# Percentages, Interest & Growth/Decay

> Mathematics · 2025-2027 CIE IGCSE
> Source: https://www.owlsprep.com/study/cie-0580-u1-percentages-interest-growth-decay/

This guide covers all Core and Extended content for percentages, interest, and growth/decay for CIE IGCSE Mathematics 0580, including core percentage operations, simple/compound interest, reverse percentages, and exponential growth/decay models.

**Prerequisites:** [Ability to perform arithmetic operations with decimals and fractions](https://www.owlsprep.com/study/cie-0580-u1-arithmetic-decimals-fractions/); [Understanding of powers/indices](https://www.owlsprep.com/study/cie-0580-u1-indices/)

## Learning objectives

- Calculate a percentage of a quantity and express one quantity as a percentage of another
- Solve percentage increase/decrease, simple and compound interest problems
- Extended only: Solve reverse percentage and repeated percentage change problems
- Extended only: Apply the $(multiplier)^n$ model to exponential growth/decay problems including depreciation and population change

## Core Foundational Percentage Calculations

**Percentage** — A fraction expressed out of 100. Convert a percentage to a decimal by dividing by 100, and convert a decimal to a percentage by multiplying by 100.

Core content covers three key percentage operations, all of which are tested in both calculator and non-calculator papers:

1. Percentage of a quantity: Multiply the quantity by the percentage in decimal form
2. Express one quantity as a percentage of another: Divide the first quantity by the second, multiply by 100
3. Percentage increase/decrease: Divide the change in value by the original value, multiply by 100

**Worked example:** A jacket costs \$40. It is reduced by 20% in a sale. Calculate the sale price, and the percentage of the original price that the sale price represents.

1. Convert 20% to decimal: $20 \div 100 = 0.2$
2. Calculate reduction amount: $40 \times 0.2 = \$8$
3. Sale price = Original price - Reduction = $40 - 8 = \$32$
4. Percentage of original price: $(32 \div 40) \times 100 = 80\%$

> **Exam tip:** Always use the original value as the denominator when calculating percentage change, not the new value. This is one of the most common mark-losing errors for this topic.

## Core Simple and Compound Interest

**Simple Interest** — Interest calculated only on the original principal amount, where $P$ = initial principal, $r$ = annual rate (decimal), $n$ = number of years.

*Notation:* $I = P \times r \times n$

**Compound Interest** — Interest calculated on the principal plus all accumulated interest from prior periods, where $A$ = final total amount.

*Notation:* $A = P(1 + r)^n$

**Worked example:** Calculate the total amount after 2 years if \$1500 is invested at 3% per annum: (a) simple interest, (b) compound interest.

1. Part (a) Simple Interest: $P=1500$, $r=0.03$, $n=2$
2. $$I = 1500 \times 0.03 \times 2 = 90$$
3. Total amount = $1500 + 90 = \$1590$
4. Part (b) Compound Interest: Substitute values into formula
5. $$A = 1500 \times (1 + 0.03)^2 = 1500 \times 1.0609 = 1591.35$$
6. Final total amount = \$1591.35 (to nearest cent)

> **Exam tip:** Check if the question asks for the *interest earned* or the *total amount*. Students often lose marks for giving the total when only the interest is required.

## Extended: Reverse Percentages & Repeated Change

**Reverse Percentage** — Calculation of the original value of a quantity before a percentage change, by dividing the new value by the percentage multiplier.

Repeated percentage change applies the same percentage change multiple times. You can calculate the total multiplier by multiplying individual multipliers, or raising a single repeated multiplier to the power of the number of periods.

**Worked example:** A phone costs \$432 after an 8% price increase. Find the original price before the increase, then calculate the price after a further 5% increase is applied to the \$432 price.

1. 8% increase multiplier = 1.08. Original price $\times 1.08 = 432$
2. $$Original Price = 432 \div 1.08 = 400$$
3. Original price = \$400
4. 5% increase multiplier = 1.05. Total multiplier for both changes = $1.08 \times 1.05 = 1.134$
5. $$Final price = 400 \times 1.134 = 453.6$$
6. Final price after both increases = \$453.60

> **Exam tip:** For reverse percentage questions, always divide by the multiplier, never multiply. Multiplying will give you an incorrect value further from the original amount.

## Extended: Exponential Growth & Decay

**Exponential Growth/Decay** — A change where a quantity increases (growth, multiplier > 1) or decreases (decay, multiplier < 1) by a fixed percentage each period. Common scenarios include depreciation, population growth, and bacterial growth.

*Notation:* $Final Value = Initial Value \times (multiplier)^n$

*Example:* 15% annual depreciation has a multiplier of $1 - 0.15 = 0.85$

**Worked example:** A new van costs \$18000. It depreciates at 12% per annum. Calculate its value after 3 years.

1. 12% depreciation multiplier = $1 - 0.12 = 0.88$
2. Number of periods $n=3$
3. $$Value after 3 years = 18000 \times (0.88)^3$$
4. $$0.88^3 = 0.681472$$
5. $$18000 \times 0.681472 = 12266.496$$
6. Value after 3 years = \$12266.50 (to nearest cent)

> **Exam tip:** For decay problems, remember to subtract the percentage rate from 1 to get the correct multiplier. A common mistake is using 1.12 instead of 0.88 for 12% depreciation.

## Common pitfalls

- **Wrong:** Using the new value as the denominator when calculating percentage increase/decrease
  - Why it fails: Percentage change is always measured relative to the original amount, not the changed amount
  - Correct: Divide the difference between new and original values by the original value, multiply by 100
- **Wrong:** Multiplying by the percentage multiplier to find the original value in reverse percentage questions
  - Why it fails: The multiplier is applied to the original value to get the new value, so you must reverse the operation
  - Correct: Divide the new value by the percentage multiplier to get the original value
- **Wrong:** Using the simple interest formula for compound interest questions
  - Why it fails: Simple interest only calculates interest on the principal, while compound interest adds interest to the principal each period
  - Correct: Use $A = P(1+r)^n$ for compound interest, and $I = Prn$ only for simple interest
- **Wrong:** Using a multiplier greater than 1 for decay problems like depreciation
  - Why it fails: Decay means the value reduces each period, so the multiplier must be less than 1
  - Correct: Subtract the percentage decay rate from 1 to get the correct multiplier, e.g., 1 - 0.15 = 0.85 for 15% decay
- **Wrong:** Giving the total amount when the question asks for only the interest earned
  - Why it fails: Questions often explicitly ask for interest, not the sum of principal and interest
  - Correct: Subtract the original principal from the final amount if only the interest is required

## Cheatsheet

| Concept | Formula/Method | Tier |
| --- | --- | --- |
| Percentage of a quantity | Quantity × (percentage ÷ 100) | Core |
| Percentage change | ($Change$ / $Original Value$) × 100 | Core |
| Simple Interest | $I = P × r × n$ | Core |
| Compound Interest | $A = P(1 + r)^n$ | Core |
| Reverse Percentage | $Original Value = New Value ÷ Multiplier$ | Extended |
| Exponential Growth/Decay | $Final Value = Initial Value × (multiplier)^n$ | Extended |

## What's next

Mastering percentages, interest, and growth/decay is a foundational skill for many other topics in CIE IGCSE Mathematics 0580, including ratio and proportion, financial mathematics, and graphical representations of functions. You can now apply these skills to solve structured exam questions, and move on to more advanced number topics including further financial arithmetic and proportional reasoning. For Extended students, these concepts will also be useful when you study graphs of exponential functions later in the syllabus. Make sure to practice past paper questions to familiarize yourself with common exam phrasing and avoid the common pitfalls listed above.

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