SPEC N8, N11 • NUMBER
⭐ HIGH PRIORITY
Fractions — Four Operations
Adding, subtracting, multiplying, dividing. Mixed numbers ↔ improper fractions. Every skill you need for Paper 1 without a calculator — and the two examiner alerts that decide the mark.
Key Definitions
Five terms define every fraction question in the spec.
Proper Fraction
Numerator less than denominator. The fraction is less than 1.
Examples: 3/5, 7/8, 1/100
Improper Fraction
Numerator greater than or equal to denominator. The fraction is 1 or more. Sometimes called "top-heavy."
Examples: 7/3, 11/4, 9/9
Mixed Number
An integer plus a proper fraction.
Examples: 2⅓, 1⅞, 5½
Reciprocal
The fraction inverted. The reciprocal of a/b is b/a.
Example: reciprocal of 2/5 is 5/2.
Simplify
Divide numerator and denominator by their HCF until no common factor above 1 remains. 12/18 = 2/3 (both divided by 6).
Mixed ↔ Improper Conversion
The first skill. Every mixed-number question starts by converting into improper form.
Mixed → Improper
Integer × denominator, plus numerator, over denominator.
2⅓ = (2 × 3 + 1) / 3 = 7/32¾ = (2 × 4 + 3) / 4 = 11/4
Improper → Mixed
Divide numerator by denominator. The quotient is the integer; the remainder is the new numerator.
17/5 → 17 ÷ 5 = 3 remainder 2 → 3⅖
The Golden Rule
For any operation involving mixed numbers, convert to improper fractions first. Do not try to work with integer and fraction parts separately — it fails as soon as carrying is needed.
Addition & Subtraction
You cannot add or subtract fractions unless the denominators are the same. That's the entire rule.
Three-step method
Find the common denominator — the LCM of the two denominators.
Rewrite each fraction with that denominator.
Add (or subtract) the numerators. Keep the denominator the same. Simplify.
Worked example: 3/4 + 2/3
LCM(4, 3) = 12 (common denominator)
3/4 = 9/12 | 2/3 = 8/129/12 + 8/12 = 17/12= 1⁵⁄₁₂
Never add across
Students write 3/4 + 2/3 = 5/7. Wrong. You cannot add numerators and denominators separately — that breaks the meaning of a fraction.
Multiplication — Straight Across
The easiest of the four operations. Multiply numerator by numerator, denominator by denominator.
a/b × c/d = (a × c) / (b × d)
Cross-cancel first — saves time
Before multiplying, cancel common factors diagonally across the two fractions. This keeps the numbers small.
Worked example: 4/9 × 3/8
Cross-cancel:
4 (top-left) and 8 (bottom-right): divide both by 4 → become 1 and 2
3 (top-right) and 9 (bottom-left): divide both by 3 → become 1 and 3
(1/3) × (1/2) = 1/6
Same answer as multiplying straight: 12/72 = 1/6. But cancelling first avoids the 72.
Division — Keep, Change, Flip
You never actually divide fractions. You turn the division into a multiplication.
a/b ÷ c/d = a/b × d/c
Keep, Change, Flip (KCF)
Keep the first fraction exactly as it is.
Change the ÷ sign to a × sign.
Flip the second fraction (take the reciprocal).
Then multiply as normal — cross-cancel first if you can.
Worked example: 3/4 ÷ 2/5
Keep: 3/4
Change: ÷ → ×
Flip: 2/5 → 5/2
3/4 × 5/2 = 15/8= 1⅞
Flip only the divisor
Students sometimes flip the first fraction instead of the second. Keep the first, flip the second. Only ever flip the divisor.
Mixed Numbers — Convert First
Rule: whenever a mixed number appears in any operation, convert to an improper fraction before calculating. Then convert back to a mixed number at the end if asked.
The standard sequence
Convert all mixed numbers → improper fractions.
Perform the operation (+, −, ×, ÷).
Simplify the result.
Convert back to mixed number form if the question asks for it.
Mark scheme insight
AQA often says "Give your answer as a mixed number in its simplest form." If you leave it as an improper fraction like 9/4 instead of 2¼, you lose the accuracy mark — even though 9/4 = 2¼. The answer must match the form requested.
Mark scheme trap — Jun23 P1 Q3
If you write both9/4and2¼ on the answer line, AQA scores B0. The mark scheme reads: "If values are given in one or more forms, all values must be correct." Inconsistent forms = zero. Write the answer once, in the form asked.
01MCE Worked Example 3 MARKS
Work out 2¾ − 1⅖ . Give your answer as a mixed number in its simplest form.
(Non-calculator, Paper 1 style — modelled on Nov24 P1 Q6)
•
MMethod. Convert each mixed number to an improper fraction. Find a common denominator. Subtract. Convert back to a mixed number.
•
CCalculate.
Step 1 — Convert to improper:
2¾ = (2 × 4 + 3)/4 = 11/4
1⅖ = (1 × 5 + 2)/5 = 7/5
Step 2 — Common denominator:
LCM(4, 5) = 20
11/4 = 55/20 | 7/5 = 28/20
Step 3 — Subtract:
55/20 − 28/20 = 27/20
Step 4 — Convert back:
27/20 = 1 remainder 7 = 1⁷⁄₂₀
•
EEvaluate.Answer: 1⁷⁄₂₀
Already in simplest form — HCF(7, 20) = 1.
Mark scheme: M1 for correct improper conversion; M1 for correct common denominator and subtraction; A1 for 1⁷⁄₂₀ as a mixed number. Leaving the answer as 27/20 loses the accuracy mark.
Weak vs Strong: Why You Convert First
Separating the integer and fraction parts often seems to work. Then it breaks.
