Comparing fractions does not always need a common denominator — a single cross-multiplication often settles it in seconds.
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Every fraction is proper, improper, or a mixed number — and converting between the last two is a skill you will use constantly.
📌 Fraction types & conversion
Proper fraction: numerator smaller than denominator (e.g. 3/4). Value is less than 1.
Improper fraction: numerator equal to or larger than denominator (e.g. 11/4). Value is 1 or more.
Mixed to improper: multiply the whole number by the denominator, add the numerator. 2¾ → (2×4)+3 = 11, so 11/4.
✨ Worked example
Mixed number
whole + fraction
3⅖ means 3 whole and 2/5 more
As improper fraction
(3×5+2)/5
17/5
Finding a common denominator works, but cross-multiplication is often faster for comparing just two fractions.
Cross-multiplication trick: to compare 3/4 and 5/7, cross-multiply: 3×7=21 and 5×4=20. Since 21 > 20, 3/4 > 5/7 — no common denominator needed.
📊 Ordering shortcuts
| Situation | Shortcut |
|---|---|
| Same denominator | Compare numerators directly — larger numerator wins |
| Same numerator | Smaller denominator means a LARGER fraction |
| Different numerator & denominator | Cross-multiply, or convert to a common denominator |
Fraction word problems almost always reduce to one operation, once you translate the words correctly.
Trap: "reduced by a fraction of itself" means you SUBTRACT that fraction of the value, not use the fraction as the final answer directly. "A number reduced by 1/4 of itself" means the result is 3/4 of the original.
12 questions on fraction types, comparison and word problems.
