Decimal to fraction and back: the quick swap

Fraction → decimal

Divide numerator by denominator.

• $\dfrac{1}{2} = 0.5$
• $\dfrac{3}{4} = 0.75$
• $\dfrac{1}{8} = 0.125$
• $\dfrac{1}{3} = 0.333\ldots$ (repeating, written $0.\overline{3}$)

Decimal → fraction

Count the digits after the decimal point, then put $10, 100, 1000, \ldots$ in the denominator.

• $0.5 = \dfrac{5}{10} = \dfrac{1}{2}$ (after simplification)
• $0.75 = \dfrac{75}{100} = \dfrac{3}{4}$
• $2.125 = \dfrac{2125}{1000} = \dfrac{17}{8}$

Table to memorize

Decimal	Fraction
$0.5$	$1/2$
$0.25$	$1/4$
$0.75$	$3/4$
$0.2$	$1/5$
$0.125$	$1/8$
$0.1$	$1/10$
$0.333\ldots$	$1/3$
$0.666\ldots$	$2/3$

Repeating decimal to fraction (bonus)

$0.\overline{3}$? Let $x = 0.333\ldots$

$10 x = 3.333\ldots = 3 + x$ ⇒ $9x = 3$ ⇒ $x = 1/3$. ✓

$0.\overline{36}$? $100 x = 36.\overline{36} = 36 + x$ ⇒ $99 x = 36$ ⇒ $x = 36/99 = 4/11$.

Trick: /10 or ×10

• Dividing by 10 = shift decimal one place left.
• Multiplying by 10 = one place right.

$2.45 \div 10 = 0.245$, $2.45 \times 100 = 245$.

Common mistakes

1. Forgetting to simplify ($75/100$ = $3/4$).
2. Miscounting denominator zeros (2 digits after decimal ⇒ /100).
3. $0.33 = 1/3$ (false: $0.33 = 33/100$, while $1/3 = 0.333\ldots$).

?À toi de jouer

Decimal of 7/8?

A.0.875B.0.78C.0.87D.0.8

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