Work out 2⅕ − 1¾
Weak student
"Subtract integer parts: 2 − 1 = 1. Subtract fraction parts: ⅕ − ¾."
⅕ − ¾ = 4/20 − 15/20 = −11/20
Answer: 1 + (−11/20) = ??
Student doesn't know what to do with a negative fraction inside a "positive" integer context. Often gives up or writes nonsense.
Strong student
"Convert both to improper first."
2⅕ = 11/5, 1¾ = 7/4
LCM(5,4) = 20 → 44/20 − 35/20
= 9/20 → 9/20
Clean. Positive. Works every time.
Why the separation method fails
It only works if the first fraction part is bigger than the second. When the second fraction part is bigger, you need to "borrow" 1 from the integer — which is exactly what converting to improper fractions does automatically.
Examiner Alert: Flipping the Wrong Fraction
What examiners see
On division questions, students correctly remember "flip" — but flip the first fraction instead of the second. The rule is Keep, Change, Flip: only the divisor gets flipped.
What to write instead
(3/4) ÷ (2/5) = (3/4) × (5/2) ✔
NOT (4/3) × (2/5) ✘
The first fraction is kept exactly. Only the second (the divisor) is flipped.
Memory hook
"KCF" order: Keep → Change → Flip. The K comes before the F. The thing you keep is the thing you see first.
Examiner Alert: Never Separate Mixed Numbers
What examiners see
Students try to multiply mixed numbers as "integer × integer" plus "fraction × fraction." 2¼ × 3 becomes "6 + ¼ = 6¼" — wrong. The correct answer is 2¼ × 3 = 9/4 × 3 = 27/4 = 6¾.
What to write instead
Always convert mixed numbers to improper fractions before any operation. Convert back only at the end, if the question asks.
2¼ × 3 = (9/4) × (3/1) = 27/4 = 6¾
Why it matters most on multiplication
Separating works by coincidence sometimes for +/−. It never works for multiplication or division. The pattern (a + b) × c ≠ ac + b — you'd need to distribute the multiplication over both parts, which is exactly what converting does.
Common Errors
Students often writeCorrect versionWhy it matters
3/4 + 2/3 = 5/7= 17/12Common denominator required (don't add across)
(3/4) ÷ (2/5) = (4/3) × (2/5)= (3/4) × (5/2)Keep first, flip second — not the other way
2¼ × 3 = 6¼= 9/4 × 3 = 27/4 = 6¾Mixed-number multiplication must use improper form
Answer left as 9/4Give as 2¼ if asked for mixed numberJun23 P1 Q3 MS: "do not allow unprocessed answers"
Both 9/4 and 2¼ on answer lineOne form onlyAQA scores B0 for inconsistent forms
Practice Question 1
Work out 5/6 ÷ 3 . Give your answer as a fraction in its simplest form.1 MARK
(Modelled on AQA 8300/1H Jun 2023 Q1(b) — Paper 1, non-calculator)
1
Rewrite 3 as a fraction: 3 = 3/1.
2
Apply KCF: 5/6 ÷ 3/1 = 5/6 × 1/3
3
Multiply straight across: (5 × 1) / (6 × 3) = 5/18
Answer: 5/18
Common wrong answers:5/2 (flipped the first fraction); 15/6 (multiplied instead of divided).
Practice Question 2
Work out 3⁴⁄₉ + 1⅚ . Give your answer as a mixed number in its simplest form.3 MARKS
1
Convert to improper fractions:
3⁴⁄₉ = (3 × 9 + 4) / 9 = 31/9
1⅚ = (1 × 6 + 5) / 6 = 11/6
2
Common denominator: LCM(9, 6) = 18.
31/9 = 62/18 | 11/6 = 33/18
3
Add: 62/18 + 33/18 = 95/18
4
Convert back: 95 ÷ 18 = 5 remainder 5 → 5⁵⁄₁₈
Answer: 5⁵⁄₁₈
Mark scheme: M1 for correct improper conversions; M1 for common denominator & correct addition; A1 for 5⁵⁄₁₈ as a mixed number in simplest form.
Practice Question 3
Three girls shared a full bottle of cola.
Karen drank ¼ of the bottle.
Rita drank ³⁄₁₀ of the bottle.
Megan drank the rest.
Work out the fraction of the bottle that Megan drank.3 MARKS
(Context problem — AO2. Modelled on AQA Higher revision guide style.)
1
Megan's fraction = 1 − Karen − Rita = 1 − 1/4 − 3/10
2
Common denominator: LCM(1, 4, 10) = 20.
1 = 20/20 | 1/4 = 5/20 | 3/10 = 6/20
3
Subtract: 20/20 − 5/20 − 6/20 = 9/20
Answer: 9/20 of the bottle
Mark scheme: M1 for setting up "1 minus both fractions"; M1 for converting to a common denominator; A1 for 9/20. HCF(9, 20) = 1, so already simplified.
Summary
The Four Operations
Add / Subtract: find common denominator (LCM), then combine numerators.
Multiply: straight across. Cross-cancel first to keep numbers small.
Divide: Keep–Change–Flip. Keep first, change ÷ to ×, flip second.
Mixed numbers: convert to improper before any operation.
Mixed ↔ Improper
Mixed → improper:a b/c = (a×c + b)/c
Improper → mixed: divide, quotient is integer, remainder is new numerator.
Examiner Alerts
Flipping the wrong fraction: KCF means keep first, flip second. Only the divisor flips.
Separating mixed numbers: fails as soon as the second fraction is bigger. Always convert first.
Answer form: if the question says "mixed number," do not leave as improper. If it says "simplest form," check HCF of numerator and denominator.
Ready for 1.06
Next: Decimals & Recurring Decimals — converting between fractions and decimals, the denominator test for termination, and the Higher-tier algebraic method for recurring decimals.
Next: Key Definitions