Recap and mixed numbers

A fraction is a number of the form $\tfrac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. It represents $a$ equal parts out of $b$ total parts.

Idea

• Proper fraction: numerator less than denominator. Value less than $1$. Example: $\tfrac{3}{5}$.
• Improper fraction: numerator greater than or equal to denominator. Value $\geq 1$. Example: $\tfrac{7}{4}$.
• Mixed number: a whole number plus a proper fraction. Example: $1\tfrac{3}{4}$, meaning $1 + \tfrac{3}{4}$.

Improper to mixed. Divide the numerator by the denominator. The quotient is the whole part, the remainder is the new numerator, and the denominator stays the same. So $\tfrac{7}{4} = 1\,\text{R}\,3 = 1\tfrac{3}{4}$.

Mixed to improper. Multiply the whole part by the denominator, add the numerator. Keep the same denominator. So $1\tfrac{3}{4} = \tfrac{1 \times 4 + 3}{4} = \tfrac{7}{4}$.

Equivalent fractions. Multiplying or dividing both numerator and denominator by the same non-zero number gives an equivalent fraction. So $\tfrac{2}{3} = \tfrac{4}{6} = \tfrac{6}{9} = \tfrac{10}{15}$, etc.

Simplest form. A fraction is in simplest form (also called lowest terms) when the numerator and denominator share no common factor other than $1$. To simplify, divide both by their HCF (highest common factor).

Adding/subtracting fractions (recap): make the denominators equal first, then add or subtract the numerators. $\tfrac{1}{2} + \tfrac{1}{3} = \tfrac{3}{6} + \tfrac{2}{6} = \tfrac{5}{6}$.

Worked examples

Example 1. Convert $\tfrac{17}{5}$ to a mixed number.

$17 \div 5 = 3$ remainder $2$. So $\tfrac{17}{5} = 3\tfrac{2}{5}$.

Example 2. Convert $4\tfrac{2}{7}$ to an improper fraction.

$4 \times 7 + 2 = 30$. So $4\tfrac{2}{7} = \tfrac{30}{7}$.

Example 3. Simplify $\tfrac{18}{24}$.

HCF of $18$ and $24$ is $6$. So $\tfrac{18}{24} = \tfrac{3}{4}$.

Example 4. Add $1\tfrac{1}{4} + \tfrac{2}{3}$.

Convert mixed to improper: $1\tfrac{1}{4} = \tfrac{5}{4}$. Common denominator with $\tfrac{2}{3}$ is $12$. So $\tfrac{15}{12} + \tfrac{8}{12} = \tfrac{23}{12} = 1\tfrac{11}{12}$.

Try it yourself

1. Convert $\tfrac{23}{6}$ to a mixed number.
2. Convert $5\tfrac{3}{4}$ to an improper fraction.
3. Simplify $\tfrac{36}{48}$.
4. Find an equivalent fraction of $\tfrac{2}{5}$ with denominator $20$.
5. Compute $\tfrac{3}{5} + \tfrac{2}{7}$.
6. Compute $\tfrac{5}{6} - \tfrac{1}{4}$.
7. Compare $\tfrac{3}{4}$ and $\tfrac{5}{7}$ , which is larger?
8. Convert $\tfrac{100}{12}$ to a mixed number in simplest form.

Activity

Fold a strip of paper into halves, quarters, eighths. Label each fold. Show $\tfrac{3}{8}$, $\tfrac{1}{2}$, $\tfrac{5}{8}$ on the strip. Convince yourself visually that $\tfrac{4}{8} = \tfrac{1}{2}$.