Formulas and equations

A formula is a rule that connects two or more quantities using pronumerals. When we know the values of every variable except one, we can find the missing value by substitution: replace each pronumeral with its number, then evaluate using the correct order of operations (brackets, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right).

Substitution works the same way whether the formula is linear (every variable appears to the power of one, such as P = 2l + 2w) or non-linear (a variable is squared, square-rooted or in a denominator, such as A = πr² or v = √(2gh)).

Good habits make substitution reliable. First, write the formula down before you substitute, so a marker can follow your reasoning and award method marks. Second, place brackets around negative numbers when you substitute them, because −3² equals −9 but (−3)² equals 9. Third, check that all quantities are in compatible units before you substitute; a formula that expects metres will give nonsense if you put in centimetres.

Fourth, only round at the final step and state the units of the answer.

Sometimes the unknown is not the variable on its own on the left-hand side. In that case you can still substitute the known values first and then solve the resulting equation using inverse operations. For example, if v = u + at and you are given v, u and t, substitute them and then solve for a. This approach is often quicker in an exam than rearranging the formula first, and it is less prone to algebraic slips.

Both methods are valid; choose the one you find most dependable.

Formulas appear across the Standard course in contexts such as perimeter and area, simple interest, energy use, temperature conversion (F = 9C/5 + 32) and sport statistics. A calculator is essential, but you should still estimate first so you can spot a keying error. For instance, if a circle has radius 10 cm, its area should be a bit more than 3 × 100 = 300 cm².