Decimals & Fractions Notes

Last reviewed 16 Sept 2026 · 5 min read

Fractions

A fraction $\frac{a}{b}$ represents $a$ parts of a whole divided into $b$ equal parts ( $b \neq = 0$ ); $a$ = numerator, $b$ = denominator.

Type	Definition	Example
Proper	Numerator < denominator	3/7
Improper	Numerator ≥ denominator	9/4
Mixed	Whole number + proper fraction	$2 \frac{1}{4}$ = 9/4
Equivalent	Same value	2/3 = 4/6 = 10/15
Like	Same denominators	2/9, 5/9
Unlike	Different denominators	1/3, 2/5
Unit	Numerator 1	1/8
Complex	Fraction in numerator/denominator	$\frac{2/3}{5/7}$
Continued	Fraction within fraction repeatedly	$1 + \frac{1}{1 + \frac{1}{2}}$

Common denominator (LCM) and compare numerators.
Cross-multiplication: for $\frac{a}{b}$ vs $\frac{c}{d}$ , compare $a d$ and $b c$ .
Convert to decimals.
Same numerators: smaller denominator → larger fraction.
Difference trick: for fractions like $\frac{3}{5}, \frac{4}{7}, \frac{5}{9}$ where numerator and denominator increase by constant amounts, compare quickly via decimals or cross-products.

FormulaFraction operations

$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$ (or use LCM of denominators)
$\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$
$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$
Mixed to improper: $p \frac{q}{r} = \frac{p r + q}{r}$

Decimal places: tenths, hundredths, thousandths.
Addition/subtraction — align decimal points.
Multiplication — multiply as integers; decimal places in answer = sum of decimal places.
Division — make the divisor an integer by multiplying both numbers by a power of 10.

Fraction → decimal: divide numerator by denominator.
Terminating decimal → fraction: write over power of 10 and simplify ( $0.375 = 375/1000 = 3/8$ ).

A fraction in lowest terms gives a terminating decimal if and only if the denominator has no prime factors other than 2 and 5.
Otherwise the decimal is recurring (repeating).
Examples: 7/40 (40 = 2³ × 5) → terminating (0.175); 7/30 (30 = 2 × 3 × 5) → recurring.

FormulaRecurring decimals to fractions

Pure recurring (all digits after decimal repeat):

0. \overline{ab} = \frac{ab}{99}, 0. \overline{a} = \frac{a}{9}

(as many 9s as repeating digits)

Mixed recurring (some digits do not repeat):

0. a \overline{b} = \frac{ab - a}{90}, 0. ab \overline{c d} = \frac{ab c d - ab}{9900}

Numerator = (all digits) − (non-repeating digits); denominator = as many 9s as repeating digits followed by as many 0s as non-repeating digits.

$1/7 = 0. \overline{142857}$ (6-digit cycle); $1/3 = 0. \overline{3}$ ; $1/11 = 0. \overline{09}$ .

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