Fractions, equivalent fractions and simplest form

What a fraction means

A chocolate bar is broken into \(6\) equal pieces and you eat \(5\) of them. You ate \(\frac{5}{6}\) of the bar.

A fraction is also a division: \(\frac{5}{6}\) means \(5 \div 6\). That's why the denominator can never be \(0\).

To find a fraction of a number, divide by the denominator and multiply by the numerator. For example, \(\frac{2}{3}\) of \(27\) is \(27 \div 3 \cdot 2 = 9 \cdot 2 = 18\).

Proper fractions, improper fractions and mixed numbers

• A proper fraction has a numerator smaller than its denominator, like \(\frac{3}{8}\). It's less than \(1\).
• An improper fraction has a numerator greater than or equal to its denominator, like \(\frac{11}{4}\). It's at least \(1\).
• A mixed number combines a whole number and a proper fraction, like \(2\frac{3}{4}\), which means \(2 + \frac{3}{4}\).

Equivalent fractions

\(\frac{1}{2}\) of a pizza and \(\frac{4}{8}\) of the same pizza are the same amount. Fractions that name the same number are equivalent.

So \(\frac{3}{5} = \frac{6}{10} = \frac{9}{15} = \frac{30}{50}\), and so on.

To test whether two fractions are equivalent, compare cross products: \(\frac{a}{b} = \frac{c}{d}\) exactly when \(a \cdot d = b \cdot c\).

Simplest form

A fraction is in simplest form (or lowest terms) when the numerator and denominator have no common factor other than \(1\). To get there in one step, divide both by their GCF.

Comparing fractions

• Same denominator: the larger numerator wins. \(\frac{7}{10} > \frac{3}{10}\).
• Same numerator: the smaller denominator wins, because the pieces are bigger. \(\frac{2}{3} > \frac{2}{5}\).
• Otherwise: rewrite both with a common denominator, ideally the LCM of the denominators.

Practice