Growth & Decay Lab

Growth & Decay Lab

Money in a savings account, a car losing value, a population rising: all are repeated percentage change. Each period you apply the same multiplier again. This lab builds straight on the multiplier method: the trick is to raise the multiplier to a power. Goggles on.

Multipliers, again

The multiplier for a percentage change is what you multiply BY, in one step. Get these automatic and the whole topic follows:

Build the multiplier

To increase by 8% you multiply by _____. To decrease by 35% you multiply by _____. To increase by 150% you multiply by _____. And to apply any of these over n years, you raise the multiplier to the _____ of n.

Compound growth

Compound interest adds interest, then earns interest on that interest. Each year you multiply by the same multiplier: £100 at 10% per year for 2 years = 100 × 1.1 × 1.1 = 100 × 1.1² = £121. The power is simply the number of years: three years would be 1.1³, ten years 1.1¹⁰. Notice you never work out the interest itself. Multiplying by 1.1 adds the 10% and keeps the original in one move.

Compound interest

£2000 is invested at 5% compound interest per year. How much is it worth after 2 years, in pounds?

Order the method

Put the steps for any repeated percentage change into the right order.

• Write the multiplier for the percentage change
• Raise it to the power of the number of periods
• Multiply the starting amount by that
• Check the answer is sensible: larger than the start for growth, smaller for decay

Compound against simple

Both start from the same rate and the same deposit, and they part company from the second year onwards. Take £100 at 10% per year:

Plot the growth

A colony of 100 bacteria DOUBLES every hour, which is a multiplier of 2. The y-axis is the population in hundreds, so 1 means 100 bacteria. Plot the population at 0, 1, 2, 3 and 4 hours.

The gap over time

After 2 years, £100 at 10% gives £121 with compound interest against £120 with simple: a difference of just £1. What happens to that gap over 20 years?

• It grows, and quickly, because compound keeps earning interest on interest while simple never does
• It stays at about £1, because the interest rate has not changed
• It shrinks, because the original deposit matters less as time passes
• Simple overtakes compound once enough years have passed

Depreciation and decay

Things that lose value work exactly the same way, but with a multiplier below 1. A car depreciating 20% a year uses × 0.8 each year: £200 for 2 years = 200 × 0.8² = 200 × 0.64 = £128. Now the trap. Two successive falls of 20% do not make a fall of 40%. 0.8 × 0.8 = 0.64, which is a 36% fall overall, because the second 20% comes off an amount that has already shrunk. The same applies to growth: two rises of 10% give 1.21, an increase of 21% rather than 20%.

Depreciation

A van worth £8000 depreciates by 25% each year. What is it worth after 2 years, in pounds?

Settle the argument

A friend says: "Simple and compound interest are basically the same thing, so it makes no difference which account I pick." Using £100 at 10% per year as your example, explain why they are wrong.

• Say what simple interest does each year, and what it is a percentage OF
• Say what compound interest does differently
• Work out both after 2 years, and state the difference
• Say what happens to that difference over a much longer period, and why

The lab rules

For repeated percentage change, raise the _____ to the power of the number of years. Compound interest beats simple interest because it applies the percentage to the _____ amount each year. A depreciation multiplier is _____ than 1, and two successive falls of 20% give an overall fall of _____ rather than 40%.