Ch 6 · 1.13, 1.17 Percentages and Exponential Growth and Decay · IGCSE Mathematics 0580

Topic 6 of 6 in Number — IGCSE Mathematics 0580 — 58 lessons.

Introduction

Sales, interest on savings, a car losing value, a city growing each year: percentages are everywhere in money and populations. The key tool is the multiplier: a 15% increase is × 1.15, a 12% decrease is × 0.88. Repeat the multiplier and you get exponential growth or decay. (Syllabus sections 1.13 and 1.17.)

Key ideas — Basic percentages

Percentages may be over 100%, and appear as deposits, discounts, profit and loss, and earnings.

🧑‍🏫 Teacher's explanation

Think of a percentage as a way of slicing a pie into 100 equal pieces. If you have 15\% of a pie, you have 15 of those tiny slices. When we calculate a percentage of a quantity, we are essentially finding out what that 'slice' is worth in real numbers. For example, 15\% of 80 is just 15/100 of 80, which simplifies to 0.15 × 80 = 12.

📖 Definitions

🔍 Worked example — Easy

Calculate the percentage decrease from 80 to 60.

🔍 Worked example — Standard

A laptop is sold for \450 after a 10\%$ discount. Calculate the original price.

🔍 Worked example — Exam-style

A population of bacteria increases by 5\% every hour. If the initial population is 2000, find the population after 3 hours.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Multipliers

Increase by r\%: multiply by 1 + r/100. Decrease by r\%: multiply by 1 - r/100. A price of \$40 increased by 15% is 40 × 1.15 = \46.

🧑‍🏫 Teacher's explanation

When we talk about percentage changes, we are essentially talking about how much of the original amount remains or how much extra is added. Instead of calculating the 'increase amount' and then adding it back to the original, we can use a single number called a 'multiplier' to do the job in one step. Think of it like a scale: if you want to increase something by 10\%, you are keeping 100\% of what you have and adding 10\%, giving you 110\% total. In decimal form, that is 1.10.

📖 Definitions

🔍 Worked example — Easy

A shop increases the price of a jacket from \80 by 15\%. Calculate the new price.

🔍 Worked example — Standard

A population of bacteria decreases by 5\% every hour. If the initial population is 5000, find the population after 3 hours.

🔍 Worked example — Exam-style

After a 12\% increase, a car's value is \15,600. Calculate the original value of the car.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Simple and compound interest

amount = P (1 + r/100 )^n

🧑‍🏫 Teacher's explanation

Imagine you are lending a friend \$100. If they agree to pay you back \$5 every year, that is like simple interest. The amount they owe you grows by the same fixed amount every single time. It is a steady, straight line.

📖 Definitions

🔍 Worked example — Easy

Calculate the simple interest earned on \$5000 at a rate of 4% per annum for 3 years.

🔍 Worked example — Standard

A bank account starts with \$1200. It earns 5% compound interest per year. Calculate the total amount in the account after 4 years.

🔍 Worked example — Exam-style

A car's value depreciates by 12% each year. If the car is worth \$25,000 now, calculate its value after 3 years.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Reverse percentages

When you know the amount after a change, divide by the multiplier to find the original. A price after a 20% discount is \$96: original = 96 ÷ 0.8 = \120. Never take 20% of \$96.

🧑‍🏫 Teacher's explanation

Imagine you are looking at a photo that has been zoomed in. You see the current size, but you want to know how big it was before the zoom happened. In maths, a percentage change is like a 'multiplier' that scales a number up or down. If something increases by 10\%, it becomes 110\% of its original size, which is a multiplier of 1.1. If it decreases by 10\%, it becomes 90\% of its original size, a multiplier of 0.9.

📖 Definitions

🔍 Worked example — Easy

A shirt is sold for \35 after a 30\% discount. Find the original price of the shirt.

🔍 Worked example — Standard

After a 15\% price increase, a laptop costs \690. Calculate the original price.

🔍 Worked example — Exam-style

A population of bacteria increases by 12\% every hour. After one hour, the population is 5600. Find the initial population.

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Exponential growth and decay

Repeated percentage change gives value = start × (multiplier)^n:

🧑‍🏫 Teacher's explanation

Imagine you are watching a snowball rolling down a hill. If the snowball picks up a fixed amount of snow every second, that is linear growth. But if the snowball picks up a percentage of its *current* size every second, the bigger it gets, the faster it grows. This is exponential growth.

Ch 6 · 1.13, 1.17 Percentages and Exponential Growth and Decay · IGCSE Mathematics 0580

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