Ordering numbers — integers, decimals and fractions
Putting numbers in size order is a foundation skill that creeps into every other topic: inequalities, statistics, probability, even algebra. The trap is that integers, decimals and fractions all need slightly different tactics, and exam questions love to mix all three in one list.
The number line — the mental model
Think of a number line stretching left (negative) to right (positive). Bigger numbers are further right. That's the only rule you ever need; everything else is a method for placing numbers correctly on the line.
Negative integers
The deeper into the negatives you go, the smaller the number. So −10 < −3 even though 10 > 3. Treat the minus sign as a direction, not a size.
Worked example: order −7, 2, −12, 0, 5 from smallest to largest. The most negative is −12, then −7, then 0, then 2, then 5. Answer: −12, −7, 0, 2, 5.
Decimals — line them up by place value
Compare digit-by-digit from the left, after lining up the decimal points. Pad shorter decimals with trailing zeros so every number has the same number of decimal places.
Worked example: order 0.7, 0.65, 0.605, 0.7001 from smallest to largest. Pad to four d.p.: 0.7000, 0.6500, 0.6050, 0.7001. Compare:
- 0.6050 < 0.6500 (third digit: 0 < 5)
- 0.6500 < 0.7000 (first digit after point: 6 < 7)
- 0.7000 < 0.7001 (fourth digit: 0 < 1)
Answer: 0.605, 0.65, 0.7, 0.7001.
Fractions — three reliable methods
Method A — convert to decimals. Divide top by bottom (calculator paper, or use known facts on non-calc: ½ = 0.5, ¼ = 0.25, ⅕ = 0.2, ⅛ = 0.125, ⅓ ≈ 0.333…).
Method B — common denominator. Pick the LCM of the denominators, scale each fraction up, then compare numerators. To compare ⅔ and ⅗: LCM(3, 5) = 15, so ⅔ = 10/15 and ⅗ = 9/15. Therefore ⅔ > ⅗.
Method C — cross-multiply. For just two fractions a/b vs c/d, compare a×d with b×c (assuming b, d > 0). For ⅔ vs ⅗: 2 × 5 = 10 and 3 × 3 = 9. The bigger product sits over the bigger fraction, so ⅔ > ⅗.
Mixing types — the standard exam question
When integers, decimals and fractions appear together, convert everything to decimals first.
Worked example: order ⅓, 0.34, 30%, 0.305 from smallest to largest.
- ⅓ = 0.333…
- 30% = 0.3
- 0.305 stays
- 0.34 stays
So we compare 0.300, 0.305, 0.333…, 0.340. Answer: 30%, 0.305, ⅓, 0.34.
⚠Common mistakes— Common mistakes (examiner traps)
- Treating −10 as bigger than −3 because "10 > 3". Sketch a number line if unsure.
- Comparing decimals by length. 0.65 is bigger than 0.605, even though it has fewer digits.
- Adding fractions instead of comparing them — re-read the question.
- Mixing percent with decimal incorrectly (forgetting 30% = 0.30, not 0.03).
- Writing the answer in the wrong order — re-read whether the question asks smallest-first or largest-first.
➜Try this— Quick check
Order from smallest to largest: −0.5, −½, ⅖, 0.4, 0.04. Answer: −0.5 = −½ (equal), then 0.04, then ⅖ = 0.4 (equal). So: −0.5 = −½ < 0.04 < ⅖ = 0.4.
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