C1.5 Ordering

Learning objectives

C1.5.1Order quantities by magnitude and demonstrate familiarity with the• Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, <, ⩾ and ⩽.

To compare quantities, express them on one common scale and then read their positions from smallest to largest. On a number line, values increase to the right; this remains true for negatives, fractions, decimals, percentages, roots and standard form.

Symbol	Meaning	Example
$=$	equal to	$0.5=50\%$
$\ne$	not equal to	$0.5\ne5\%$
$>$	greater than	$0.5>5\%$
$<$	less than	$-7<-5$
$\ge$	greater than or equal to	$x\ge3$ includes 3
$\le$	less than or equal to	$x\le3$ includes 3

Choose a common form that preserves enough accuracy: convert percentages to decimals, divide numerator by denominator for fractions, and evaluate roots or standard-form values. Keep extra decimal places until the order is secure; do not round two close values to the same comparison value.

Original value	Comparable decimal
$58\%$	$0.58$
$\frac{7}{12}$	$0.5833\ldots$
$0.6$	$0.6$
$\frac{8}{13}$	$0.6153\ldots$
$\frac{2}{3}$	$0.6666\ldots$

Therefore $58\%<\frac{7}{12}<0.6<\frac{8}{13}<\frac{2}{3}$ . A chain uses a comparison between every neighbouring pair; each inequality must point consistently from smaller to larger.

To find a fraction between two fractions, a common denominator can expose a gap. Since $\frac{3}{25}=\frac{6}{50}$ and $\frac{4}{25}=\frac{8}{50}$ , the value $\frac{7}{50}$ lies between them.

The symbols $\ge$ and $\le$ include equality, while $>$ and $<$ do not. For negative values, the number with the greater absolute size can be smaller: $-8<-3$ because $-8$ lies further left on the number line.

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