Lesson 17 · Ratio, Proportion and Percentage

Percentage

MathematicsSubject
14 minEstimated read
A common bottom that everybody agreed on

Per cent means out of a hundred. 17 out of 20 and 41 out of 50 become 85 and 82, and the comparison takes a second.

The three conversions

From To What to do
FractionPercentageMultiply by 100: 3/4 × 100 = 75%
PercentageFractionPut it over 100 and reduce: 75/100 = 3/4
DecimalPercentageMove the point two places right: 0.75 becomes 75%

Moving the point two places is not a separate rule. Multiplying by a hundred moves every digit two columns left. So 1.25 is 125 per cent, which is perfectly legal.

Finding a percentage of an amount

Find ten per cent by moving the point one place, then build what you need. Ten per cent of 800 is 80, five per cent is 40, so fifteen per cent is 120.

Percentage increase and decrease

Change divided by the ORIGINAL amount, times a hundred. 400 to 460 is a change of 60, and 60 over 400 is 15 per cent.

A rise of 20% then a fall of 20% does not bring you back

100 rises 20% to 120. A 20% fall is then calculated on 120, not 100, so it removes 24 and leaves 96. The two percentages were taken from different starting points.

Percentages of different wholes cannot be added

60% of 20 girls and 50% of 40 boys is not 55%. Count the people: 12 and 20 passed, which is 32 out of 60, about 53%.

Working backwards to the original

After a 20% discount, 720 is 80% of the original, not 100%. One per cent is 9, so the original was 900. Adding 20% back onto 720 gives the wrong answer.

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Worked Example 1

Sita scored 17 out of 20 in maths and 41 out of 50 in science.

The wholes are differentA mark out of 20 is worth more than a mark out of 50.
Rewrite both out of a hundredMaths 85%, science 82%
AnswerMaths, even though 41 looks larger than 17.
Worked Example 2

A bag costs Rs 800, with a 15% discount.

Find ten per cent first10% is 80, so 5% is 40
Build the fifteen per cent80 + 40 = Rs 120 discount
Take it offRs 680
A faster route85% remains, so 0.85 × 800 = 680 in one step.
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Activity 17.1

Test the discount claims

What you needThree or four real shop adverts showing discounts, a pencil and paper.
MethodWork out the actual saving in rupees, the price you would pay, and what percentage of the original that is.
The interesting oneIf you find a discount on top of a discount, work out the single percentage it really equals.
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1

Per cent means out of a hundred. It is the common bottom everybody agreed on.

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13 more points to remember - sign in to see the rest.

1What does per cent actually mean, and why is it so useful?
2Why does a 20% rise followed by a 20% fall not bring a price back to where it started?
3A shirt costs Rs 720 after a 20% discount. Why is the original price not Rs 864?
4Why can percentages of different groups not simply be averaged?
5How would you find 15% of Rs 800 without a calculator, and why does that method work?

Question 1 of 10

1Per cent means
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Slide 1 of 4
Unit 17 · Percentage

Percentage

Mathematics · Grade 7

What You Will Be Able To Do

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Convert freely between fractions, decimals and percentages

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Find a percentage of any amount, with or without a calculator

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Work out percentage increase and decrease, and work backwards

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Check whether a shop discount claim is really what it appears to be

Percentage is just the common bottom trick from fractions, with everybody agreeing in advance to use a hundred.

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Presenter notes: Open with two marks on the board: 17 out of 20 and 41 out of 50. Ask which is better and take a vote. Many will pick 41 because it is the larger number, which is exactly the difficulty percentages exist to solve. Then ask why every shop sign, every exam result and every news report about elections uses percentages rather than raw counts. By the end of the lesson they will be able to answer that in one sentence.