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Cambridge IGCSE Mathematics — 0580 Extended

Topic 1.4: Number — Fractions, Decimals & Percentages + Ordering

Fraction types and conversion

Proper: numerator \(<\) denominator (\(\dfrac{3}{5}\)). Improper: numerator \(\geq\) denominator (\(\dfrac{7}{4}\)). Mixed: whole + proper (\(1\dfrac{3}{4}\)).

Method — mixed to improper

  1. Multiply the whole number by the denominator.
  2. Add the numerator; keep the same denominator.

Write \(2\dfrac{3}{5}\) as an improper fraction.

Worked solution: converting 2 and three fifths to thirteen fifths
Mixed to improper: \(2\dfrac{3}{5} = \dfrac{13}{5}\)

FDP conversions

Fraction → decimal

Divide numerator by denominator.

Fraction → percentage

Find the decimal, then \(\times 100\%\), or multiply by \(\dfrac{100}{1}\).

Percentage → fraction

Write over \(100\) and simplify.

Express \(\dfrac{3}{8}\) as a percentage.

Worked solution: three eighths as a percentage equals 37.5 percent
Fraction to percentage: \(\dfrac{3}{8} = 37.5\%\)

Recurring decimals

Dot notation marks the repeating block: \(0.\dot{1} = 0.111\ldots\), \(0.1\dot{7}\dot{} = 0.1777\ldots\) (1 non-recurring, 7 recurring).

Method — pure recurring \(0.\dot{a}\)

  1. Let \(x = 0.\dot{a}\) (one digit repeats).
  2. Then \(x = \dfrac{a}{9}\). Example: \(0.\dot{1} = \dfrac{1}{9}\).

Method — general recurring (algebra)

  1. Let \(x\) equal the decimal; multiply by \(10^n\) to shift the recurring block.
  2. Subtract to eliminate the recurring part; solve for \(x\).

Write \(0.\dot{1}\) as a fraction.

Worked solution: converting 0.1 recurring to one ninth using 10x minus x
Recurring decimal: \(0.\dot{1} = \dfrac{1}{9}\)

Write \(0.1\dot{7}\dot{} = 0.1777\ldots\) as a fraction.

Worked solution: converting 0.1777 recurring to eight forty-fifths
Mixed recurring: \(0.1\dot{7}\dot{} = \dfrac{8}{45}\)

Try this

Convert \(0.\dot{3}\) to a fraction in simplest form.

Show answer
Answer

  1. Let \(x\) equal the recurring decimal

    \[ x = 0.333\ldots \]

  2. Multiply by 10 (one recurring digit)

    \[ 10x = 3.333\ldots \]

  3. Subtract to clear the recurring part

    \[ 10x - x = 3 \quad \Rightarrow \quad 9x = 3 \]

  4. Simplify

    \[ x = \dfrac{3}{9} = \dfrac{1}{3} \]

  5. Answer

    \[ \dfrac{1}{3} \]

Ordering numbers

Use \(=\) equal, \(\neq\) not equal, \(>\) greater, \(<\) less, \(\geq\) greater or equal, \(\leq\) less or equal.

Method

  1. Convert all values to the same form (usually decimals) for comparison.
  2. Compare place values from left to right.
  3. Write the order using inequality symbols or smallest-to-largest list.

Put in order, smallest first: \(\dfrac{5}{8}\), \(0.62\), \(63\%\).

Worked solution: ordering five eighths, 0.62 and 63 percent smallest first
Smallest first: \(0.62,\ \dfrac{5}{8},\ 63\%\)

Exam Traps

  • \(0.\dot{3}\) is exactly \(\dfrac{1}{3}\), not a rounded \(0.33\) — use algebra for recurring conversions.
  • When ordering, check whether the question asks for smallest first or largest first.

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