Fractions
Theory
A fraction \(\dfrac{a}{b}\) has a numerator \(a\) (top) and a denominator \(b\) (bottom). Equivalent fractions name the same amount, such as \(\dfrac{3}{4}\) and \(\dfrac{6}{8}\). This Year 7 maths page covers proper and improper fractions, mixed numerals, equivalent fractions, simplest form using the HCF, and comparing fractions.
In \(\dfrac{a}{b}\), the numerator \(a\) is the top number and the denominator \(b\) is the bottom number.
A proper fraction has a numerator less than its denominator: \(\dfrac{5}{7}\), \(\dfrac{41}{100}\). An improper fraction has a numerator equal to or bigger than its denominator: \(\dfrac{7}{5}\), \(\dfrac{100}{41}\).
A mixed numeral is a whole number and a proper fraction: \(3\dfrac{5}{6}\), \(10\dfrac{1}{7}\).
Equivalent fractions name the same amount. Multiply or divide the numerator and the denominator by the same number: \(\dfrac{3}{4} = \dfrac{6}{8} = \dfrac{9}{12} = \dfrac{12}{16}\).
The highest common factor (HCF) is the largest number that divides both numbers. A fraction is in simplest form when the only common factor of the top and bottom is \(1\).
Converting and simplifying:
How to write a mixed numeral as an improper fraction
- Multiply the whole number by the denominator.
- Add the numerator, and keep the denominator: \(6\dfrac{3}{4} = \dfrac{27}{4}\).
How to write an improper fraction as a mixed numeral
- Divide the numerator by the denominator.
- The remainder is the new numerator: \(\dfrac{45}{7} = 6\dfrac{3}{7}\).
How to simplify a fraction
- Find the HCF of the numerator and denominator.
- Divide both by the HCF.
| \(\dfrac{2}{3}\) | \(=\) | \(\dfrac{2 \times 3}{3 \times 3}\) |
| \(\) | \(=\) | \(\dfrac{6}{9}\) |
\(3 \times 3 = 9\), so multiply the top by \(3\) too: \(x = 6\).
(i) \(\dfrac{4}{7} = \dfrac{\square}{28}\)
(ii) \(\dfrac{3}{8} = \dfrac{15}{\square}\)
(i) \(28 \div 7 = 4\), so multiply the top by \(4\).
(ii) \(15 \div 3 = 5\), so multiply the bottom by \(5\).
(i) \(16\)
(ii) \(40\)
Factors of \(18\): \(1, 2, 3, 6, 9, 18\). Factors of \(30\): \(1, 2, 3, 5, 6, 10, 15, 30\). The HCF is \(6\).
| \(\dfrac{18}{30}\) | \(=\) | \(\dfrac{18 \div 6}{30 \div 6}\) |
| \(\) | \(=\) | \(\dfrac{3}{5}\) |
\(\dfrac{3}{5}\)
| \(\dfrac{24}{5}\) | \(=\) | \(\dfrac{20}{5} + \dfrac{4}{5}\) |
| \(\) | \(=\) | \(4\dfrac{4}{5}\) |
\(24 \div 5\) is \(4\) remainder \(4\), so \(4\dfrac{4}{5}\).
(ii) Write \(\dfrac{45}{12}\) in simplest form, then as a mixed numeral.
| \(4\dfrac{2}{3}\) | \(=\) | \(\dfrac{12}{3} + \dfrac{2}{3} = \dfrac{14}{3}\) |
| \(\dfrac{45}{12}\) | \(=\) | \(\dfrac{15}{4} = 3\dfrac{3}{4}\) |
(i) \(\dfrac{14}{3}\)
(ii) HCF \(3\): \(\dfrac{15}{4} = 3\dfrac{3}{4}\)
| \(\dfrac{14}{24}\) | \(=\) | \(\dfrac{7}{12} = \dfrac{49}{84}\) |
| \(\dfrac{15}{28}\) | \(=\) | \(\dfrac{45}{84}\) |
The LCD of \(12\) and \(28\) is \(84\). \(49 > 45\), so 7A has the larger fraction.
Common pitfalls
Frequently asked questions
What is an equivalent fraction?
A fraction that names the same amount, made by multiplying or dividing the top and bottom by the same number. \(\dfrac{3}{4} = \dfrac{6}{8}\).
How do you simplify a fraction?
Divide the top and bottom by their HCF. \(\dfrac{18}{30} = \dfrac{3}{5}\).
How do you change a mixed numeral to an improper fraction?
Multiply the whole number by the denominator and add the numerator. \(6\dfrac{3}{4} = \dfrac{27}{4}\).
What is an improper fraction?
A fraction whose numerator is equal to or bigger than its denominator, such as \(\dfrac{13}{5}\).
How do you compare two fractions?
Write them with the same denominator and compare the numerators.