NumberFoundation + Higher★ Core topic
Fractions.
Adding, subtracting, ×, ÷.
A fraction shows part of a whole. The top (numerator) is how many parts you have; the bottom (denominator) is how many parts make a whole. Four operations — all boil down to common denominators or flip and multiply.
I · Key definitions
Numerator
The top number. How many parts you have.
in 3/4, the numerator is 3
Denominator
The bottom. How many parts make a whole.
in 3/4, denominator is 4
Proper fraction
Numerator smaller than denominator.
3/4, 2/5, 7/8
Improper
Numerator larger than denominator.
7/4, 11/3
Mixed number
A whole plus a fraction.
1¾ means 1 + 3/4
Equivalent
Fractions with the same value, different form.
2/4 = 1/2 = 4/8
II · The four operations · rules
Adding / Subtracting
a/b ± c/d = (ad ± bc)/bd
1. Common denominator first
2. Add/subtract numerators, keep denominator
3. Simplify
1/3 + 1/4 = 4/12 + 3/12 = 7/12
Multiplying
a/b × c/d = ac/bd
1. Multiply tops
2. Multiply bottoms
3. Simplify (or cancel first)
2/3 × 3/5 = 6/15 = 2/5
Dividing
a/b ÷ c/d = a/b × d/c
Keep, flip, multiply (KFC)
Then simplify
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Mixed numbers
Convert to improper first:
a b/c = (ac + b)/c
Then operate · convert back if needed
2¼ = (2×4+1)/4 = 9/4
III · Fraction wall · equivalents
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
Reading across: 1/2 = 2/4 = 3/6 = 6/12. All equivalent. Simplifying = dividing top and bottom by the same number.
IV · Worked examples
Add two fractions
3 marks
Work out: 2/3 + 3/4
i.Find common denominator: LCM(3, 4) = 12
ii.2/3 = 8/12 · 3/4 = 9/12
iii.Add: 8/12 + 9/12 = 17/12
iv.Convert improper to mixed: 17/12 = 1 5/12
2/3 + 3/4 = 17/12 = 1 5/12
Divide mixed numbers
3 marks
Work out: 1½ ÷ ⅔
i.Convert mixed to improper: 1½ = 3/2
ii.Keep, flip, multiply: 3/2 ÷ 2/3 = 3/2 × 3/2
iii.Multiply: 9/4
iv.Convert back: 9/4 = 2¼
1½ ÷ ⅔ = 9/4 = 2¼
V · Common mistakes & examiner tips
Common mistakes
Adding bottoms together. 1/3 + 1/4 ≠ 2/7. Use common denominators.
Forgetting to simplify. 6/8 = 3/4 — always check.
Not converting mixed numbers before operating. Make them improper first.
KFC done backwards. Keep first, flip the second fraction only.
Cancelling before you should. You can cancel across × but NOT across +.
Examiner tips
Cancel before multiplying — smaller numbers, fewer mistakes.
Leave as improper if the next step needs it (e.g. in algebra).
Common denominator = LCM. Don't just multiply the two bottoms.
Mixed numbers → improper → operate → back. Always this order.