Percentages of amounts

15% of $80 is 0.15 × 80 = $12. Take it off for a discount ($68, or 80 × 0.85 in one step), add it on for an increase ($92, or 80 × 1.15). And 18 out of 24 is 18 ÷ 24 = 75%.

A jacket is $80, and the sign says 15% off. Before you reach the counter, can you work out what you will pay?

A percentage is a number of hundredths: $15\% = \tfrac{15}{100} = 0.15$ . So 15% of an amount is $0.15 \times$ the amount, the same "of means multiply" as for fractions.

Three questions come up again and again:

A percentage of an amount: 15% of $80 is 0.15 × 80 = $12.
An increase or decrease: add it on or take it off, or multiply by what is left ( $100\% - 15\% = 85\%$ , so $\times 0.85$ ).
What percentage one amount is of another: divide, then $\times 100$ .

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.

15% off

10% of $80 is $8. What is 5%?

A jacket costs $80. It is 15% off. What do you pay?$68$65$12$92

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

Take it off

The orange part is taken off; the blue part is what you pay.

Try it yourselfYour edits are saved in this browser.

The bar is the $80 price. Set the discount to 15%. What do you pay? Was your prediction right?

Use the board. This ticks when you get there.

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

Of, off and out of

Here is each kind with $80.

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

$10\% \text{ of } 80 = 8, \quad 5\% \text{ of } 80 = 4$ Mental: 10% is a tenth, 5% is half of that.
$15\% \text{ of } 80 = 0.15 \times 80 = 12$ Or as a decimal: 15% = 0.15.
$80 - 12 = 68 \quad \text{or} \quad 80 \times 0.85 = 68$ A decrease: take it off, or keep 85% in one step.
$80 \times 1.15 = 92$ An increase: 100% + 15% = 115%, so × 1.15.
$18 \div 24 = 0.75 = 75\%$ What percentage: divide, then × 100.

Now try one: 35% of 60.

Read Grace's working. Is $10 a sensible price?

Find 35% of 60.

Grace finds the sale price of a $50 game that is 20% off. Click the line where her working goes wrong.

Ari scored 18 out of 24. What percentage is that?

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing.

Need a hint?

GST in Australia is 10%.

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A TV is $250 before GST. Add 10% GST. What is the price?

Use the board. This ticks when you get there.

Increase 60 by 25%.

Decrease 45 by 40%.

In a class of 30, 12 students walk to school. What percentage walk?

Edge case: a $100 price goes up 10%, then the new price goes down 10%. What is it now?$90$99$101$100

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

Percentages

Find 12.5% of 64. (12.5% is $\tfrac{1}{8}$ .)

What do you multiply by to decrease an amount by 30%?

A plant 120 mm tall grows 5% taller. How tall is it now, in mm?

Review: work out $\tfrac{2}{3} \times \tfrac{3}{4}$ in simplest form.

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

Save $15

Then find the pair with the lowest sale price.

Try it yourselfYour edits are saved in this browser.

Need a hint?

10% of $150 is $15. Find another.

Another hint

Try 25% of something, or 50%.

Set a price and a discount that save exactly $15.

Use the board. This ticks when you get there.

For teachers

Students use a percentage bar to find a percentage of an amount, apply increases and decreases (in two steps and with one multiplier), and express one amount as a percentage of another (AC9M7N06, node 3). About 45 minutes.

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

Students will be able to

I can find a percentage of an amount, mentally and with a decimal.
I can increase or decrease an amount by a percentage, including with one multiplier.
I can write one amount as a percentage of another.

Before class

Do the Run board yourself first. Students need percentages as decimals (Fractions, decimals and percentages) and multiplying by a fraction less than 1 (previous lesson).

The one idea: a percentage is hundredths, and "of" means multiply. For a decrease, multiply by what is left (Grace multiplies by what is taken off). Varied structures matter: of, off, on, and out of.

Calculators are fine for the decimals; the mental route (10%, 5%, 1%) builds number sense and checks calculator answers.

Running the lesson

Predict: students who chose $12 stopped at the discount; $65 took 15 dollars off.
Run: ask what percentage the blue part is, and how that gives a one-step multiplier.
Investigate: compare the mental route and the decimal route for 35% of 60.
Modify: for up then down, take a vote before working it out.
Make: collect pairs on the board; order them by sale price.

The hard part (on purpose)

Choosing the multiplier for a decrease. Ask what percentage is left after the discount before students touch the calculator.

When students get stuck

A student multiplies by the percentage taken off for a decrease (Grace's mistake). Ask: Is your answer the price or the saving? Fix: Multiply by what is left: 100% − 20% = 80%.
A student takes off the number, not the percentage (80 − 15 = 65). Ask: 15 what: dollars or hundredths? Fix: Find 15% of 80 first.
A student gives a decimal for 'what percentage' (0.75). Ask: Out of what? Fix: × 100 for a percentage.

Stretch and support

Stretch: A shop increases a price by 25%, then offers 20% off the new price. Is the final price more, less or the same as the start? Explain with multipliers.

Support: Use the bar for every problem; start from 10% and build.

Where it leads

Next: Choosing a strategy (node 4), choosing between fractions, decimals and percentages.

Where it leads: financial maths (profit, GST, discounts in AC9M7N09), and percentage change with repeated multipliers (Year 9 and 10).

What each step is for

15% off: Committing to a sale price shows whether they find the discount and take it off, stop at the discount, or add it.
take it off: Sliding the percentage to 15 shows the discount as part of the bar and the price as what is left: 85% of the whole.
of, off and out of: A percentage of an amount is a multiplication; Grace multiplies by the part taken off instead of what is left. Finding a percentage divides first.
one change at a time: One change at a time: an increase on the board, an increase by a multiplier, a decrease, a percentage of a group, then increase then decrease.
percentages: A friendly percentage, a multiplier, an increase and a percentage of a group, mixed; the multiplier for a decrease is the planted slip; the review keeps multiplying fractions fresh.
save $15: Finding a price and a discount that save exactly $15 shows that many pairs work, and that the saving is percentage × price.

Now you can

Find a percentage of an amount, mentally and with a decimal.
Increase or decrease an amount by a percentage, including with one multiplier.
Write one amount as a percentage of another.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson15. Choosing a strategyThe same calculation can be quick or slow. Change the order (4 × 2.7 × 25 = 100 × 2.7), change the form (0.75 × 36 = 3/4 of 36), or split a number (12 × 2.5 = 24 + 6). Then check the answer is about right.

Source: content/maths/number/percentages/lesson.md