NSW HSC Mathematics Standard

## Formulas and pronumerals A **formula** is a rule that links quantities, written with letters called pronumerals. The formula $A = \tfrac{1}{2}h(a + b)$ gives the area of a trapezium, where $a$ and $b$ are the parallel sides and $h$ is the distance between them. Each letter stands for a number that changes from one case to the next, and the formula fixes how the quantities depend on each other. Units matter as much as numbers. If $h$ is in centimetres and $a$ and $b$ are in metres, the formula gives nonsense, so every quantity has to be in a compatible unit before it is used. ## Substituting into a formula To **substitute**, replace each pronumeral with its value and then evaluate, following the order of operations: brackets first, then powers, then multiplication and division, then addition and subtraction. For the trapezium with $a = 5$, $b = 9$ and $h = 6$: $$ \begin{aligned} A &= \tfrac{1}{2} \times 6 \times (5 + 9) \\ &= 3 \times 14 \\ &= 42 \end{aligned} $$ The same steps work for a non-linear formula. A ball dropped from rest falls a distance $s = 4.9t^2$ metres in $t$ seconds, so after $t = 3$ seconds: $$ s = 4.9 \times 3^2 = 4.9 \times 9 = 44.1 \text{ metres} $$ The power applies to the 3 alone, so $3^2$ is worked out before the multiplication by 4.9. A negative value goes into the formula inside brackets. To convert using $C = \tfrac{5}{9}(F - 32)$ when $F = 14$: $$ C = \tfrac{5}{9}(14 - 32) = \tfrac{5}{9} \times (-18) = -10 $$ A scientific calculator does this…