Ch 5 · 1.11, 1.12, 1.15, 1.16 Ratio, Rates, Time and Money · IGCSE Mathematics 0580

Topic 5 of 6 in Number — IGCSE Mathematics 0580 — 49 lessons.

Introduction

Scaling a recipe, reading a map, choosing the cheaper pack, catching a connecting flight in another time zone: everyday problems that all come down to ratio and rate. This lesson collects the skills, with care over units, which is where most marks are lost. (Syllabus sections 1.11, 1.12, 1.15 and 1.16.)

Key ideas — Ratio

Simplify by dividing every part by the HCF: 20 : 30 : 40 = 2 : 3 : 4 (make units the same first: 50 cm : 2 m = 50 : 200 = 1 : 4).

🧑‍🏫 Teacher's explanation

A ratio is a way of comparing quantities to show how much of one thing there is compared to another. Think of it like a recipe for orange juice: if the ratio of concentrate to water is 1:3, it means for every 1 cup of juice, you must add 3 cups of water. The total mixture will always have 4 parts (1+3=4).

📖 Definitions

🔍 Worked example — Easy

Simplify the ratio 45 : 75 to its simplest form.

🔍 Worked example — Standard

Divide \$500 between Alice and Bob in the ratio 2 : 3.

🔍 Worked example — Exam-style

A map has a scale of 1 : 50,000. A road on the map measures 4 cm. Calculate the actual length of the road in kilometers.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

(a) Write the ratio of flour to milk in its simplest form. [2] (b) How much flour is needed for 12 people? [2] [4 marks]

Key ideas — Proportional reasoning

Key ideas — Rates

A rate compares two quantities with different units: pay per hour, litres per minute, km per litre, an exchange rate. Other rates use a formula given in the question:

🧑‍🏫 Teacher's explanation

A rate is simply a way of describing how one quantity changes in relation to another. While a ratio compares two similar things (like 2 boys to 3 girls), a rate compares two different things, like distance and time. Think of it like a conversion factor that tells you how much of 'Thing A' you get for every single unit of 'Thing B'.

📖 Definitions

🔍 Worked example — Easy

A car travels 240 km in 3 hours. Calculate its average speed in km/h.

🔍 Worked example — Standard

A metal block has a mass of 540 g and a volume of 60 cm^3. Calculate its density.

🔍 Worked example — Exam-style

A plane leaves at 22:45 on Monday. It flies for 5 hours and 30 minutes. At what time (in 24-hour clock) does it land?

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Average speed

You must know:

🧑‍🏫 Teacher's explanation

Imagine you are driving from one city to another. You might speed up to pass a truck or slow down for a traffic light, but your speedometer is constantly changing. Average speed doesn't care about those tiny fluctuations; it only cares about the 'big picture'. It asks: 'If you had traveled at one single, constant speed for the entire journey to cover that same distance in that same time, what would that speed be?'

📖 Definitions

🔍 Worked example — Easy

A cyclist travels 45 km in 3 hours 45 minutes. Calculate the average speed in km/h.

🔍 Worked example — Standard

A train travels 120 km at a speed of 80 km/h. How long does the journey take in hours and minutes?

🔍 Worked example — Exam-style

A car travels 250 m in 12 seconds. Calculate its average speed in m/s. Then, convert this speed to km/h.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Time and money

🧑‍🏫 Teacher's explanation

Think of time and money as 'rates of change'. Time is the measurement of how long an event lasts, while money is the value we assign to goods or services. When we deal with rates, we are essentially asking: 'How much of X happens for every 1 unit of Y?' For example, if you earn \$10 per hour, your rate of pay is \$10/h.

Ch 5 · 1.11, 1.12, 1.15, 1.16 Ratio, Rates, Time and Money · IGCSE Mathematics 0580

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