Percentage change

An increase of p% multiplies by (1 + p/100); a decrease multiplies by (1 − p/100). One change after another means multiplying the multipliers, so up 20% then down 20% is × 1.2 × 0.8 = × 0.96: a 4% fall overall, not back to the start.

A shop puts a price up 20% in March and has a 20% sale in April. The sign says 'back to the old price'. Is it?

A percentage change is a multiplication by a multiplier:

an increase of $p\%$ : multiply by $1 + \frac{p}{100}$ (up 15%: × 1.15);
a decrease of $p\%$ : multiply by $1 - \frac{p}{100}$ (down 15%: × 0.85).

When one change follows another, the second is a percentage of the new amount, so multiply the multipliers: up 20% then down 20% is $1.2 \times 0.8 = 0.96$ , a 4% decrease overall. Ask "a percentage of what?" every time.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Up then down

A $100 price goes up 10%, then the new price goes down 10%. What is the final price?$101$90$99$100

TryTry it on the board and compare the result with your prediction.

Two changes

Try it yourselfYour edits are saved in this browser.

Set up 10%, then down 10%. Was your prediction right? Then try 20% each way.

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Multipliers

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$\text{up } 20\%$ \times 1.2
$\text{down } 20\%$ \times 0.8
$1.2 \times 0.8 = 0.96$ Multiply the multipliers.
$0.96 = 96\%$ 4% less than the start: an overall decrease of 4%.

Read Sam's working. 10% of which price?

What do you multiply by for a 15% increase?

Sam works out the jacket's final price. Click the line where his working goes wrong.

A price is multiplied by 0.85. Has it gone up or down, and by how much?Up 15%Down 15%Down 85%Up 85%

ModifyChange the problem on purpose and see what happens.

Back to the start, and discounts

Need a hint?

Up 25% is × 1.25.

Another hint

What times 1.25 is 1? Try down 20%.

Try it yourselfYour edits are saved in this browser.

Go up 25%, then find the decrease that brings it back to exactly $100.

Use the board. This ticks when you get there.

Set two discounts of 20% in a row. Is that the same as 40% off?

Use the board. This ticks when you get there.

A price goes up 50% and then down 50%. What is the overall percentage change? (A decrease is negative.)

A shop buys a hat for $40 and marks it up 30%. Set the board. What is the selling price?

Use the board. This ticks when you get there.

PractiseA few on your own, to check you have it.

Percentage change

Increase $80 by 15%.

What do you multiply by for a 35% decrease?

A $250 bike goes up 20%, then has a 10% discount. What does it cost now, in dollars?

Review: what is 15% of 60?

A price is $80 before GST. GST adds 10%. What is the price with GST, in dollars?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

An overall change

Then explain why the order of the two changes does not matter to the final price.

Try it yourselfYour edits are saved in this browser.

Need a hint?

−28% overall means × 0.72.

Another hint

0.9 × 0.8 = 0.72.

Find two changes that make an overall decrease of exactly 28%.

Use the board. This ticks when you get there.

For teachers

Students model percentage increases and decreases as multipliers, chain two changes, and explain why equal percentages up and down do not cancel (AC9M8N05, node 1). About 35 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can use a multiplier for a percentage increase or decrease.
I can work out the result of two changes, one after the other.
I can explain why up 20% then down 20% is not back to the start.

Before class

Do the Run board yourself first. Students need percentages of amounts and increases and decreases (Number Foundations).

The one idea: each change multiplies the current amount, so successive changes multiply. The common mistake is taking the second percentage of the original amount (Sam's mistake).

The board draws the three amounts as bars with each multiplier between them; the Mark-up task uses the percentage bar.

Wider: population or prices changing by the same percentage each year (repeated multipliers: informal exponential growth).

Running the lesson

Predict: ask '20% of what?' for the second change.
Run: ask whether any pair of equal ups and downs gets back to $100.
Investigate: ask Sam what 10% of $220 is.
Modify: ask whether two 20% discounts are better or worse for the shopper than one of 40%.
Make: ask why the order does not matter (multiplying in either order).

The hard part (on purpose)

Asking 'a percentage of what?'. Ask students to name the amount each percentage is taken of before they calculate.

When students get stuck

A student takes the second percentage of the original. Ask: What is the price just before the second change? Fix: Each percentage is of the current amount.
A student multiplies by 0.15 for a 15% increase. Ask: Have you found the new amount, or just the increase? Fix: New amount: × (1 + 0.15).
A student adds the percentages (−10 and −20 gives −30). Ask: What is 0.9 × 0.8? Fix: Multiply the multipliers: 0.72, a 28% fall.

Stretch and support

Stretch: A town's population grows 10% a year for 3 years. Is that a 30% increase? Find the true overall change.

Support: Write each multiplier under its arrow, and multiply with a calculator.

Where it leads

This starts AC9M8N05.

Where it leads: money models (next lesson), compound interest and exponential growth (Years 9 and 10).

What each step is for

up then down: Committing to the final price after up 10% then down 10% shows whether they take the second percentage of the new amount.
two changes: Trying pairs of changes on the bars shows that equal percentages up and down do not cancel.
multipliers: Each change is a multiplier on the current amount (Sam takes 10% of the old price); the overall multiplier gives the overall change.
back to the start, and discounts: Finding a change that undoes another, comparing two discounts with one, and a mark-up show the multiplier model at work.
percentage change: Multipliers both ways and a two-step change, mixed; taking the second percentage of the old amount is the planted slip; the percentage-of review keeps the basics fresh.
an overall change: Searching for two changes with a given overall multiplier uses the product of multipliers as the model.

Now you can

Use a multiplier for a percentage increase or decrease.
Work out the result of two changes, one after the other.
Explain why up 20% then down 20% is not back to the start.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson26. Money modelsModel a money problem by choosing what to calculate, then check the answer makes sense. Pay is the rate times the hours. Income tax is worked out in brackets: each part of the income is taxed at its own rate, so for incomes above $18 200 the rate on the last dollar (the marginal rate) is higher than the tax as a share of all the income (the effective rate).

Source: content/maths/number/percentage-change/lesson.md