Hello students!

Welcome to this comprehensive study guide on a foundational topic in IGCSE Extended Mathematics: Fractions, Decimals, and Percentages. My name is your dedicated curriculum guide, and I've designed this report to be your ultimate resource for mastering the art of converting between these three essential forms of representing numbers.

This topic isn't just about passing an exam; it's about building number fluency. It's the language of finance, shopping, statistics, and science. By the end of this guide, you won't just know how to convert these numbers—you'll understand why the methods work, where to apply them, and how to solve complex problems with confidence.

We have a lot to cover, and I've made sure every detail is explained clearly with plenty of examples. So, grab a notepad, a calculator (for checking your work!), and let's dive into the world of numerical equivalence.

IGCSE EXTENDED MATHEMATICS (0580) - STUDY REPORT

CHAPTER: Number

TOPIC: 1.4 Fractions, Decimals, and Percentages

SUBTOPIC: Recognise equivalence and convert between these forms

1. Introduction to the Subtopic

Imagine you have a friend named Alexander. Sometimes you call him "Alex," and his family might call him "Sasha." Alexander, Alex, and Sasha are three different names for the same person. In the world of mathematics, Fractions, Decimals, and Percentages are just like that—they are three different names or "languages" for representing the same value, specifically, a value that is a part of a whole.

This subtopic, "Recognise equivalence and convert between these forms," is all about learning to translate between these three mathematical languages fluently. You will learn how to take a value written as a fraction (like 1/2) and express it as a decimal (0.5) or a percentage (50%), and to do this for any conversion path.

Mastering this skill is absolutely critical for success in IGCSE Maths. It forms the bedrock for many other topics, including ratios, probability, interest, and statistics.

Key Terms and Their Importance:

Fraction: A fraction represents a part of a whole. It is written in the form a/b, where 'a' is the numerator (the number of parts you have) and 'b' is the denominator (the total number of equal parts the whole is divided into).
Importance: Fractions provide an exact way to represent values that might be messy as decimals (like 1/3). They are fundamental in algebra, probability, and everyday tasks like sharing or scaling recipes.
Decimal: A decimal is a number written using the base-10 system, where a decimal point separates the whole number part from the fractional part. Each digit to the right of the decimal point represents a power of ten (tenths, hundredths, thousandths, etc.).
Importance: Decimals are the language of calculators and computers. They are essential for precise measurements in science and engineering and are the standard for money calculations.
Percentage: A percentage is a special type of fraction where the denominator is always 100. The word "percent" literally comes from the Latin "per centum," which means "out of one hundred." The symbol for percent is %.
Importance: Percentages are the language of comparison. They make it incredibly easy to understand proportions and changes. Discounts in shops, interest rates at banks, your test scores, and statistics in the news are all commonly expressed as percentages because they provide a universal, relatable scale.

Equivalence: This is the central idea of the subtopic. It means that even though 1/4, 0.25, and 25% look different, they all represent the exact same value or quantity. Our goal is to become experts at proving and using this equivalence.

2. Detailed Explanation with Examples

Let's build a mental "conversion map." Think of Fractions, Decimals, and Percentages as three cities. There are roads connecting each city to the other two. We need to learn how to travel on all six of these roads.

The Conversion Triangle:

          FRACTIONS
         /         \
        /           \
  (A, B) /             \ (E, F)
      /                 \
     /                   \
DECIMALS ------------ PERCENTAGES
          (C, D)

Path A: Fraction to Decimal
Path B: Decimal to Fraction
Path C: Decimal to Percentage
Path D: Percentage to Decimal
Path E: Fraction to Percentage
Path F: Percentage to Fraction

We will now explore each path in exhaustive detail.

Path A: Converting Fractions to Decimals

This is one of the most straightforward conversions. The fraction bar ('/') simply means "divide."

The Golden Rule: To convert a fraction to a decimal, you divide the numerator by the denominator.

Types of Decimal Outcomes:

Terminating Decimals: These are decimals that end.
Recurring (or Repeating) Decimals: These are decimals that go on forever with a repeating pattern.

Example 1: A Simple Terminating Decimal

Convert 3/4 to a decimal.

Step 1: Set up the division: 3 ÷ 4.
Step 2-4: Perform long division.

    0.75
  ____
4 | 3.00
   -2.8
   ----
    0.20
   -0.20
   ----
     0

Result: 3/4 = 0.75

Example 2: A Simple Recurring Decimal

Convert 2/3 to a decimal.

Notice the pattern where the remainder '2' and quotient '6' repeat.

We use dot notation (0.˙6) or bar notation (0.¯6) to show the repeating digit.

Result: 2/3 = 0.˙6

Example 3: A More Complex Recurring Decimal

Convert 5/11 to a decimal.

The division shows the digits '45' repeat.

We use dot notation (0.˙4˙5) or bar notation (0.¯45).

Result: 5/11 = 0.¯45

Example 4: Converting a Mixed Number

Convert 4 3/8 to a decimal.

Method 2 (Recommended): Separate the whole and fractional parts.

The whole number is 4.
Convert the fraction part: 3/8 = 3 ÷ 8 = 0.375.
Combine them: 4 + 0.375 = 4.375.

Result: 4 3/8 = 4.375

Path B: Converting Decimals to Fractions

This process involves understanding place value.

The Mantra: "Say it, Write it, Simplify it."

