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Adding & Subtracting
Fractions

Master how to add and subtract fractions with unlike denominators by finding equivalent fractions with a common denominator.

Pre-Algebra

What Is a Fraction?

A fraction represents a part of a whole. It has two parts: the numerator (top number) tells how many parts you have, and the denominator (bottom number) tells how many equal parts the whole is divided into.

\(\frac{3}{4}\)
Numerator (parts you have) Denominator (total parts) 1 2 3 4 3 out of 4 parts are shaded

Like vs. Unlike Denominators

Before adding or subtracting fractions, you need to know whether the denominators are the same or different. This distinction determines your approach.

Like Denominators
Both fractions have the same bottom number. You can add or subtract the numerators directly: \(\frac{2}{7} + \frac{3}{7}\)
Unlike Denominators
The fractions have different bottom numbers. You must find a common denominator first: \(\frac{1}{3} + \frac{1}{4}\)

Why Can't We Just Add Across?

When denominators differ, the pieces are different sizes. Adding numerators directly would be like adding apples and oranges: the numbers don't represent the same unit.

\(\frac{1}{3}\)
\(\frac{1}{4}\)
✕ Wrong: \(\frac{1}{3} + \frac{1}{4} \neq \frac{2}{7}\)
The pieces are different sizes: you cannot just add 1+1 and 3+4
✓ Correct: \(\frac{1}{3} + \frac{1}{4} = \frac{7}{12}\)

The Power of Equivalent Fractions

An equivalent fraction is a fraction that represents the same value but uses different numbers. We create one by multiplying or dividing both the numerator and denominator by the same nonzero number.

\(\frac{1}{3}\)
=
\(\frac{2}{6}\)
=
\(\frac{4}{12}\)
\(\times 2\)
\(\times 2\)
\(\times 2\)
\(\times 2\)
Multiply top and bottom by the same number: the value stays the same!

The Core Rule for Unlike Denominators

\(\frac{a}{b} \pm \frac{c}{d} = \frac{a \cdot \frac{\text{LCD}}{b} \pm c \cdot \frac{\text{LCD}}{d}}{\text{LCD}}\)

LCD = Least Common Multiple of the two denominators.

Finding the LCD

The Least Common Denominator is the smallest number that both denominators divide into evenly. Two methods to find it:

List Multiples
Write out multiples of each denominator until you find the smallest match.

Multiples of 3: 3, 6, 9, 12, ...
Multiples of 4: 4, 8, 12, ...
LCD = 12
Prime Factorization
Factor each denominator into primes. Take the highest power of each prime.

\(6 = 2 \times 3\), \(8 = 2^3\)
LCD = \(2^3 \times 3 = \) 24

Same Denominator vs. Different Denominator

Like Denominators Unlike Denominators
Example \(\frac{2}{5} + \frac{1}{5}\) \(\frac{1}{3} + \frac{1}{4}\)
Find LCD? No: already the same Yes: required first step
Steps Add numerators, keep denominator Find LCD → rewrite → then add
Result \(\frac{3}{5}\) \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\)

How to Apply the Process

1
Find the LCD of the two denominators (list multiples or use prime factorization).
2
Rewrite each fraction as an equivalent fraction with the LCD as the new denominator.
3
Add or subtract the numerators. Keep the LCD as the denominator.
4
Simplify the result if possible (divide numerator and denominator by their GCF).

Example 1: \(\frac{1}{3} + \frac{1}{4}\)

STEP 1: LCD LCD of 3 and 4 = 12 STEP 2: REWRITE
\(\frac{1}{3}\)
\(\times \frac{4}{4}\)
=
\(\frac{4}{12}\)
\(\frac{1}{4}\)
\(\times \frac{3}{3}\)
=
\(\frac{3}{12}\)
STEP 3: ADD NUMERATORS
\(\frac{4}{12} + \frac{3}{12} = \)
\(\frac{7}{12}\)
STEP 4: 7 and 12 share no common factors → already simplified ✓

Example 2: \(\frac{5}{6} - \frac{1}{4}\)

1
Find LCD: multiples of 6: 6, 12; multiples of 4: 4, 8, 12. LCD = 12.
2
Rewrite: \(\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \) \(\frac{10}{12}\) and \(\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \) \(\frac{3}{12}\).
3
Subtract: \(\frac{10}{12} - \frac{3}{12} = \frac{10 - 3}{12} = \frac{7}{12}\)
4
Simplify: GCF of 7 and 12 is 1, so \(\frac{7}{12}\) is already in simplest form.

Example 3: \(\frac{3}{8} + \frac{1}{6}\)

1
Find LCD: \(8 = 2^3\), \(6 = 2 \times 3\). LCD = \(2^3 \times 3 =\) 24.
2
Rewrite: \(\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}\) and \(\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}\).
3
Add: \(\frac{9}{24} + \frac{4}{24} = \frac{13}{24}\).
4
Simplify: 13 is prime and doesn't divide 24, so \(\frac{13}{24}\) is in simplest form.

Example 4: \(\frac{3}{10} + \frac{1}{6}\) (Simplify!)

1
LCD of 10 and 6: \(10 = 2 \times 5\), \(6 = 2 \times 3\). LCD = \(2 \times 3 \times 5 = \) 30.
2
Rewrite: \(\frac{3}{10} = \frac{9}{30}\) and \(\frac{1}{6} = \frac{5}{30}\).
3
Add: \(\frac{9}{30} + \frac{5}{30} = \frac{14}{30}\).
4
Simplify: GCF of 14 and 30 is 2. \(\frac{14 \div 2}{30 \div 2} = \) \(\frac{7}{15}\).

Seeing It Visually

Let's visualize \(\frac{1}{3} + \frac{1}{4} = \frac{7}{12}\) with fraction bars.

Whole (12 twelfths)
\(\frac{1}{3} = \frac{4}{12}\)
\(\frac{1}{4} = \frac{3}{12}\)
Sum = \(\frac{7}{12}\)

Common Mistakes to Avoid

✕ Adding Denominators
Writing \(\frac{1}{3} + \frac{1}{4} = \frac{2}{7}\). You never add or subtract the denominators: they describe the size of the pieces.
✕ Forgetting to Multiply the Numerator
Changing the denominator but leaving the numerator unchanged: \(\frac{1}{3} \neq \frac{1}{12}\). Both top and bottom must be multiplied.
✕ Not Simplifying
Leaving \(\frac{4}{12}\) as the final answer instead of simplifying to \(\frac{1}{3}\). Always check for common factors.

Summary Checklist

Step What to Do Example (\(\frac{2}{5} + \frac{1}{3}\))
1. Find LCD LCM of denominators LCM(5, 3) = 15
2. Rewrite Equivalent fractions \(\frac{6}{15} + \frac{5}{15}\)
3. Combine Add/subtract numerators \(\frac{6+5}{15} = \frac{11}{15}\)
4. Simplify Divide by GCF GCF = 1 → \(\frac{11}{15}\)

Question 1 of 5

What is \(\frac{1}{4} + \frac{1}{6}\)?

Question 2 of 5

What is \(\frac{5}{8} - \frac{1}{3}\)?

Question 3 of 5

What is the LCD of \(\frac{2}{9}\) and \(\frac{5}{12}\)?

Question 4 of 5

Simplify: \(\frac{3}{10} + \frac{2}{5}\)

Question 5 of 5

A recipe calls for \(\frac{2}{3}\) cup of flour and \(\frac{1}{4}\) cup of sugar. How much more flour is needed than sugar?