Number, estimation and rates

Every practical calculation in Foundation Mathematics starts with numbers written in one of several forms: integers, fractions, decimals, percentages, ratios and rates. Being fluent means you can move between these forms quickly and choose the one that makes a problem easiest. An integer is a whole number that may be positive, negative or zero, such as −7, 0 or 215.

Negative integers appear whenever we measure below a reference point, for example temperatures below freezing, an overdrawn bank balance or a depth below sea level.

A fraction describes part of a whole: the denominator tells you how many equal parts the whole is split into and the numerator tells you how many of those parts you have. A decimal is a fraction whose denominator is a power of ten, and a percentage is a fraction out of 100. So 3/4 = 0.75 = 75%. To convert a fraction to a decimal, divide the numerator by the denominator.

To convert a decimal to a percentage, multiply by 100. To convert a percentage to a fraction, write it over 100 and simplify, for example 35% = 35/100 = 7/20.

When adding or subtracting fractions you need a common denominator; when multiplying you multiply numerators together and denominators together; when dividing you multiply by the reciprocal of the second fraction. Mixed numbers such as 2½ are usually easier to handle if you first convert them to improper fractions (5/2).

Percentages are used constantly in everyday life: discounts, GST, interest, test scores and survey results. Three standard calculations are worth memorising. To find a percentage of an amount, convert the percentage to a decimal and multiply. To express one quantity as a percentage of another, divide and multiply by 100. To increase or decrease by a percentage, multiply by a single multiplier: increasing by 15% means multiplying by 1.15, and decreasing by 15% means multiplying by 0.85.

Notice that the same question could be done in two steps (find 30% of 180 = $54, then subtract), but the multiplier method is quicker and works well for repeated changes. For example, a price that rises 10% and then falls 10% is multiplied by 1.1 × 0.9 = 0.99, so it ends up 1% lower, not back where it started. Students often expect the two changes to cancel, and the multiplier method shows clearly why they do not.

Fluency with these forms underpins every later topic, including budgeting, measurement and data.