← Home

Common Fractions

Grade 8
Step 1
INTRODUCTION
1
Ready? Try the Quiz →

Full lesson notes

Everything covered in this lesson, in one place - useful for revision or printing.

\( \dfrac{3}{4} \)

Common fractions (also called vulgar fractions) are one of the most useful tools in Mathematics — they let us describe parts of a whole exactly, without rounding.

In this lesson you will learn to:

We will work with common fractions throughout — no decimals needed.

1. What Is a Common Fraction?

\( \dfrac{3}{4} \;\longleftarrow\; \text{numerator} \over \text{denominator} \)

A common fraction is written \( \dfrac{a}{b} \), where:

• the numerator (\(a\)) — how many parts we have
• the denominator (\(b\)) — how many equal parts the whole is split into
In \( \dfrac{3}{4} \), the whole is cut into 4 equal parts, and we have 3 of them.

\( \dfrac{2}{5} \quad \text{vs} \quad \dfrac{7}{5} \)

A proper fraction has numerator smaller than denominator, e.g. \( \dfrac{2}{5} \) — it represents less than one whole.

An improper fraction has numerator equal to or larger than denominator, e.g. \( \dfrac{7}{5} \) — it represents one whole or more.

\( \dfrac{7}{5} = 1\dfrac{2}{5} \)

A mixed number combines a whole number and a proper fraction, e.g. \( 1\dfrac{2}{5} \).

Improper → mixed: divide numerator by denominator. \( 7 \div 5 = 1 \) remainder \( 2 \), so \( \dfrac{7}{5} = 1\dfrac{2}{5} \)
Mixed → improper: \( 1\dfrac{2}{5} = \dfrac{(1\times5)+2}{5} = \dfrac{7}{5} \)

\( 2\dfrac{3}{4} = \dfrac{?}{4} \)

Quick check: convert \( 2\dfrac{3}{4} \) to an improper fraction.

2. Equivalent & Simplest Form

\( \dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6} = \dfrac{4}{8} \)

Equivalent fractions represent the same value. Multiply (or divide) the numerator AND denominator by the SAME non-zero number to get an equivalent fraction.

\( \dfrac{1}{2} \times \dfrac{4}{4} = \dfrac{4}{8} \)  — still one half, written differently.

\( \dfrac{12}{18} = \dfrac{12\div6}{18\div6} = \dfrac{2}{3} \)

To write a fraction in its simplest form, divide the numerator and denominator by their HCF (Highest Common Factor).

HCF of 12 and 18 is 6.
\( \dfrac{12}{18} = \dfrac{12\div6}{18\div6} = \dfrac{2}{3} \)
A fraction is fully simplified when the HCF of numerator and denominator is 1.

\( \dfrac{24}{36} \)

Simplify \( \dfrac{24}{36} \):

HCF(24, 36) = 12
\( \dfrac{24}{36} = \dfrac{24\div12}{36\div12} = \dfrac{2}{3} \)

\( \dfrac{15}{20} \) in simplest form?

Quick check: simplify \( \dfrac{15}{20} \) to lowest terms.

3. Comparing & Ordering

\( \dfrac{3}{7} \; ? \; \dfrac{5}{7} \)

When fractions have the same denominator, just compare the numerators — the bigger numerator is the bigger fraction.

\( \dfrac{3}{7} < \dfrac{5}{7} \)  because \( 3 < 5 \)

\( \dfrac{2}{3} \; ? \; \dfrac{3}{4} \)

To compare fractions with different denominators, first rewrite them with a Lowest Common Denominator (LCD) — the LCM of the denominators.

LCM(3, 4) = 12
\( \dfrac{2}{3} = \dfrac{8}{12} \)  and  \( \dfrac{3}{4} = \dfrac{9}{12} \)
Since \( 8 < 9 \), \( \dfrac{2}{3} < \dfrac{3}{4} \)

\( \dfrac{1}{2}, \; \dfrac{2}{3}, \; \dfrac{5}{6} \)

Order \( \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{5}{6} \) from smallest to largest:

LCD of 2, 3, 6 is 6.
\( \dfrac{1}{2}=\dfrac{3}{6} \),   \( \dfrac{2}{3}=\dfrac{4}{6} \),   \( \dfrac{5}{6}=\dfrac{5}{6} \)
So the order is \( \dfrac{1}{2} < \dfrac{2}{3} < \dfrac{5}{6} \)

\( \dfrac{3}{5} \; ? \; \dfrac{5}{8} \)

Quick check: which is bigger, \( \dfrac{3}{5} \) or \( \dfrac{5}{8} \)? Type <, > or =.

Hint: LCD of 5 and 8 is 40. \( \dfrac{3}{5}=\dfrac{24}{40} \), \( \dfrac{5}{8}=\dfrac{25}{40} \).

4. Adding & Subtracting

\( \dfrac{2}{9} + \dfrac{4}{9} = \dfrac{6}{9} = \dfrac{2}{3} \)

With the same (like) denominator, add or subtract the numerators and keep the denominator. Always simplify the answer.

\( \dfrac{2}{9} + \dfrac{4}{9} = \dfrac{6}{9} = \dfrac{2}{3} \)

\( \dfrac{1}{4} + \dfrac{1}{6} \)

With unlike denominators, first find the LCD, rewrite both fractions, then add/subtract numerators.

