Year 7NumberNumber structures and operationsYear 7 Maths

Fractions, decimals and percentages

What this lesson covers

  • A fraction can describe a proportional relationship between two amounts.
  • Every fraction can be represented by an infinite set of equivalent fractions that occupy the same point on the number line.
  • Fractions can be converted to decimals using division, and the result can be a terminating decimal or a recurring decimal.
Key idea

A half plus a third is not two fifths

  • Adding the tops and the bottoms feels right.
  • But two fifths is less than a half.
  • We just added something to a half. It cannot shrink.
A half plus a third is not two fifths
Key idea

What a fraction is

  • Three quarters means three out of every four.
  • It compares two amounts, not one.
  • So it works on 4, on 40, on anything.
What a fraction is
Key idea

Same amount, different cuts

  • One half. Two quarters. Four eighths.
  • Cut it finer and take more pieces.
  • The shaded width never changes.
Same amount, different cuts
Key idea

They are the same number

  • Not just the same amount: the same point.
  • One place on the line, several names.
  • 0.5 and 50% are two more names for it.
They are the same number
Key idea

Four names, one number

  • A fraction. An equivalent fraction. A decimal.
  • A percentage. All the same value.
  • You pick whichever suits the question.
FormWritten
Fraction1/2
Equivalent2/4, 5/10, 50/100
Decimal0.5
Percentage50%
Key idea

Making an equivalent fraction

  • Multiply the top and the bottom by the same number.
  • Both, or the value changes.
  • There are infinitely many of them.
FromDo thisGet
1/2× 2 top and bottom2/4
1/2× 5 top and bottom5/10
1/2× 50 top and bottom50/100
Worked example

Simplest form: 18/24

  • Go the other way: divide instead.
  • The HCF of 18 and 24 is 6.
  • 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so it is 3/4.
StepWorkingResult
18/24HCF of 18 and 24 is 6
÷ 618 ÷ 6 = 3, 24 ÷ 6 = 43/4
Check3 and 4 share nothingdone
Key idea

Fraction to decimal: just divide

  • The line in a fraction means divide.
  • 5/16 means 5 ÷ 16.
  • Which is 0.3125, and it stops there.
FractionDivideDecimal
5/165 ÷ 160.3125 (stops)
7/87 ÷ 80.875 (stops)
Key idea

Sometimes it does not stop

  • 1 ÷ 3 gives 0.333 and it keeps going.
  • 2 ÷ 11 gives 0.1818, repeating forever.
  • That is normal. The division still worked.
FractionDivideDecimal
5/165 ÷ 160.3125 (stops)
7/87 ÷ 80.875 (stops)
1/31 ÷ 30.333... (never stops)
2/112 ÷ 110.1818... (repeats)
Key idea

Percent means out of a hundred

  • 50% is 50 hundredths, which is 0.5.
  • The % sign is shorthand for "out of 100".
  • So a percentage is just a decimal in disguise.
Percent means out of a hundred
Key idea

The ones worth knowing cold

  • These five come up constantly.
  • Knowing them saves working every time.
  • And a third is the only messy one.
FractionDecimalPercentage
1/20.550%
1/40.2525%
3/40.7575%
1/50.220%
1/30.333...33.3%
Worked example

Which is bigger: 3/5 or 0.62?

  • You cannot compare different forms directly.
  • Turn them all into decimals first.
  • 3 ÷ 5 = 0.6, so 0.62 is bigger.
ValueWorkingAs a decimal
3/53 ÷ 50.6
0.62already a decimal0.62
Bigger0.62so 0.62 wins
Key idea

Why you cannot just add the tops

  • Halves and thirds are different sized pieces.
  • You cannot count them together.
  • Cut them both into the same size first.
Why you cannot just add the tops
Worked example

1/2 + 1/3 in sixths

  • Sixths work for both: 2 and 3 both go into 6.
  • A half is 3 sixths. A third is 2 sixths.
  • 3 + 2 = 5 sixths.
1/2 + 1/3 in sixths
Worked example

The same thing written down

  • Find a denominator both go into.
  • Rewrite each fraction with it.
  • Add the tops. Leave the bottom alone.
FractionStepBecomes
1/2becomes3/6
1/3becomes2/6
Add the pieces3 + 2 = 55/6
Worked example

Mixed numbers: 2 1/4 + 1 5/6

  • Turn both into improper fractions first.
  • Twelfths: 27/12 + 22/12 = 49/12.
  • 49 ÷ 12 is 4 remainder 1, so 4 and 1/12.
StepWorkingResult
2 1/4as an improper fraction9/4
1 5/6as an improper fraction11/6
Twelfths27/12 + 22/1249/12
Back to mixed49 ÷ 12 = 4 r 14 1/12
Worked example

Adding decimals

  • Much easier: the pieces already match.
  • Line up the decimal points, not the last digits.
  • 0.73 + 0.80 = 1.53.
Adding decimals
Your turn

Your turn: four questions

  • Pause here and try all four.
  • For question 3, find a denominator both go into.
  • For question 4, turn the fraction into a decimal.
Question
1Write 18/24 in its simplest form.
2Convert 7/8 to a decimal.
3Work out 2/3 + 1/4.
4Which is bigger: 5/8 or 0.6?
Show answer

1. 3/4. The HCF of 18 and 24 is 6, so divide both by 6. 2. 0.875, from 7 ÷ 8. 3. 11/12. Twelfths: 8/12 + 3/12. It is already in simplest form. 4. 5/8, because 5 ÷ 8 = 0.625, which is more than 0.6.

Worked example

The answers

  • 18/24 simplifies to 3/4. And 7 ÷ 8 = 0.875.
  • 2/3 + 1/4 in twelfths is 8/12 + 3/12 = 11/12.
  • 5/8 is 0.625, so it beats 0.6.
Answer
13/4
20.875
311/12
45/8, since it is 0.625
Summary

What to take away

  • Equivalent fractions are one point on the line.
  • To add, cut both into the same size pieces first.
  • Never add the tops and the bottoms.
Remember
Equivalent fractionsthe same point on the line
To simplifydivide top and bottom by their HCF
To decimalisedivide the top by the bottom
To addcut both into the same size pieces first
Neveradd the tops and add the bottoms

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