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Arithematics

Arithematics

Fractions

Fractions Paper 1 + 2 9 min read

On this page
  1. In this note
  2. Formula bank
  3. What a fraction means
  4. Adding and subtracting
  5. Multiplying and dividing
  6. Fractions in problems
  7. Ways they test it
  8. Mistakes to avoid
  9. Summary
  10. Practice
  11. Can I do this?
  12. Quiz

In this note

Syllabus 1.4 and 1.6: fractions
  • 1.4Use proper and improper fractions and mixed numbers; write fractions in their simplest form.
  • 1.6Add, subtract, multiply and divide fractions and mixed numbers without a calculator, showing every step.
  • 1.6Find a fraction of an amount, the whole from a part, and solve "fraction of the remainder" problems.

Before you start: factors and multiples. Recurring decimals: recurring decimals.

Formula bank

Mixed → improperLearn it\(a\tfrac bc=\dfrac{a\times c+b}{c}\)\(4\tfrac35=\tfrac{23}5\).Use it when before ANY operation.
Add / subtractLearn it\(\dfrac ab\pm\dfrac cd=\dfrac{ad\pm cb}{bd}\) (or use the LCM)same-size parts first: a common denominator.Use it when + or −.
MultiplyLearn it\(\dfrac ab\times\dfrac cd=\dfrac{ac}{bd}\)cancel any top with any bottom first.Use it when ×, or "of".
DivideLearn it\(\dfrac ab\div\dfrac cd=\dfrac ab\times\dfrac dc\)keep, change, flip (the second one only).Use it when ÷.
Fraction of an amountLearn it\(\dfrac ab\) of N = N ÷ b × aUse it when "find 3/8 of 40".
Whole from a partLearn itwhole = part ÷ \(\dfrac ab\)Use it when "12 is 3/13 of the number".

What a fraction means

\(\tfrac34\) means split one whole into 4 equal parts and take 3. Equivalent fractions are the same amount cut into different-sized parts: \(\tfrac34=\tfrac68\). Multiplying or dividing top and bottom by the same number never changes the value.

3/4 and 6/8 are the same amount3/46/8
The same length shaded, cut into 4 or 8 parts.

General methodHow to simplify and convert

  1. Simplest form: divide top and bottom by their HCF.
  2. Mixed → improper: whole × bottom + top, over the bottom.
  3. Improper → mixed: top ÷ bottom = whole number, remainder over the bottom.

Try it yourself

1
Simplify \(\tfrac{28}{42}\).
Show answer

Answer \(\tfrac23\)

2
Write \(3\tfrac25\) as an improper fraction.
Show answer

Answer \(\tfrac{17}5\)

3
Write \(\tfrac{29}6\) as a mixed number.
Show answer

Answer \(4\tfrac56\)

Adding and subtracting

You can only add parts of the SAME size: thirds and quarters must both become twelfths first.

General methodHow to add or subtract any fractions

  1. Change mixed numbers to improper fractions.
  2. Find the LCM of the bottoms (the common denominator).
  3. Rewrite each fraction over it (multiply top and bottom by the same number).
  4. Add or subtract the tops; keep the bottom.
  5. Simplify; change to a mixed number if asked.

Example Work out \(1\tfrac56+2\tfrac34\).

Improper, common denominator 12, add, back to a mixed number.

  1. \(\tfrac{11}6+\tfrac{11}4\)
  2. \(=\tfrac{22}{12}+\tfrac{33}{12}=\tfrac{55}{12}\)

Answer \(4\tfrac7{12}\)

Try it yourself

1
\(\tfrac23+\tfrac16\)
Show answer

Answer \(\tfrac56\)

2
\(\tfrac56-\tfrac7{12}\)
Show answer

Answer \(\tfrac14\)

3
\(4\tfrac12-2\tfrac56\) (as a mixed number)
Show answer

Answer \(1\tfrac23\)

Multiplying and dividing

General methodHow to multiply any fractions

  1. Change mixed numbers to improper fractions.
  2. Cancel: divide any top and any bottom by a common factor.
  3. Multiply the tops, multiply the bottoms.