LCD of 4, 6 is 12.
\( \dfrac{1}{4}=\dfrac{3}{12} \),   \( \dfrac{1}{6}=\dfrac{2}{12} \)
\( \dfrac{3}{12}+\dfrac{2}{12}=\dfrac{5}{12} \)

\( 2\dfrac{1}{3} + 1\dfrac{3}{4} \)

With mixed numbers: convert to improper fractions first, use the LCD, then simplify (and convert back if needed).

\( 2\dfrac{1}{3} = \dfrac{7}{3} \),   \( 1\dfrac{3}{4} = \dfrac{7}{4} \)
LCD of 3, 4 is 12: \( \dfrac{28}{12} + \dfrac{21}{12} = \dfrac{49}{12} = 4\dfrac{1}{12} \)

\( 3\dfrac{1}{5} - 1\dfrac{3}{5} \)

When subtracting mixed numbers and the second fraction is bigger, convert to improper fractions to avoid borrowing mistakes.

\( 3\dfrac{1}{5} = \dfrac{16}{5} \),   \( 1\dfrac{3}{5} = \dfrac{8}{5} \)
\( \dfrac{16}{5} - \dfrac{8}{5} = \dfrac{8}{5} = 1\dfrac{3}{5} \)

\( \dfrac{2}{5} + \dfrac{3}{10} \)

Quick check: calculate \( \dfrac{2}{5} + \dfrac{3}{10} \) in simplest form.

5. Multiplying

\( \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{8}{15} \)

To multiply fractions: multiply the numerators together, and multiply the denominators together. No need for a common denominator!

\( \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{2\times4}{3\times5} = \dfrac{8}{15} \)

\( \dfrac{3}{8} \times \dfrac{4}{9} \)

It is often easier to cancel common factors before multiplying.

\( \dfrac{3}{8} \times \dfrac{4}{9} \)  — divide 3 & 9 by their HCF 3, and 4 & 8 by their HCF 4:
\( \dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6} \)

\( 1\dfrac{1}{2} \times 2\dfrac{2}{3} \)

Convert mixed numbers to improper fractions BEFORE multiplying — never multiply the whole number and fraction parts separately.

\( 1\dfrac{1}{2} = \dfrac{3}{2} \),   \( 2\dfrac{2}{3} = \dfrac{8}{3} \)
\( \dfrac{3}{2} \times \dfrac{8}{3} = \dfrac{24}{6} = 4 \)

\( \dfrac{3}{5} \; \text{of} \; 20 \)

"Of" means multiply. Write the whole number as a fraction over 1.

\( \dfrac{3}{5} \; \text{of} \; 20 = \dfrac{3}{5} \times \dfrac{20}{1} = \dfrac{60}{5} = 12 \)

\( \dfrac{2}{7} \times \dfrac{7}{8} \)

Quick check: calculate \( \dfrac{2}{7} \times \dfrac{7}{8} \), simplified.

6. Dividing

\( \dfrac{3}{5} \;\longrightarrow\; \dfrac{5}{3} \)

The reciprocal of a fraction is found by swapping (flipping) the numerator and denominator.

Reciprocal of \( \dfrac{3}{5} \) is \( \dfrac{5}{3} \).
A fraction multiplied by its own reciprocal always equals 1.

\( \dfrac{2}{3} \div \dfrac{4}{5} \)

To divide by a fraction: KEEP the first fraction, CHANGE \( \div \) to \( \times \), and FLIP the second fraction (use its reciprocal).

\( \dfrac{2}{3} \div \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6} \)

\( 2\dfrac{1}{4} \div 1\dfrac{1}{2} \)

As with multiplication, convert mixed numbers to improper fractions FIRST, then keep-change-flip.

\( 2\dfrac{1}{4} = \dfrac{9}{4} \),   \( 1\dfrac{1}{2} = \dfrac{3}{2} \)
\( \dfrac{9}{4} \div \dfrac{3}{2} = \dfrac{9}{4} \times \dfrac{2}{3} = \dfrac{18}{12} = \dfrac{3}{2} = 1\dfrac{1}{2} \)

\( \dfrac{4}{5} \div 2 \)

Write the whole number as a fraction over 1, then keep-change-flip as usual.

\( \dfrac{4}{5} \div 2 = \dfrac{4}{5} \div \dfrac{2}{1} = \dfrac{4}{5} \times \dfrac{1}{2} = \dfrac{4}{10} = \dfrac{2}{5} \)

\( \dfrac{3}{4} \div \dfrac{9}{8} \)

Quick check: calculate \( \dfrac{3}{4} \div \dfrac{9}{8} \), simplified.

\[ \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd} \qquad \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} \]

Common Fractions — full summary:

• Equivalent: multiply/divide numerator & denominator by the same number
• Simplify: divide by the HCF
• Compare/order: rewrite over the LCD, then compare numerators
• Add/subtract: use the LCD; convert mixed numbers to improper fractions first
• Multiply: numerator × numerator, denominator × denominator; cancel first if possible
• Divide: keep, change, flip — multiply by the reciprocal
• Always convert mixed numbers to improper fractions before multiplying or dividing
You are ready to attempt the quiz!