M4.2.3 Chapter 3

Simple and compound interest

Two savings clubs both add 5% a year to the coins I put in. Why does one of them give more?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A One adds 5% of the start amount each year. The other adds 5% of the amount that has grown.
You picked A. Read on. At the end, the chapter comes back to your guess.
B They cannot differ. 5% a year is 5% a year.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The one that ends with more must have a higher rate.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Work out simple interest on 2 000 coins at 4% a year
Find the value after 3 years of 5% compound interest
Compare simple and compound interest in a table over 10 years

A saver puts 2 000 coins into a savings club. Each year the club adds interest, a percent of the savings. Here the rate is 4% a year.

Simple interest is the same amount every year. It is always 4% of the start amount. 4% of 2 000 is 80, so the saver gains 80 coins each year.

Simple interest on 2 000 coins at 4% a year. The interest is the same every year.

Worked example

Work out the simple interest on 2 000 coins at 4% a year for 5 years.

One year of interest: 4% of 2 000 = 0.04 × 2 000 = 80.
5 years of interest: 5 × 80 = 400.
The total is 2 000 + 400 = 2 400 coins.

Answer: 400 coins of interest

To find simple interest, multiply the start amount by the rate as a decimal, and then by the number of years.

Compound interest is different. Each year the interest is worked out on the whole amount, including interest already added. The amount is multiplied by the same multiplier every year. For 5% the multiplier is 1.05.

8 000 coins at 5% compound interest. The interest grows because the start of each year is bigger.

Worked example

8 000 coins earn 5% compound interest a year. How many coins are there after 3 years?

Each year the amount is multiplied by 1 + 0.05 = 1.05.
After year 1: 8 000 × 1.05 = 8 400.
After year 2: 8 400 × 1.05 = 8 820.
After year 3: 8 820 × 1.05 = 9 261.

Answer: 9 261 coins

The three steps multiply by 1.05 three times. That is 1.05 × 1.05 × 1.05, written 1.05³. So the value after n years is the start amount times the multiplier, n times over.

See it three ways

With things

Put in 1 000 coins at 10% a year. After one year add a tenth: 1 100. After year two add a tenth of 1 100: 1 210.

As a picture

In symbols

1 000 × 1.1 × 1.1 = 1 000 × 1.21 = 1 210.

Now compare the two over 10 years. Take 8 000 coins at 5% a year, both ways. In year 1 they are equal. After that, compound interest pulls ahead.

Draw it first

Simple and compound on one graph

1 Simple interest adds the same amount each year, so its graph is a straight line.
2 Compound interest adds more each year, so its graph bends upward. The curve starts on the line, then rises above it.

Compound values are rounded to the nearest coin. The two match in year 1, and then the gap widens each year.

Worked example

8 000 coins earn 5% a year for 10 years. How many more coins does compound interest give than simple interest?

Simple interest for 10 years: 8 000 × 0.05 × 10 = 4 000.
So the simple total is 8 000 + 4 000 = 12 000.
A calculator multiplies 8 000 by 1.05 ten times and gives 13 031, to the nearest coin.
Compound gives more by 13 031 − 12 000 = 1 031.

Answer: 1 031 coins more

Try it

Make your own table

You need: paper, a pencil.

Start with 1 000 coins and a rate of 10% a year. Draw three columns: year, simple and compound.
For simple interest, add 100 each year.
For compound interest, multiply the last amount by 1.1 each year. Round to the nearest coin.

What you will see: The two columns match in year 1. In year 2 compound is already ahead, with 1 210 against 1 200, and the gap grows.

If you cannot try it, this is what happens.

A wrong idea many people have

8 000 coins at 5% compound interest for 3 years earns 3 × 5% = 15%, so the total is 9 200.

Why it feels right: Three years of 5% looks like 15%, and it is the answer simple interest would give.

The test that breaks it: Work year by year: 8 000 → 8 400 → 8 820 → 9 261. The total is 9 261, which is more than 9 200.

What is true: Each year's interest is worked out on a bigger amount. Multiply by 1.05 three times, not by 1.15 once.

Key words

interest: extra amount added to savings each year, worked out as a percent.
simple interest: interest that is always a percent of the start amount, so it is the same every year.
compound interest: interest worked out on the amount so far, including interest already added.

Back to your guess

The answer is A: One adds 5% of the start amount each year. The other adds 5% of the amount that has grown.

The rate can be the same and the totals still differ. Simple interest is always worked out on the start amount. Compound interest is worked out on the amount that has grown, so it earns interest on interest.

Keep it

Copy this into your exercise book.

Simple interest: start × rate × years, the same each year. Compound interest: multiply by 1 + rate once for every year. Compound is ahead after year 1.

Draw: A graph with a straight line named simple and a curve named compound above it.