Unit 1: Numbers and Calculations with Numbers
- Jul 26
- 3 min read
Updated: Jul 27
⬇️ Download the PDF notes at the end of the module! ⬇️
1.1 Using a Calculator and Order of Operations
📌 REMEMBER | BODMAS determines the order of operations: Brackets → Orders (powers/roots) → Division & Multiplication → Addition & Subtraction. Always work from LEFT to RIGHT. |
The order is:
B | Brackets () – do everything inside brackets first |
O | Orders – powers, square roots, cube roots |
D / M | Division and Multiplication – left to right |
A / S | Addition and Subtraction – left to right |
💡 TIP | Always estimate your answer first by rounding numbers – this helps you spot a big mistake. Always do a calculation TWICE. If the two answers differ, redo the calculation. |
1.2 Common Fractions
A fraction shows how something is divided into parts.
Fraction = Numerator / Denominator e.g.: 5/6 means 5 parts out of a total of 6 |
Proper fraction | Numerator SMALLER than denominator → e.g. 2/10 or 3/5 |
Improper fraction | Numerator LARGER than denominator → e.g. 5/2 or 7/4 |
Mixed number | Whole number + fraction → e.g. 4½ or 1⅓ |
1.3 Decimals
A decimal fraction is a fraction where the denominator is a power of ten.
Place value table | Hundreds | Tens | Units ● tenths | hundredths | thousandths |
Multiply by 10 | Decimal comma shifts ONE place to the right |
Multiply by 100 | Decimal comma shifts TWO places to the right |
Multiply by 1 000 | Decimal comma shifts THREE places to the right |
💡 TIP | 1 ÷ 10 = 0.1 1 ÷ 100 = 0.01 1 ÷ 1 000 = 0.001 The more zeros, the smaller the number. |
1.4 Percentages
Percentage means a rate per 100.
Percentage of a number: % ÷ 100 × whole Percentage of two numbers: (part ÷ whole) × 100 % increase: (increase ÷ original) × 100 % decrease: (decrease ÷ original) × 100 |
💡 TIP | Always identify WHICH part is UNKNOWN: the percentage itself, the whole, or the new amount? Then choose the correct formula. |
1.5 Ratio, Proportion and Rate
A ratio shows how one number relates to another. It is written as 3:5 or as the fraction 3/5.
Rate = quantity ÷ time (e.g. km/h or litres/100km) Direct proportion: as x increases, y also increases Indirect proportion: as x increases, y decreases |
📌 REMEMBER | To divide in a ratio: add the parts together, divide the total by the number of parts, multiply each part by the quotient. |
1.6 Rounding According to Context
Nearest whole number | Look at the first decimal place: ≥5 → round up, <5 → round down |
Nearest cent | Look at the third decimal place: ≥5 → round up, <5 → round down |
Context determines | e.g. people: always round UP (you can't have 0.8 of a person) |
1.7 Squares and Cube Roots
Square: n² = n × n e.g. 7² = 49 Cube root: ∛n e.g. ∛125 = 5 Square root: √n e.g. √144 = 12 |
1.8 Time
24-hour → 12-hour | Hours ≥ 13: subtract 12 and add PM; hours < 12: add AM |
12-hour → 24-hour | PM times: add 12 (except 12:xx PM) |
Time elapsed | Work in minutes: end time − start time (in minutes) ÷ 60 = hours |
💡 TIP | Remember: 1 hour = 60 minutes; 1 day = 24 hours; 1 week = 7 days; 1 year = 365 days or 52 weeks. |