × 1.1025 = £882
Interest = £882 − £800 = £82
Simple: 5% of £800 = £40 a year, so £80
Watch out
Read the question: "how much interest" wants £82 here, not the £882 total.
Growth & Decay
Depreciation
How do you work out the value of something that loses a percentage of its value each year?
Answer it in your head, then turn the card over.
Depreciation
Key points
- Depreciation is a loss of value each year: use a multiplier below 1.
- Losing 15% a year leaves 85%, so multiply by 0.85 once a year.
- After n years: value = starting value × 0.85ⁿ.
£12 000 car, losing 15% a year, 3 years
12 000 × 0.85³ = 12 000 × 0.614125
= £7369.50
Watch out
Taking 15% of the original (£1800) off every year gives £6600: each year loses 15% of a smaller value.
Growth & Decay
Why decay slows down
Why does a car losing 20% a year lose less money in its second year than its first?
Answer it in your head, then turn the card over.
Why decay slows down
Key points
- Each year's 20% is of that year's value, which keeps falling.
- So the amount lost shrinks every year: the value falls fast at first, then more slowly.
- Multiplying by 0.8 each year handles this for you.
£9000 car, losing 20% a year
Year 1: loses 20% of £9000 = £1800, leaving £7200
Year 2: loses 20% of £7200 = £1440, leaving £5760
In one step: 9000 × 0.8² = £5760
Watch out
Year 2 loses 20% of £7200, not of £9000: taking £1800 off twice gives £5400, which is too low.
Growth & Decay
The growth and decay formula
How do you write any repeated percentage change as one expression?
Answer it in your head, then turn the card over.
The growth and decay formula
Key points
- Amount after n periods = P × mⁿ.
- P is the starting amount, m the multiplier for one period, and n the number of periods.
- Write the expression first: then it is one calculator step.
Formula
amount = P × mⁿ
5000 people, growing 8% a year
After n years: 5000 × 1.08ⁿ
After 3 years: 5000 × 1.08³ = 6298.56
So about 6299 people
Watch out
The years go in the power: 5000 × 1.08 × 3 gives 16 200, which should look far too big.
Growth & Decay
Reading a growth or decay expression
What does an expression like 2000 × 0.85ⁿ tell you about how an amount changes?
Answer it in your head, then turn the card over.
Reading a growth or decay expression
Key points
- The number in front is the starting amount.
- The multiplier gives the change each period: above 1 is growth, below 1 is decay.
- The power counts the periods, such as years or months.
Example
2000 × 0.85ⁿ: starts at 2000, keeps 85% each period,
so it falls by (1 − 0.85) × 100 = 15% a period
400 × 1.12ⁿ: starts at 400, grows by 12% a period
Watch out
A multiplier of 0.85 is an 85% share left, not an 85% fall: the fall is 15%.
Growth & Decay
How long to pass a target
How do you find how many years an investment takes to pass a target amount?
Answer it in your head, then turn the card over.
How long to pass a target
Key points
- Try increasing powers of the multiplier until the amount passes the target.
- On a calculator, enter the start, then press × 1.05 = again and again, counting the presses.
- Write down the values either side of the target: they prove your answer.
£3000 at 5% a year, to pass £4000
3000 × 1.05⁵ = £3828.84, still short
3000 × 1.05⁶ = £4020.29, past £4000
So it takes 6 years.
Watch out
Stopping a year short: after 5 years it is still under £4000, so the answer is 6, not 5.
Growth & Decay
How long to fall below a target
How do you find how many years a value takes to fall below a target?
Answer it in your head, then turn the card over.
How long to fall below a target
Key points
- Use the decay multiplier and try increasing powers until the value drops below the target.
- The answer is the first whole number of years that gets there.
- Show the year before (still above) and the year it drops below.
£8000 machine, 20% a year, below £4000
8000 × 0.8³ = £4096, still above £4000
8000 × 0.8⁴ = £3276.80, below it
So after 4 years.
Watch out
20% a year does not halve the value in 2.5 years: each loss is of a smaller value, so here it takes 4.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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