E1.5 Ordering

Learning objectives

E1.5.1Order quantities by magnitude and demonstrate familiarity with the• Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, <, ⩾ and ⩽.

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

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: divide numerator by denominator for fractions and divide percentages by 100. Keep extra decimal places until the order is secure; rounding close values too early can make unequal quantities appear equal.

Original value	Comparable decimal
$58\%$	$0.58$
$\frac7{12}$	$0.5833\ldots$
$0.6$	$0.6$
$\frac8{13}$	$0.6153\ldots$
$\frac23$	$0.6666\ldots$

Therefore $58\%<\frac7{12}<0.6<\frac8{13}<\frac23$ . In an inequality chain, every neighbouring comparison must point consistently from the smaller value towards the larger value.

A common denominator can expose a value between two fractions. Since $\frac3{25}=\frac6{50}$ and $\frac4{25}=\frac8{50}$ , $\frac7{50}$ lies strictly between them.

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

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