Year 11 Standard · Unit 3 Sequences, formulae and proportion · Lesson 5 of 8

Why does the same percentage give more each year?

Interest and depreciation change an amount by the same percentage year after year. This lesson asks what that does to the total, and how to reach it in one calculation.

Number and AlgebraScientific calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 7: find 10% of 4500.
    Answer450.
  2. From Year 8: increase 200 by 20%.
    Answer240. 200 × 1.2.
  3. From lesson 4: find the common ratio of 3, 12, 48.
    Answer4.
Words used in this lesson
multiplier
the number you multiply by to apply a percentage change. For example: 1.07 for a 7% rise.
compound interest
interest that is added to the balance and then earns interest itself. For example: 3% a year on a growing balance.
depreciation
a fall in value as something ages. For example: a van losing 12% a year.
growth factor
another name for the multiplier when it is more than 1. For example: 1.2.
decay factor
the multiplier when it is less than 1. For example: 0.7.

Is 5% for eight years the same as 40%?

Wanjiru in Nairobi puts 4000 shillings into a savings account that adds 5% of the balance at the end of every year. She leaves the money there and does not touch it for eight years. Her brother says that is the same as getting 8 × 5% = 40% extra.

Predict first: will the extra be more than 40%, exactly 40%, or less than 40%? Say why. Then use the figure: set the rate to 5 and move the years slider, and compare the curve with the dashed straight line.

10
0
×1.1multiplier each year
1000value at that year

Start 1000, change +10% a year (multiplier 1.1): after 0 years the value is 1000. Adding the same amount each year would give 1000.

Show me the steps
  1. Change +10% a year, 0 years: the value is 1000 and the multiplier is 1.1.
  2. Change +10% a year, 3 years: the value is 1000 × 1.13 = 1331. A rule that added 100 each year would give 1300, so the curve is above the dashed line.
  3. Change −10% a year, 3 years: the multiplier is 0.9 and the value is 1000 × 0.93 = 729. Each year takes 10% of a smaller amount, so the falls get smaller.
  4. Any change p% a year: the value after t years is the start × (1 + p/100)t, a geometric sequence whose ratio is the multiplier.

A percentage change is a multiplier, and repeated change is a power. A rise of p% is multiplied by 1 + p/100 (a 6% rise is ×1.06) and a fall of p% by 1 − p/100 (a 15% fall is ×0.85). After t years the amount is start × multipliert.

This is a geometric sequence with first term the start and ratio the multiplier. The curve bends because each year's change is a percentage of an amount that has already changed.

Two common mistakes. Adding the percentages: two years at 7% is 1.072 = 1.1449, which is 14.49% extra, not 14%. And using the rate as the multiplier: a 15% fall keeps 85%, so it is ×0.85, not ×0.15.

Worked examples

1. 40 000 rupees earns 5% compound interest a year. What is it worth after 3 years?

  1. The multiplier is 1 + 0.05 = 1.05.
  2. 40 000 × 1.053 = 40 000 × 1.157625 = 46 305 rupees.

2. A tractor worth 30 000 loses 10% of its value each year. What is it worth after 4 years?

  1. The multiplier for a 10% fall is 1 − 0.1 = 0.9.
  2. 30 000 × 0.94 = 30 000 × 0.6561 = 19 683.

3. A price rises from 800 to 1058 in 2 years, by the same percentage each year. Find the percentage.

  1. The total multiplier is 1058 ÷ 800 = 1.3225, which is multiplier2.
  2. The multiplier is the square root, 1.15, so the rise is 15% a year.

Your go. The first step is given. 6000 pesos earns 3% compound interest a year. How much is it worth after 2 years, to the nearest centavo? The multiplier is 1.03.

  1. The amount after 2 years is 6000 × 1.032.
  2. Work it out on a calculator, keeping all the digits until the end, and round to two decimal places.
Check the answer

1.032 = 1.0609, so 6000 × 1.0609 = 6365.4, which is 6365.40 pesos.

With a calculator

Find the multiplier first and write it down: a 6% rise is 1.06 and a 15% fall is 0.85. Then use the power key, usually marked xy. To find 3000 × 1.064, type 3000 × 1.06, press the power key, type 4 and press equals.

Estimate first. Four years of about 6% a year is a little more than 24% extra, so a little over 3720. If the display shows something close, you pressed the right keys. Keep every decimal in the calculator until the end, and round only the final answer.

Check before you go on

1. A laptop loses 35% of its value each year. By what number is its value multiplied each year?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A town has 8000 people and its population grows by 5% each year. How many people are there after 2 years?

3. A car is worth 6000 and loses 20% of its value each year. What is it worth after 3 years?

Level 2 · Building

4. 2500 pesos is saved at 4% compound interest a year. How many pesos are there after 2 years?

5. 4000 rupees earns 10% compound interest a year. After how many whole years is it first worth more than 5000?

Level 3 · Stretch

6. The number of members of an online club grows from 7000 to 15 379 in 3 years, by the same percentage each year. What is the yearly percentage increase?

7. A phone costs 24 000. It loses 25% of its value in the first year and then 10% of its new value in the second year. What is it worth after 2 years?

Something to think about

A price rises by 10% and then falls by 20%. Does it matter if the 20% fall comes first? Is the final price 10% lower than the start?

One way to think about it

The order does not matter, because 1.1 × 0.8 = 0.8 × 1.1 = 0.88. But the final price is 12% lower, not 10%: the 20% is taken from a bigger amount after the rise, so the percentages cannot just be added.

Exit ticket

8. A value of 600 rises by 8% each year for 5 years. Which calculation gives its value at the end?