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Concept 11 of 11Foundation
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📌 Formula Sheet — Fractions

Every operation, with the reason it works.

a/b + c/d = (a·d + b·c) / (b·d)
Means: rewrite both over common denominator b·d, then add topsUse when: adding/subtracting unlike fractions
Why: You can only add things of the same kind. Both fractions must be split into the same number of equal parts (common denominator) before counting.
⚠ Trap: Never add denominators. 1/2 + 1/3 ≠ 2/5. Find LCM first.
a/b × c/d = (a·c) / (b·d)
Means: multiply tops; multiply bottomsUse when: multiplying any two fractions
Why: a/b means "a of b equal parts." Taking c/d of that splits each of those parts further into d, and takes c of them. Total parts = b·d; taken = a·c.
⚠ Trap: Cancel common factors BEFORE multiplying, else numbers get big.
a/b ÷ c/d = a/b × d/c (Keep-Change-Flip)
Means: dividing = multiplying by the reciprocalUse when: any fraction division
Why: Dividing by a fraction asks "how many c/d's fit into a/b?" Flipping makes the count straightforward — 6 ÷ (1/2) = 12 because each whole fits 2 halves.
⚠ Trap: Flip ONLY the DIVISOR (second fraction). Don't flip the first.
Mixed → Improper: whole × den + num, over den
Example: 2¾ = (2·4 + 3)/4 = 11/4Use when: before any + − × ÷ with mixed numbers
Why: Each whole contains (den) parts of size 1/den. Two wholes = 2·4 = 8 quarters. Add the 3 extra quarters → 11 quarters total.
⚠ Trap: Beginners add the whole with the numerator ignoring the denominator — always MULTIPLY the whole by the denominator first.
Cross-multiplication test: a/b vs c/d → compare a·d with b·c
Rule: if a·d > b·c then a/b > c/dUse when: quickly comparing two fractions
Why: Multiplying both fractions by bd (positive) gives ad and bc. Comparing whole numbers is easier.
⚠ Trap: Works only when both denominators are POSITIVE. If a denominator is negative, flip the inequality.
Simplest form: divide top and bottom by HCF
Means: no common factor leftUse when: final answer should be simplified
Why: A fraction's value doesn't change if you divide numerator and denominator by the same number. HCF ensures you do it in one step.
⚠ Trap: Dividing by just a common factor (not the HCF) gives a partially simplified fraction — keep reducing until HCF = 1.
Decimal ↔ fraction: move decimal, put over a power of 10
Examples: 0.7 = 7/10, 0.25 = 25/100 = 1/4, 1.5 = 3/2Use when: switching between decimal and fraction form
Why: Every digit after the decimal point is a fractional part with 10, 100, 1000, … as denominator. Simplify by HCF if needed.
⚠ Trap: To convert fraction to decimal, divide numerator by denominator — use long division if needed.
Decimal multiplication: place decimal so answer has as many decimal places as total in inputs
Example: 0.3 × 0.02 = 0.006 (1 + 2 = 3 decimal places)Use when: multiplying decimal numbers
Why: 0.3 = 3/10 and 0.02 = 2/100. Product = 6/1000 = 0.006. The denominator's zeros stack up.
⚠ Trap: Count the TOTAL decimal places across both numbers — don't just use one.
💡 Tip:'of' usually means multiply. 'per' usually means divide. These two words hide 90% of fraction word problems.
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