Home / IGCSE Maths 0580 / Number / Ordering quantities and inequality symbols
Cambridge IGCSE · Topic 1.5
IGCSE Maths 0580 / Topic 1 / IGCSE

Same form, then compare.

Fractions, decimals, percentages, negatives, roots and measurements in different units can all be put in order once they are written the same way. Then the symbols \(<\), \(>\), \(\le\), \(\ge\) and \(\ne\) say exactly how they compare.

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The key idea

To order mixed quantities, convert every one into the same form (usually a decimal to at least 3 or 4 places, or the same unit), compare, then write the answer using the ORIGINAL forms in the order asked for.

−3−2−1 012 −3−1.25−0.52 ← smaller (further left) larger (further right) →
Fig. 1: on a number line, further right means larger, so −3 < −1.25 < −0.5 < 2 even though 3 looks bigger than 1.25
Section 01

Line them up.

In the lab, enter each fraction as numerator and denominator and read off its decimal. Write the decimals in a column with the points lined up, then compare digit by digit from the left.

Worked example Fractions, decimals and percentages
Write in order of size, smallest first: \(\ \dfrac{5}{8},\ 0.62,\ 61\%,\ 0.6\).
  1. Convert each to a decimal: \(\dfrac{5}{8} = 0.625\), \(61\% = 0.61\).
  2. Line up to 3 places: \(0.625,\ 0.620,\ 0.610,\ 0.600\).
  3. Smallest first: \(0.600,\ 0.610,\ 0.620,\ 0.625\).
  4. Answer in the original forms.
Answer: \(0.6,\ 61\%,\ 0.62,\ \dfrac{5}{8}\)
Worked example Negative numbers
Write in order, smallest first: \(\ -0.5,\ 2,\ -3,\ -1.25,\ 0\).
  1. Negative numbers are smaller than 0, and the further from 0 the smaller (Fig. 1).
  2. Among the negatives: \(-3 < -1.25 < -0.5\).
  3. Then 0, then 2.
Answer: \(-3,\ -1.25,\ -0.5,\ 0,\ 2\)
Worked example Different units
Write these lengths in order, smallest first: \(\ 1.2\) m, \(115\) cm, \(1180\) mm, \(0.0011\) km.
  1. Convert to one unit, say cm: \(1.2 \text{ m} = 120\) cm, \(1180 \text{ mm} = 118\) cm.
  2. \(0.0011 \text{ km} = 0.0011 \times 100\,000 = 110\) cm.
  3. Order: 110, 115, 118, 120 cm.
Answer: \(0.0011\) km, \(115\) cm, \(1180\) mm, \(1.2\) m
Examiner trap

More digits does not mean bigger: \(0.405 < 0.45 < 0.5\). Compare place by place from the left (or pad with zeros: 0.405, 0.450, 0.500).

Section 02

Inequality symbols.

The symbol always opens towards the larger number: the wide end faces the bigger side.

SymbolMeaningExample
\(=\)is equal to\(25\% = \dfrac{1}{4}\)
\(\ne\)is not equal to\(0.3 \ne \dfrac{1}{3}\)
\(<\)is less than\(-5 < -2\)
\(>\)is greater than\(0.7 > 0.69\)
\(\le\)is less than or equal to\(x \le 4\): 4 is allowed
\(\ge\)is greater than or equal to\(x \ge -1\): \(-1\) is allowed
Worked example Choosing the symbol
Write \(<\), \(>\) or \(=\) between (a) \(\dfrac{3}{7}\) and \(0.43\) (b) \(-5\) and \(-2\).
  1. (a) \(\dfrac{3}{7} = 0.4285\dots\), which is less than \(0.4300\).
  2. (b) \(-5\) is further left on the number line than \(-2\).
Answer: (a) \(\dfrac{3}{7} < 0.43\)   (b) \(-5 < -2\)
Worked example Integers in a range
List the integers \(n\) such that \(-3 < n \le 2\).
  1. \(-3 < n\): \(-3\) itself is NOT included.
  2. \(n \le 2\): 2 IS included.
Answer: \(-2,\ -1,\ 0,\ 1,\ 2\)
Your turn
Write as an inequality: \(x\) is at least 4 but less than 9. Then list the integer values of \(x\).
Examiner trap

\(-5 > -2\) is FALSE. With negatives, the number that looks bigger is smaller: \(-5\) is 3 below \(-2\).

Stage 1 · Learn

Check your understanding

Four quick checks. Each correct answer earns XP and records this skill on your profile.

Quick check+10 XP

Which number is the largest?

Quick check+10 XP

Which symbol makes this true? \(\ -4 \ \square\ -7\)

Quick check+10 XP

What does \(x \ge 3\) mean?

Quick check+10 XP

Which number is the smallest?

Section 03

Roots, pi and close calls.

When values are very close, a calculator decides it. Write each one to at least 4 decimal places, then order; 2 places is often not enough to separate them.

Worked example Very close values
Write in order, smallest first: \(\ \sqrt{10},\ 3.1,\ \pi,\ \dfrac{22}{7}\).
  1. \(\sqrt{10} = 3.1623\dots\), \(\pi = 3.1416\dots\), \(\dfrac{22}{7} = 3.1429\dots\), \(3.1 = 3.1000\).
  2. To 2 d.p. \(\pi\) and \(\dfrac{22}{7}\) both show 3.14, so go to the third place: \(3.1416 < 3.1429\).
  3. Order: \(3.1000,\ 3.1416,\ 3.1429,\ 3.1623\).
Answer: \(3.1,\ \pi,\ \dfrac{22}{7},\ \sqrt{10}\)
Examiner trap

Read the instruction: 'starting with the largest' reverses the order. And always give the final list in the original forms, not as your decimals.

Stage 2 · Exam

Exam-style questions

Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 2 and Paper 4 standard.

Finish the four checks above to unlock the exam questions
Exam style+20 XP

Write in order of size, smallest first: \(\ \dfrac{2}{3},\ 0.67,\ 66\%,\ \dfrac{13}{20}\).

Exam style+20 XP

Which statement is true?

Exam style+20 XP

How many integers \(n\) satisfy \(-3 < n \le 2\)?

Exam style+20 XP

Which of these lengths is the longest? \(\ 0.035\) km, \(3400\) cm, \(35\,100\) mm, \(34.5\) m

Skill unlocked

Ordering, sorted.

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Go deeper · practice
Six original Cambridge-style questions
Inequality symbols, negatives, fractions against decimals and percentages, close calculator values, integer ranges, and race times.