Case 1: Terminating Decimals

Example: Convert 0.85 to a fraction

Say it: "Eighty-five hundredths."
Write it: 85/100.
Simplify it: Divide top and bottom by 5 to get 17/20.

Result: 0.85 = 17/20

Case 2: Recurring Decimals (An IGCSE Extended Speciality)

We must use algebra. This is a key skill.

Example: Convert 0.4˙8 (0.4888...) to a fraction

Let x = 0.4888...
Multiply by 10 to isolate the recurring part:
10x = 4.888... (Equation A)
Multiply Equation A by 10 to shift past one repeating digit:
100x = 48.888... (Equation B)
Subtract Equation A from Equation B:
100x - 10x = 48.888... - 4.888...
90x = 44
Solve for x and simplify:
x = 44/90 = 22/45

Result: 0.4˙8 = 22/45

Path C & D: Decimals and Percentages

Decimal to Percentage: Multiply by 100 (move decimal point 2 places right).
Example: 0.65 → 65%

Percentage to Decimal: Divide by 100 (move decimal point 2 places left).
Example: 38% → 0.38

Watch out! 9% = 0.09 (not 0.9), and 0.4 = 40% (not 4%).

Path E & F: Fractions and Percentages

Fraction to Percentage: Convert to a decimal first, then to a percentage (e.g., 3/8 = 0.375 = 37.5%). Or, make the denominator 100 (e.g., 13/20 = 65/100 = 65%).

Percentage to Fraction: Write over 100 and simplify.
Example: 40% = 40/100 = 2/5

Example: Convert 12.5% to a fraction

Write over 100: 12.5 / 100.
Eliminate the decimal by multiplying top and bottom by 10: 125 / 1000.
Simplify the fraction: 125/1000 = 1/8.

Result: 12.5% = 1/8

3. Key Concepts and Principles

Equivalence: 0.5, 50%, and 1/2 are identical in value.
The "Whole" as a Benchmark: The number 1.0 is equivalent to 100% and any fraction x/x.
"Per Centum": "Percent" means "out of 100," which is the key to converting to/from percentages.
The Fraction Bar: The line in a/b means a ÷ b.
Terminating vs. Recurring Rule: A simplified fraction a/b gives a terminating decimal if the prime factors of the denominator b are only 2s and/or 5s. Otherwise, it is recurring.
Example: 7/40 (denominator 40 = 2³×5) is terminating. 2/15 (denominator 15 = 3×5) is recurring because of the '3'.

4. Practical Applications

These conversions are used everywhere, every day.

Retail: Comparing a "1/3 off" deal vs. a "30% off" deal. (1/3 is 33.˙3%, so it's a better deal).
Finance: To calculate 4.5% interest on $2000, you must use the decimal 0.045. ($2000 × 0.045 = $90).
Statistics: To report that 18 out of 40 students own a pet, you convert 18/40 to a percentage. (18/40 = 9/20 = 45/100 = 45%).

5. Use Cases and Problem-Solving

Problem 1: Multi-step Comparison

In a bag of 80 marbles, 3/10 are red, 35% are blue, and the rest are green. (a) Which colour has the most marbles? (b) How many green marbles are there?

Solution:

Step 1: Convert to a common format (percentages). Red: 3/10 = 30/100 = 30%. Blue: 35%.

Step 2: Compare. 35% (Blue) > 30% (Red).

Step 3: Find the percentage of green. 100% - (30% + 35%) = 100% - 65% = 35%.

Step 4: Calculate the number of green marbles. 35% of 80 = 0.35 × 80 = 28.

Final Answer: (a) Blue and Green are tied for the most marbles (35% each). (b) There are 28 green marbles.

6. Summary Points for Quick Revision

Common Equivalents to Memorize:

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%
1/8	0.125	12.5%
1/3	0.˙3	33 1/3 % or 33.˙3%
2/3	0.˙6	66 2/3 % or 66.˙6%

7. Questions and Answers

Q1: What is numerical equivalence between fractions, decimals, and percentages?

A: It is the concept that a single value can be written in three different forms—as a fraction, a decimal, or a percentage—without changing its quantitative value. For example, 1/2, 0.5, and 50% are all equivalent.

Q2: Why do we multiply by 100 to convert a decimal to a percentage?

A: "Percent" means "out of 100." A decimal like 0.75 means 75 hundredths. Multiplying by 100 scales the value from a "part of 1" to a "part of 100," giving the percentage.

Q3: A student converts 6% to a decimal and gets 0.6. What is their mistake?

A: The mistake was moving the decimal point only one place instead of two. To convert a percentage to a decimal, you must divide by 100. 6 ÷ 100 = 0.06. The value 0.6 is equivalent to 60%, not 6%.

Q4: Without doing the full division, how can you predict if the fraction 9/24 will be a terminating or a recurring decimal?

A: First, simplify the fraction to 3/8. Next, find the prime factors of the denominator (8), which are 2 × 2 × 2. Since the only prime factor is '2', the decimal will be terminating.

Q5: If a price is increased by 25%, what single decimal number would you multiply the original price by to find the new price?

A: The new price is 100% + 25% = 125% of the original. To use this in a calculation, convert 125% to a decimal by dividing by 100, which gives 1.25. You would multiply by 1.25.

Study Notes

Study Notes

Recognise equivalence and convert between these forms: