Year 7NumberNumber structures and operationsYear 7 Maths

Percentages, fractions of amounts, and proportion

What this lesson covers

  • Multiplying a whole number by a fraction and finding that fraction of that whole number have the same result.
  • Percentages are decimal fractions with denominators of 100; they are represented using the percent symbol %.
Key idea

40% of a number is 28

  • What is the number?
  • Most people try to guess and check.
  • There is a picture that just tells you.
40% of a number is 28
Key idea

First, a fraction of an amount

  • Divide by the bottom, multiply by the top.
  • That order, every time.
  • It works even when the fraction is top-heavy.
QuestionStepWorking
5/3 of 186divide by the bottom186 ÷ 3 = 62
multiply by the top62 × 5 = 310
Answerbigger than 186310
Key idea

5/3 of 186 is bigger than 186

  • Five thirds is more than one whole.
  • So taking five thirds of something grows it.
  • 310, not something smaller.
5/3 of 186 is bigger than 186
Key idea

A percentage is hundredths

  • 40% means 40 out of every 100.
  • Which is the fraction 40/100, or 0.4.
  • So percentages are fractions with a fixed bottom.
A percentage is hundredths
Key idea

The ones you can do in your head

  • 50% is half. 25% is half of that.
  • 10% is a shift of one column.
  • And 5% is half of 10%.
PercentageHowSame as
50%halve it÷ 2
25%halve it twice÷ 4
10%shift one column right÷ 10
5%half of 10%÷ 20
1%shift two columns right÷ 100
Key idea

Find 10% first, then scale it

  • 10% is the easiest one to get.
  • Every other percentage is built from it.
  • 30% is three lots. 5% is half a lot.
PercentageHowSame as
50%halve it÷ 2
25%halve it twice÷ 4
10%shift one column right÷ 10
5%half of 10%÷ 20
1%shift two columns right÷ 100
Worked example

15% of 60

  • 10% of 60 is 6.
  • 5% is half of that, so 3.
  • 6 + 3 = 9.
StepWorkingAnswer
10% of 6060 ÷ 106
5% of 60half of 63
15%6 + 39
Key idea

Two scales, side by side

  • Percentages on top, the real amount below.
  • Both start at zero and end together.
  • Every tick lines up with its partner.
Two scales, side by side
Worked example

Reading it downwards: 40% of 70

  • Find 40% on the top scale.
  • Look straight down.
  • 28.
Reading it downwards: 40% of 70
Worked example

The same thing in numbers

  • 10% of 70 is 7.
  • 40% is four lots of 7.
  • 28.
StepWorkingAnswer
10% of 7070 ÷ 107
40%4 lots of 728
Worked example

Now upwards: 40% is 28

  • Same picture, different end known.
  • 28 sits at the 40% tick.
  • Read along to 100%: the whole is 70.
Now upwards: 40% is 28
Worked example

In numbers, through 10%

  • If 40% is 28, then 10% is 28 ÷ 4 = 7.
  • 100% is ten lots of 10%.
  • 7 × 10 = 70.
StepWorkingAnswer
40% is28
10% is28 ÷ 47
100% is7 × 1070
Worked example

5/4 of a set is 85. How big is the set?

  • Five quarters is 85, so one quarter is 17.
  • The whole set is four quarters.
  • 17 × 4 = 68.
StepWorkingAnswer
5/4 of the set is85
So 1/4 is85 ÷ 517
And 4/4 is17 × 468
Key idea

Why 68 is smaller than 85

  • Five quarters is MORE than the whole set.
  • So the whole set has to be less than 85.
  • If your answer came out bigger, check it.
Why 68 is smaller than 85
Key idea

3 red for every 2 blue

  • Not 3 red and 2 blue in total.
  • A repeating group: 3 red with 2 blue, over and over.
  • So count the groups.
3 red for every 2 blue
Worked example

12 red. How many blue?

  • 12 red is 12 ÷ 3 = 4 groups.
  • Each group has 2 blue.
  • 2 × 4 = 8 blue.
StepWorkingAnswer
For every3 red2 blue
12 red is12 ÷ 3 = 4 groups
So blue is2 × 48
Your turn

Your turn: four questions

  • Pause here and try all four.
  • On question 3, find 10% before anything else.
  • On question 4, count the groups.
Question
1Find 3/8 of 96.
2Find 35% of 80.
360% of a number is 48. What is the number?
45 apples for every 3 oranges. With 20 apples, how many oranges?
Show answer

1. 36. 96 ÷ 8 = 12, then 12 × 3 = 36. 2. 28. 10% is 8, so 30% is 24; 5% is 4; 24 + 4 = 28. 3. 80. If 60% is 48 then 10% is 8, so 100% is 80. 4. 12. 20 apples is 4 groups of 5, so oranges are 3 × 4 = 12.

Worked example

The answers

  • 3/8 of 96 is 36. 35% of 80 is 28.
  • 60% is 48, so 10% is 8 and the number is 80.
  • 20 apples is 4 groups, so 12 oranges.
Answer
136
228
380
412
Summary

What to take away

  • Find 10% first; everything else is built from it.
  • Going backwards, get to 10%, then up to 100%.
  • The double number line reads in both directions.
Remember
A fraction of an amountdivide by the bottom, times the top
10% firstthen scale it to any percentage
Going backwardsget to 1% or 10%, then up to 100%
Double number lineworks in both directions
Ratioscount the groups, then scale

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