Mastering Fraction Operations
Step-by-step methods to add, subtract, multiply, and divide fractions with confidence.
Fractions show parts of a whole, and you can add, subtract, multiply, and divide them by following clear steps.
You will learn how to find common denominators, simplify answers, and handle mixed numbers confidently.
With practice, fraction operations become quick and reliable.
🔢 Fraction Basics and Simplifying
Key words
Numerator: the top number, counts parts.
Denominator: the bottom number, shows equal parts of the whole.
Simplify: write a fraction in simplest form by dividing the numerator and denominator by their HCF.
To simplify $\frac{a}{b}$, divide top and bottom by the HCF of $a$ and $b$.
Example
$$\frac{18}{24}$$
$$\text{HCF}(18,24)=6$$
$$\frac{18 \div 6}{24 \div 6}=\frac{3}{4}$$
Equivalent fractions represent the same value.
$$\frac{2}{3}=\frac{2 \times 5}{3 \times 5}=\frac{10}{15}$$
➕➖ Add and Subtract Fractions
Same denominator
Add or subtract numerators and keep the denominator, then simplify.
$$\frac{3}{8}+\frac{1}{8}=\frac{4}{8}=\frac{1}{2}$$
Different denominators
Find a common denominator, usually the LCM of the denominators.
Rewrite each fraction with that denominator.
Add or subtract numerators, keep the denominator, and simplify.
Example
Compute $\frac{1}{6}+\frac{1}{4}$.
LCM of $6$ and $4$ is $12$.
$$\frac{1}{6}=\frac{2}{12} \quad \frac{1}{4}=\frac{3}{12}$$
$$\frac{2}{12}+\frac{3}{12}=\frac{5}{12}$$
Subtraction with mixed numbers
Compute $2\tfrac{1}{5}-\tfrac{3}{5}$.
Write $2\tfrac{1}{5}=\frac{11}{5}$.
$$\frac{11}{5}-\frac{3}{5}=\frac{8}{5}=1\tfrac{3}{5}$$
✖️➗ Multiply and Divide Fractions
Multiplication rule
Multiply numerators together and denominators together, then simplify.
$$\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}$$
Example
$$\frac{3}{4} \times \frac{2}{9}=\frac{6}{36}=\frac{1}{6}$$
Cross simplification
You can simplify before multiplying by cancelling common factors across a numerator and the other denominator.
$$\frac{5}{12} \times \frac{9}{10}$$
$5$ and $10$ share factor $5$ so $5 \div 5=1$ and $10 \div 5=2$.
$9$ and $12$ share factor $3$ so $9 \div 3=3$ and $12 \div 3=4$.
Now multiply
$$\frac{1}{4} \times \frac{3}{2}=\frac{3}{8}$$
Division rule
To divide by a fraction, multiply by its reciprocal.
$$\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}$$
Example
$$\frac{7}{9} \div \frac{14}{27}=\frac{7}{9} \times \frac{27}{14}$$
Simplify $7$ with $14$ to get $1$ and $2$.
Simplify $27$ with $9$ to get $3$ and $1$.
Now multiply
$$\frac{1}{1} \times \frac{3}{2}=\frac{3}{2}=1\tfrac{1}{2}$$
Quick checks
Simplify at the end, and also before multiplying if possible.
For addition and subtraction, never add denominators; use an LCM to find a common denominator.
Mixed numbers are easier if you convert them to improper fractions first, unless subtraction is straightforward.
Practice Problems
Try these problems to build fluency. Simplify all answers.
1) Simplify $frac{42}{56}$.
Show answer
Find HCF of $42$ and $56$ which is $14$.
$$frac{42 div 14}{56 div 14}=frac{3}{4}$$
2) Compute $frac{5}{12}+frac{7}{18}$.
Show answer
LCM of $12$ and $18$ is $36$.
$$frac{5}{12}=frac{15}{36} quad frac{7}{18}=frac{14}{36}$$
$$frac{15}{36}+frac{14}{36}=frac{29}{36}$$
3) Compute $3tfrac{1}{4}-1tfrac{5}{8}$.
Show answer
Convert to improper fractions.
$3tfrac{1}{4}=frac{13}{4}$ and $1tfrac{5}{8}=frac{13}{8}$.
LCM of $4$ and $8$ is $8$.
$$frac{13}{4}=frac{26}{8}$$
$$frac{26}{8}-frac{13}{8}=frac{13}{8}=1tfrac{5}{8}$$
4) Multiply $frac{6}{35} times frac{14}{9}$ with cross simplification.
Show answer
Simplify $14$ with $35$ using $7$ to get $2$ and $5$.
Simplify $6$ with $9$ using $3$ to get $2$ and $3$.
Now multiply
$$frac{2}{5} times frac{2}{3}=frac{4}{15}$$
5) Divide $frac{11}{12} div frac{22}{27}$ and give the answer as a mixed number if improper.
Show answer
Turn division into multiplication by the reciprocal.
$$frac{11}{12} times frac{27}{22}$$
Simplify $11$ with $22$ to get $1$ and $2$.
Simplify $27$ with $12$ using $3$ to get $9$ and $4$.
Now multiply
$$frac{1}{4} times frac{9}{2}=frac{9}{8}=1tfrac{1}{8}$$
Interactive Quiz
Test your understanding with these quick questions.