Ch 3 · 1.4–1.6 Fractions, Decimals, Ordering and the Four Operations · O Level Mathematics 4024

Topic 3 of 6 in Number — O Level Mathematics 4024 — 58 lessons.

Introduction

A recipe asks for tfrac34 of a cup, a shop offers 15% off, a bank pays 0.025 interest: the same idea in three forms. This lesson converts between fractions, decimals and percentages, handles recurring decimals, and makes sure calculations follow the right order. Paper 1 is non-calculator, so these skills must work by hand. (Syllabus sections 1.4, 1.5 and 1.6.)

Key ideas — Forms and conversions

🧑‍🏫 Teacher's explanation

Think of fractions, decimals, and percentages as three different languages used to describe the same thing: a part of a whole. Imagine you have a pizza. You could say you ate 3/4 of it (fraction), you ate 0.75 of it (decimal), or you ate 75\% of it (percentage). They are all describing the exact same amount of pizza, just using different 'dialects'.

📖 Definitions

🔍 Worked example — Easy

Convert 23/5 into a decimal and a percentage.

🔍 Worked example — Standard

Convert the recurring decimal 0.4dot7 into a fraction in its simplest form.

🔍 Worked example — Exam-style

Arrange the following in descending order: 0.65, 5/8, 62\%.

⚠️ Common errors

🧠 Quick check 1

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🎯 Exam question

Key ideas — Recurring decimals

A dot shows the repeating part: 0.1dot7 = 0.1777dots, 0.dot12dot3 = 0.123123dots, 0.1dot2dot3 = 0.12323dots

🧑‍🏫 Teacher's explanation

Imagine you are walking towards a wall, but every step you take is only a fraction of the remaining distance. You might never actually touch the wall, but you get closer and closer, creating a pattern that repeats forever. In mathematics, some numbers don't end; they enter a loop where the same digits repeat infinitely. We call these recurring decimals.

📖 Definitions

🔍 Worked example — Easy

Convert 0.dot5 to a fraction in its simplest form.

🔍 Worked example — Standard

Convert 0.1dot6 to a fraction in its simplest form.

🔍 Worked example — Exam-style

Show that 0.dot12dot3 can be written as $ rac{41}{333}$.

⚠️ Common errors

🧠 Quick check 1

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🎯 Exam question

Key ideas — Ordering

Convert to the same form (usually decimals) before comparing. Symbols: = equal, ≠ not equal, > greater than, < less than, ≥ greater than or equal to, ≤ less than or equal to. With negatives, -7 < -2 (further left on the number line).

🧑‍🏫 Teacher's explanation

Comparing numbers is like comparing the heights of people. If one person is 1.75m and another is 1.8m, it is hard to tell who is taller until you look at the same unit. In maths, comparing a fraction like 3/4 to a decimal like 0.8 is tricky because they are in different 'languages'. To compare them fairly, you should translate them into the same language—usually decimals.

📖 Definitions

🔍 Worked example — Easy

Arrange the following in ascending order: 2/5, 0.35, 3/8

🔍 Worked example — Standard

Fill in the missing symbol (<, >, =) to make the statement true: -1.5 dots -1.25

🔍 Worked example — Exam-style

Given x = 0.666... (recurring) and y = 2/3, determine if x = y, x < y, or x > y.

⚠️ Common errors

🧠 Quick check 1

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🎯 Exam question

Key ideas — Order of operations

Brackets, then indices (powers and roots), then × and ÷ (left to right), then + and - (left to right): 3 - 2 × (4 - 7)^2 = 3 - 2 × 9 = -15.

🧑‍🏫 Teacher's explanation

Imagine you are following a recipe. If you don't follow the steps in the correct order—like adding the flour before you've even turned on the oven—the cake won't rise. Mathematics has a strict 'recipe' for solving expressions to ensure everyone gets the same answer. This set of rules is called the Order of Operations (often remembered by the acronyms BIDMAS or BODMAS).\n\nWhen you see a long string of numbers and symbols, you cannot simply work from left to right like reading a sentence. You must prioritize. Brackets come first, then indices (the little numbers representing powers), then multiplication and division, and finally addition and subtraction. Think of brackets as a 'VIP zone' where the numbers inside get processed before they are allowed to interact with the rest of the equation.

📖 Definitions

🔍 Worked example — Easy

Calculate 10 + 5 × 2

🔍 Worked example — Standard

Evaluate 4^2 - (3 + 1) × 2

🔍 Worked example — Exam-style

Simplify 12 - 2 × 3/2^2

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Fractions and negative numbers

🧑‍🏫 Teacher's explanation

Think of fractions as 'slices' of a whole. If you have a pizza cut into 8 slices, each slice is 1/8. When we add fractions, we can't just add the slices unless the slices are the same size (the same denominator). If one pizza is cut into 4 pieces and another into 8, we must convert the 4-piece pizza into 8ths so we are comparing 'apples to apples'.

Ch 3 · 1.4–1.6 Fractions, Decimals, Ordering and the Four Operations · O Level Mathematics 4024

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