Example Work out \(2\tfrac23\times1\tfrac18\).

Improper, cancel, multiply.

  1. \(\tfrac83\times\tfrac98\)
  2. cancel 8 with 8 and 9 with 3: \(\tfrac11\times\tfrac31\)

Answer 3

General methodHow to divide any fractions

  1. Change mixed numbers to improper fractions.
  2. Keep the first, change ÷ to ×, flip the SECOND fraction.
  3. Cancel, then multiply.

Why it works: \(\tfrac34\div\tfrac18\) asks how many eighths fit into \(\tfrac34\): six. And \(\tfrac34\times\tfrac81=6\).

Example Work out \(2\tfrac14\div1\tfrac78\).

Improper, flip the second, cancel.

  1. \(\tfrac94\div\tfrac{15}8=\tfrac94\times\tfrac8{15}\)
  2. cancel: \(\tfrac31\times\tfrac25=\tfrac65\)

Answer \(1\tfrac15\)

Try it yourself

1
\(\tfrac35\times\tfrac{10}{21}\)
Show answer

Answer \(\tfrac27\)

2
\(\tfrac47\div8\)
Show answer

Answer \(\tfrac1{14}\)

3
\(1\tfrac13\div\tfrac8{15}\) (as a mixed number)
Show answer

Answer \(2\tfrac12\)

Fractions in problems

General methodHow to answer fraction word problems

  1. Fraction of an amount: ÷ bottom × top.
  2. One amount as a fraction of another: part over whole (same units), then simplify.
  3. Whole from a part: part ÷ the fraction (or find one part, then all of them).
  4. "Of the remainder": multiply the fractions to get each as a fraction of the WHOLE, then see what is left.

Example Kate eats \(\tfrac14\) of some grapes, then \(\tfrac23\) of the rest. 15 are left. How many were there?

Track the fraction left at each step.

  1. after the first: \(\tfrac34\) left
  2. she eats \(\tfrac23\times\tfrac34=\tfrac12\) of the whole
  3. left: \(\tfrac34-\tfrac12=\tfrac14\) = 15

Answer 60 grapes

Try it yourself

1
Find \(\tfrac38\) of 40.
Show answer

Answer 15

2
12 tins is \(\tfrac3{13}\) of the stock. How many tins were there?
Show answer

Answer 52

3
Write 45 minutes as a fraction of an hour.
Show answer

Answer \(\tfrac34\)

Ways they test it

Type 1 Show all your working (non-calculator)

"Without a calculator, work out \(1\tfrac56+2\tfrac34\). Show all your working; answer as a mixed number in its simplest form." (3 marks)

Spot it "you must show all your working".

  1. Improper fractions written down (M1).
  2. The common-denominator line (M1).
  3. Simplest mixed number (A1).

Type 2 Multiply / divide mixed numbers

"Work out \(2\tfrac25\div1\tfrac1{15}\)."

Spot it × or ÷ with mixed numbers.

  1. Improper.
  2. Flip the second if ÷.
  3. Cancel, multiply, convert back.

For example \(2\tfrac25\div1\tfrac1{15}\)

  1. \(\tfrac{12}5\times\tfrac{15}{16}=\tfrac94\)

Answer \(2\tfrac14\)

Type 3 Fraction of the remainder

"Sara spends \(\tfrac25\) on clothes and \(\tfrac13\) of the remainder on food. $36 left." (3 marks)

Spot it "of the remainder", "of the rest".

  1. Each step as a fraction of the WHOLE.
  2. Fraction left = the given amount, then find the whole.

For example \(\tfrac25\), then \(\tfrac13\) of the rest

  1. food \(=\tfrac13\times\tfrac35=\tfrac15\)
  2. left \(=\tfrac25\) = $36

Answer $90

Type 4 How many to buy

"A pet eats \(\tfrac38\) of a tin a day. How many tins for 14 days?"

Spot it whole items needed.

