Fractions, decimals and percentages
Section: Number 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Fractions, Decimals and Percentages
Fractions, decimals and percentages are three different ways of writing the same value. Being able to switch between them, and to recognise which form is which, is a skill used throughout the rest of the course.
- A proper fraction has a numerator smaller than its denominator, e.g. 3/8
- An improper fraction has a numerator equal to or greater than its denominator, e.g. 11/4
- A mixed number combines a whole number with a proper fraction, e.g. 2¾
- Fractions should always be given in their simplest form - divide the numerator and denominator by their highest common factor
| To convert... | ...to a decimal | ...to a percentage |
|---|---|---|
| from a fraction | divide the numerator by the denominator | convert to a decimal, then × 100 |
| from a decimal | - | × 100 |
| from a percentage | ÷ 100 | - |
Worked Example: Improper Fractions and Mixed Numbers
- Question: Write 17/5 as a mixed number, and write 3⅝ as an improper fraction.
- Step 1: 17 ÷ 5 = 3 remainder 2, so (17/5) = 3(2/5)
- Step 2: For 3⅝, multiply the whole number by the denominator and add the numerator: 3 × 8 + 5 = 29
- Answer: (17/5) = 3(2/5), 3(5/8) = (29/8)
Worked Example: Fraction to Decimal to Percentage
- Question: Write 7/8 as a decimal, and as a percentage.
- Step 1: Divide the numerator by the denominator: 7 ÷ 8 = 0.875
- Step 2: Multiply the decimal by 100 to get a percentage: 0.875 × 100 = 87.5
- Answer: 0.875 = 87.5%
Worked Example: Decimal to Fraction in Simplest Form
- Question: Write 0.44 as a fraction in its simplest form.
- Step 1: Write the decimal over a power of 10: 0.44 = (44/100)
- Step 2: Divide numerator and denominator by their HCF, 4: (44 ÷ 4/100 ÷ 4) = (11/25)
- Answer: 11/25
Common Mistakes
MistakeLeaving a converted fraction unsimplified, e.g. giving 44/100 as a final answer instead of 11/25
Fixalways divide the numerator and denominator by their HCF as a last step, until no common factor remains
MistakeMultiplying by 10 instead of 100 when converting a decimal to a percentage, e.g. writing 0.875 = 8.75%
Fixconverting to a percentage always means × 100, not × 10: 0.875 × 100 = 87.5%
MistakeConverting a mixed number to a decimal by converting only the fraction part and forgetting to add the whole number
Fixconvert the fraction part first, then add it to the whole number: for 3⅝, work out 5 ÷ 8 = 0.625, then add 3 to get 3.625
Recurring Decimals
Extended Only
A recurring decimal has a block of digits that repeats forever. Dot notation shows exactly which digits repeat, and a short algebraic trick converts any recurring decimal back into an exact fraction.
- A dot is placed above a repeating digit; if a block of digits repeats, a dot is placed above the first and last digit of the block
- 0.6 = 0.6666... (the digit 6 repeats forever)
- 0.45 = 0.454545... (the block "45" repeats forever)
- 0.321 = 0.3212121... (the "3" does not repeat, but the block "21" does)
Worked Example: Converting a Recurring Decimal to a Fraction
- Question: Write 0.6 as a fraction.
- Step 1: Let x = 0.6666...
- Step 2: Only 1 digit repeats, so multiply both sides by 10: 10x = 6.6666...
- Step 3: Subtract to eliminate the repeating part: 10x - x = 6.6666... - 0.6666..., so 9x = 6
- Step 4: Solve and simplify: x = (6/9) = (2/3)
- Answer: 0.6 = (2/3)
Worked Example: Converting a Two-Digit Recurring Block
- Question: Write 0.45 as a fraction.
- Step 1: Let x = 0.454545...
- Step 2: The block "45" has 2 digits, so multiply both sides by 100: 100x = 45.454545...
- Step 3: Subtract: 100x - x = 45.454545... - 0.454545..., so 99x = 45
- Step 4: Solve and simplify by dividing by 9: x = (45/99) = (5/11)
- Answer: 0.45 = (5/11)
Common Mistakes
MistakeMultiplying by the wrong power of 10, e.g. multiplying by 10 for a two-digit repeating block instead of 100
Fixthe power of 10 must match the number of repeating digits: multiply by 10 for a 1-digit block, by 100 for a 2-digit block, by 1000 for a 3-digit block
MistakeLeaving the final fraction unsimplified, e.g. giving 45/99 instead of 5/11
Fixalways simplify the resulting fraction by dividing by the HCF of the numerator and denominator
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