Operations with fractions

Adding Fractions

When working with fractions, the main idea in any operation is to pay attention to both the numerator (top number) and the denominator (bottom number), and to keep the meaning of “parts of a whole” in mind.

For addition and subtraction, the key step is to work with like denominators (the same denominator).

Like denominators

If the denominators are already the same, you can add the numerators directly and keep the denominator.

Example:
$$
\frac{2}{7} + \frac{3}{7} = \frac{2 + 3}{7} = \frac{5}{7}
$$

Here you are adding “sevenths” together. The size of each part (a seventh) stays the same; only the count of parts changes.

Unlike denominators: finding a common denominator

If the denominators are different, you first rewrite the fractions so they have the same denominator. This new denominator is called a common denominator.

A simple (though not always smallest) choice is to use the product of the denominators.

Example:
$$
\frac{1}{3} + \frac{1}{4}
$$

1. Find a common denominator. Using the product:
   $$
   3 \times 4 = 12
   $$
   so we choose $.
2. Rewrite each fraction with denominator $12$:
   $$
   \frac{1}{3} = \frac{4}{12} \quad(\text{multiply top and bottom by } 4)
   $$
   $$
   \frac{1}{4} = \frac{3}{12} \quad(\text{multiply top and bottom by } 3)
   $$
3. Add the new fractions:
   $$
   \frac{4}{12} + \frac{3}{12} = \frac{7}{12}
   $$

Using the least common denominator (LCD)

Often there is a smaller common denominator, called the least common denominator (LCD): the smallest positive number that both denominators divide into.

Finding the LCD:

• For $4$ and $6$, the LCD is $12$ (not $24$).
• For $3$ and $5$, the LCD is $15$.
• For $8$ and $12$, the LCD is $24$.

Example using the LCD:
$$
\frac{5}{6} + \frac{1}{4}
$$

1. LCD of $6$ and $4$ is $12$.
2. Rewrite:
   $$
   \frac{5}{6} = \frac{10}{12} \quad(\times 2)
   $$
   $$
   \frac{1}{4} = \frac{3}{12} \quad(\times 3)
   $$
3. Add:
   $$
   \frac{10}{12} + \frac{3}{12} = \frac{13}{12}
   $$

Often you then rewrite improper fractions or simplify them, but that is treated elsewhere.

Adding mixed numbers

Mixed numbers combine a whole number and a fraction (for example, $2\frac{1}{3}$).

There are two common methods for addition.

Method 1: Add whole parts and fractional parts separately

Example:
$$
2\frac{1}{3} + 1\frac{2}{3}
$$

1. Add the whole numbers: $2 + 1 = 3$.
2. Add the fractions:
   $$
   \frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1
   $$
3. Combine:
   $$
   3 + 1 = 4
   $$

If the fractional parts have different denominators, first convert them to like denominators, then add.

Example:
$$
3\frac{1}{4} + 2\frac{2}{3}
$$

1. Whole numbers: $3 + 2 = 5$.
2. Fractions:
• LCD of $4$ and $3$ is $12$.
• $\dfrac{1}{4} = \dfrac{3}{12}$ and $\dfrac{2}{3} = \dfrac{8}{12}$.
• Add: $\dfrac{3}{12} + \dfrac{8}{12} = \dfrac{11}{12}$.
3. Result:
   $$
   5\frac{11}{12}
   $$

Method 2: Convert mixed numbers to improper fractions

Example:
$$
2\frac{1}{3} + 1\frac{2}{3}
$$

1. Convert:
   $$
   2\frac{1}{3} = \frac{7}{3}, \quad 1\frac{2}{3} = \frac{5}{3}
   $$
2. Add:
   $$
   \frac{7}{3} + \frac{5}{3} = \frac{12}{3} = 4
   $$

This method is especially convenient when there are several mixed numbers or more complex denominators.

Subtracting Fractions

Subtraction of fractions mirrors addition: use common denominators and subtract the numerators.

Like denominators

If denominators are the same:
$$
\frac{5}{9} - \frac{2}{9} = \frac{5 - 2}{9} = \frac{3}{9}
$$

You might then simplify the result.

Unlike denominators

Find a common denominator, rewrite, then subtract.

Example:
$$
\frac{3}{4} - \frac{1}{6}
$$

1. LCD of $4$ and $6$ is $12$.
2. Rewrite:
   $$
   \frac{3}{4} = \frac{9}{12} \quad(\times 3)
   $$
   $$
   \frac{1}{6} = \frac{2}{12} \quad(\times 2)
   $$
3. Subtract:
   $$
   \frac{9}{12} - \frac{2}{12} = \frac{7}{12}
   $$

Subtracting mixed numbers

Again, there are two main methods.