  1. Multiply.
  2. Round UP to a whole number.

Mistakes to avoid

  • ✗ \(\tfrac13+\tfrac14=\tfrac27\)✓ \(\tfrac7{12}\)Add only same-size parts.
  • ✗ \(2\tfrac23\times1\tfrac18=2\tfrac2{24}\)✓ 3Improper fractions first.
  • ✗ \(\tfrac94\div\tfrac{15}8=\tfrac49\times\tfrac{15}8\)✓ \(\tfrac94\times\tfrac8{15}\)Flip the second fraction only.
  • ✗ \(\tfrac{55}{12}\) when a mixed number is asked✓ \(4\tfrac7{12}\)Give the form asked.
  • ✗ 5.25 tins → 5 tins✓ 6 tinsYou can't buy part of a tin.

Summary

  • Mixed numbers → improper fractions before any operation.
  • + −: common denominator (LCM).
  • ×: cancel first, then tops × tops, bottoms × bottoms.
  • ÷: multiply by the reciprocal of the second fraction.
  • "Of the remainder": multiply, then track what is left.

If you see …

  • "show all your working"write every line.
  • "of"multiply.
  • "of the rest"multiply by what was left.

Practice

Practice: easy, medium, hard

Easy

1
\(\tfrac23+\tfrac16\)
Show working
  1. \(\tfrac46+\tfrac16\)

Answer \(\tfrac56\)

2
\(\tfrac35\times\tfrac{10}{21}\)
Show working
  1. cancel 5 and 10, 3 and 21

Answer \(\tfrac27\)

3
18 is \(\tfrac35\) of a number. Find the number.
Show working
  1. \(18\div3\times5\)

Answer 30

Medium

1
\(3\tfrac15-1\tfrac23\) (as a mixed number)
Guide me

What is it asking? the difference

What do we know? denominators 5 and 3

Plan improper, over 15, subtract

Show working
  1. \(\tfrac{16}5-\tfrac53=\tfrac{48}{15}-\tfrac{25}{15}=\tfrac{23}{15}\)

Answer \(1\tfrac8{15}\)

2
\(3\tfrac13\times1\tfrac45\)
Guide me

What is it asking? the product

What do we know? two mixed numbers

Plan improper, cancel, multiply

Show working
  1. \(\tfrac{10}3\times\tfrac95=6\)

Answer 6

3
A pet eats \(\tfrac38\) of a tin a day. How many tins must be bought for 14 days?
Guide me

What is it asking? whole tins

What do we know? \(\tfrac38\) a day for 14 days

Plan multiply, then round up

Show working
  1. \(\tfrac38\times14=5\tfrac14\)
  2. round up: 6

Answer 6

Hard

1
Without a calculator, \(1\tfrac13\div\tfrac8{15}\). Answer as a mixed number.
Guide me

What is it asking? the quotient

What do we know? a mixed number ÷ a fraction

Plan improper, flip, cancel

Show working
  1. \(\tfrac43\times\tfrac{15}8=\tfrac52\)

Answer \(2\tfrac12\)

PartAnswer / workingMarksGuidance
\(\tfrac43\div\tfrac8{15}\)B1
\(\tfrac43\times\tfrac{15}8\)M1
\(2\tfrac12\)A1
Total: 3 marks
2
Sara spends \(\tfrac25\) of her money, then \(\tfrac13\) of the remainder. She has $36 left. How much did she start with?
Guide me

What is it asking? the starting amount

What do we know? a fraction of the remainder

Plan fraction left of the whole = 36

Show working
  1. remainder \(\tfrac35\); food \(\tfrac15\)
  2. left \(\tfrac25\) = 36
  3. whole = 36 ÷ 2 × 5

Answer $90

PartAnswer / workingMarksGuidance
\(\tfrac13\times\tfrac35\)M1
\(\tfrac25\) leftM1
90A1
Total: 3 marks
3
\(2\tfrac25\div1\tfrac1{15}\) (as a mixed number)
Guide me

What is it asking? the quotient

What do we know? two mixed numbers

Plan improper, flip the second, cancel

Show working
  1. \(\tfrac{12}5\times\tfrac{15}{16}=\tfrac94\)

Answer \(2\tfrac14\)

PartAnswer / workingMarksGuidance
\(\tfrac{12}5\times\tfrac{15}{16}\)M1
\(\tfrac94\)M1
\(2\tfrac14\)A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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