Method 1: Subtract whole and fractional parts (with borrowing if needed)

Example without borrowing:
$$
5\frac{3}{8} - 2\frac{1}{8}
$$

1. Whole numbers: $5 - 2 = 3$.
2. Fractions:
   $$
   \frac{3}{8} - \frac{1}{8} = \frac{2}{8}
   $$
3. Result:
   $$
   3\frac{2}{8}
   $$

Often you then simplify the fractional part.

Example with borrowing:
$$
6\frac{1}{5} - 2\frac{3}{5}
$$

You cannot do $\dfrac{1}{5} - \dfrac{3}{5}$ directly (the top would be negative), so borrow $1$ from the whole number part.

1. Rewrite $6\frac{1}{5}$:
• Borrow $1$ from $6$: it becomes $5$.
• Convert the borrowed $1$ to fifths: $1 = \dfrac{5}{5}$.
• Add to the existing $\dfrac{1}{5}$:
  $$
  \frac{5}{5} + \frac{1}{5} = \frac{6}{5}
  $$
• Now $6\frac{1}{5} = 5\frac{6}{5}$.

So rewrite the subtraction as:
$$
5\frac{6}{5} - 2\frac{3}{5}
$$

2. Whole numbers: $5 - 2 = 3$.
3. Fractions:
   $$
   \frac{6}{5} - \frac{3}{5} = \frac{3}{5}
   $$
4. Result:
   $$
   3\frac{3}{5}
   $$

Method 2: Convert to improper fractions

Example:
$$
4\frac{2}{3} - 1\frac{5}{6}
$$

1. Convert:
   $$
   4\frac{2}{3} = \frac{14}{3}, \quad 1\frac{5}{6} = \frac{11}{6}
   $$
2. Find a common denominator (LCD of $3$ and $6$ is $6$):
   $$
   \frac{14}{3} = \frac{28}{6}
   $$
3. Subtract:
   $$
   \frac{28}{6} - \frac{11}{6} = \frac{17}{6}
   $$
4. Optionally, convert back to a mixed number.

Multiplying Fractions

Multiplication of fractions is more direct: you multiply numerators together and denominators together.

Basic rule

If
$$
\frac{a}{b} \text{ and } \frac{c}{d}
$$
are fractions (with $b \neq 0$ and $d \neq 0$), their product is
$$
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
$$

Example:
$$
\frac{2}{5} \times \frac{3}{7} = \frac{2 \cdot 3}{5 \cdot 7} = \frac{6}{35}
$$

You usually simplify the result if possible.

Cross-cancelling before multiplying

To avoid large numbers and simplify work, you can look for common factors between any numerator and any denominator before multiplying. This is often called cross-cancelling.

Example:
$$
\frac{6}{14} \times \frac{7}{9}
$$

Look for common factors:

• $6$ and $9$ have common factor $3$.
• $14$ and $7$ have common factor $7$.

1. Divide $6$ by $3$ and $9$ by $3$:
   $$
   6 \to 2, \quad 9 \to 3
   $$
2. Divide $14$ by $7$ and $7$ by $7$:
   $$
   14 \to 2, \quad 7 \to 1
   $$
3. Now multiply the simplified numbers:
   $$
   \frac{2}{2} \times \frac{1}{3} = \frac{2}{6} = \frac{1}{3}
   $$

This gives the same final result as multiplying first and simplifying later, but usually with easier arithmetic.

Multiplying by whole numbers

A whole number $n$ can be written as a fraction $\dfrac{n}{1}$.

So:
$$
n \times \frac{a}{b} = \frac{n}{1} \times \frac{a}{b} = \frac{na}{b}
$$

Example:
$$
3 \times \frac{4}{5} = \frac{3 \cdot 4}{5} = \frac{12}{5}
$$

This can be left improper or rewritten as a mixed number.

Multiplying mixed numbers

You typically convert mixed numbers to improper fractions first, then multiply.

Example:
$$
2\frac{1}{2} \times 1\frac{1}{3}
$$

1. Convert:
   $$
   2\frac{1}{2} = \frac{5}{2}, \quad 1\frac{1}{3} = \frac{4}{3}
   $$
2. Multiply:
   $$
   \frac{5}{2} \times \frac{4}{3} = \frac{20}{6}
   $$
3. Simplify:
   $$
   \frac{20}{6} = \frac{10}{3}
   $$
4. Optionally convert to a mixed number later.

Dividing Fractions

Division with fractions uses the idea of multiplying by a reciprocal.

The reciprocal of $\dfrac{a}{b}$ (assuming $a \neq 0$) is $\dfrac{b}{a}$.

Basic rule: multiply by the reciprocal

To divide by a fraction, multiply by its reciprocal:
$$
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
$$
provided $c \neq 0$.

This is sometimes remembered as “keep, change, flip”:

• Keep the first fraction.
• Change division to multiplication.
• Flip (take the reciprocal of) the second fraction.

Example:
$$
\frac{3}{4} \div \frac{2}{5}
$$

1. Keep the first fraction: $\dfrac{3}{4}$.
2. Change division to multiplication.
3. Flip the second fraction (reciprocal of $\dfrac{2}{5}$ is $\dfrac{5}{2}$).

So:
$$
\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}
$$

You can then simplify or convert to a mixed number if desired.

Dividing a fraction by a whole number

Again, write the whole number as a fraction and then use the rule.

Example:
$$
\frac{5}{6} \div 2
$$

1. Write $2$ as $\dfrac{2}{1}$:
   $$
   \frac{5}{6} \div \frac{2}{1}
   $$
2. Multiply by the reciprocal of $\dfrac{2}{1}$, which is $\dfrac{1}{2}$:
   $$
   \frac{5}{6} \times \frac{1}{2} = \frac{5}{12}
   $$

So dividing by $2$ halves the fraction.

Dividing a whole number by a fraction

Example:
$$
4 \div \frac{1}{3}
$$

1. Write $4$ as $\dfrac{4}{1}$:
   $$
   \frac{4}{1} \div \frac{1}{3}
   $$
2. Multiply by the reciprocal of $\dfrac{1}{3}$, which is $\dfrac{3}{1}$:
   $$
   \frac{4}{1} \times \frac{3}{1} = \frac{12}{1} = 12
   $$

This matches the idea “How many thirds are in 4?” There are $12$ thirds in $4$.

Dividing mixed numbers

As with multiplication, convert mixed numbers to improper fractions first.

Example:
$$
3\frac{1}{2} \div 1\frac{3}{4}
$$

1. Convert:
   $$
   3\frac{1}{2} = \frac{7}{2}, \quad 1\frac{3}{4} = \frac{7}{4}
   $$
2. Divide:
   $$
   \frac{7}{2} \div \frac{7}{4} = \frac{7}{2} \times \frac{4}{7}
   $$
3. Simplify before multiplying:
• $7$ in the numerator and $7$ in the denominator cancel to $1$.
• Result becomes:
  $$
  \frac{1}{2} \times 4 = \frac{4}{2} = 2
  $$

Order of Operations with Fractions

Fractions can appear in longer expressions that also include whole numbers, parentheses, and other operations.

When that happens, you still follow the usual order of operations:

1. Parentheses.
2. Multiplication and division (from left to right).
3. Addition and subtraction (from left to right).

Example:
$$
\frac{1}{2} + \frac{3}{4} \times \frac{2}{5}
$$

1. Handle multiplication first:
   $$
   \frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}
   $$
2. Now add:
   $$
   \frac{1}{2} + \frac{3}{10}
   $$
• LCD of $2$ and $10$ is $10$.
• $\dfrac{1}{2} = \dfrac{5}{10}$.
• Add:
  $$
  \frac{5}{10} + \frac{3}{10} = \frac{8}{10} = \frac{4}{5}
  $$

Using the same order-of-operations rules keeps fraction calculations consistent with other arithmetic.

Common Pitfalls and How to Avoid Them

When doing operations with fractions, some mistakes show up often. Being aware of them helps you avoid them.

• Adding or subtracting denominators:
• Wrong: $\dfrac{1}{3} + \dfrac{1}{4} = \dfrac{2}{7}$.
• Denominators represent the size of each part; you must first make that size the same.
• Forgetting to change to a common denominator for addition or subtraction:
• Always check whether denominators match before adding or subtracting.
• Dividing instead of multiplying when multiplying fractions:
• For multiplication, do not combine denominators or numerators by adding or subtracting; use product top–top and bottom–bottom.
• Flipping the wrong fraction in division:
• In division $\dfrac{a}{b} \div \dfrac{c}{d}$, only flip the second fraction.
• Ignoring the order of operations:
• In expressions mixing fractions and other operations, multiplication and division still come before addition and subtraction.

Keeping these rules in mind will make operations with fractions consistent and reliable.