How to Add Fractions with Unlike Denominators

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After working through this page, you will be able to add any two fractions that have different denominators. You will know how to find the Least Common Denominator, convert each fraction, add the numerators, and simplify the result. These steps work every time, whether the denominators are small numbers or larger values.

Theory

Why the denominators must match

A fraction's denominator tells you the size of each part. When you look at $\frac{1}{3}$, each part is one-third of the whole. When you look at $\frac{1}{4}$, each part is one-fourth of the whole. Because thirds and fourths are different-sized pieces, you cannot combine them directly. You first need to rewrite both fractions so they describe the same-sized pieces -- that means giving them a common denominator.

Finding the Least Common Denominator (LCD)

The Least Common Denominator is the smallest number that both denominators divide into evenly. It is the same as the Least Common Multiple (LCM) of the two denominators.

There are two reliable methods to find the LCD:

Method 1 -- List multiples:

Write out the multiples of each denominator until you find the first number that appears in both lists.

For denominators 6 and 8:

• Multiples of 6: 6, 12, 18, 24, 30, ...
• Multiples of 8: 8, 16, 24, 32, ...

The LCD is 24.

Method 2 -- Prime factorization:

Break each denominator into prime factors. Take the highest power of every prime that appears, then multiply them together.

$$\text{LCD} = \text{LCM}(b, d)$$

For denominators 12 and 18:

• $12 = 2^2 \times 3$
• $18 = 2 \times 3^2$
• LCD $= 2^2 \times 3^2 = 4 \times 9 = 36$

Method 2 is especially helpful when the denominators are larger, because listing multiples becomes tedious.

The four-step process

Once you understand the reasoning, every problem follows the same four steps:

$$\frac{a}{b} + \frac{c}{d} \;=\; \frac{a \times \frac{\text{LCD}}{b}}{{\text{LCD}}} + \frac{c \times \frac{\text{LCD}}{d}}{{\text{LCD}}} \;=\; \frac{a \times \frac{\text{LCD}}{b} \;+\; c \times \frac{\text{LCD}}{d}}{\text{LCD}}$$

In plain language:

1. Find the LCD of the two denominators.
2. Convert each fraction to an equivalent fraction with the LCD as the new denominator. Multiply each numerator by whatever factor you multiplied its denominator by.
3. Add the numerators. Keep the LCD as the denominator.
4. Simplify the result by dividing the numerator and denominator by their Greatest Common Divisor (GCD). Convert to a mixed number if the numerator is larger than the denominator.

When one denominator is a multiple of the other

A useful shortcut: if one denominator divides evenly into the other, the larger denominator is already the LCD. You only need to convert the fraction with the smaller denominator.

For example, with denominators 4 and 12, the LCD is 12 because $12 \div 4 = 3$ with no remainder. You only convert the fraction with denominator 4.

Worked Examples

Example 1: Adding fractions with small coprime denominators (easy)

Problem: Calculate $\dfrac{1}{3} + \dfrac{1}{5}$.

Step 1: Find the LCD of 3 and 5. Since 3 and 5 share no common factors (they are coprime), the LCD is their product: $\text{LCD} = 3 \times 5 = 15$

Step 2: Convert each fraction. $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$ $\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}$

Step 3: Add the numerators. $\frac{5}{15} + \frac{3}{15} = \frac{5 + 3}{15} = \frac{8}{15}$

Step 4: Simplify. The GCD of 8 and 15 is 1, so the fraction is already in lowest terms.

Answer: $\dfrac{8}{15}$

Example 2: One denominator is a multiple of the other (medium)

Problem: Calculate $\dfrac{5}{6} + \dfrac{3}{4}$.

Step 1: Find the LCD of 6 and 4.

• Multiples of 6: 6, 12, 18, ...
• Multiples of 4: 4, 8, 12, 16, ...

The LCD is 12.

Step 2: Convert each fraction. $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$ $\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

Step 3: Add the numerators. $\frac{10}{12} + \frac{9}{12} = \frac{19}{12}$

Step 4: Simplify and convert. The GCD of 19 and 12 is 1, so the fraction cannot be reduced. Since $19 > 12$, convert to a mixed number: $\frac{19}{12} = 1\frac{7}{12}$

Answer: $\dfrac{19}{12}$ or $1\dfrac{7}{12}$

Example 3: Larger denominators requiring simplification (medium)

Problem: Calculate $\dfrac{7}{10} + \dfrac{4}{15}$.

Step 1: Find the LCD of 10 and 15. Using prime factorization:

• $10 = 2 \times 5$
• $15 = 3 \times 5$
• LCD $= 2 \times 3 \times 5 = 30$

Step 2: Convert each fraction. $\frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30}$ $\frac{4}{15} = \frac{4 \times 2}{15 \times 2} = \frac{8}{30}$

Step 3: Add the numerators. $\frac{21}{30} + \frac{8}{30} = \frac{29}{30}$

Step 4: Simplify. The GCD of 29 and 30 is 1 (29 is prime), so the fraction is already in lowest terms.

Answer: $\dfrac{29}{30}$

Example 4: Adding mixed numbers with unlike denominators (challenging)

Problem: Calculate $2\dfrac{3}{8} + 1\dfrac{5}{6}$.

Step 1: Convert each mixed number to an improper fraction. $2\frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{19}{8}$ $1\frac{5}{6} = \frac{1 \times 6 + 5}{6} = \frac{11}{6}$

Step 2: Find the LCD of 8 and 6.

• $8 = 2^3$
• $6 = 2 \times 3$
• LCD $= 2^3 \times 3 = 24$

Step 3: Convert each fraction. $\frac{19}{8} = \frac{19 \times 3}{8 \times 3} = \frac{57}{24}$ $\frac{11}{6} = \frac{11 \times 4}{6 \times 4} = \frac{44}{24}$

Step 4: Add the numerators. $\frac{57}{24} + \frac{44}{24} = \frac{101}{24}$

Step 5: Simplify and convert to a mixed number. The GCD of 101 and 24 is 1 (101 is prime), so the fraction cannot be reduced. $\frac{101}{24} = 4\frac{5}{24}$

Answer: $4\dfrac{5}{24}$

Example 5: Adding three fractions with different denominators (challenging)

Problem: Calculate $\dfrac{2}{3} + \dfrac{1}{4} + \dfrac{5}{6}$.

Step 1: Find the LCD of 3, 4, and 6.

• $3 = 3$
• $4 = 2^2$
• $6 = 2 \times 3$
• LCD $= 2^2 \times 3 = 12$

Step 2: Convert each fraction. $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$ $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$ $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

Step 3: Add all the numerators. $\frac{8}{12} + \frac{3}{12} + \frac{10}{12} = \frac{8 + 3 + 10}{12} = \frac{21}{12}$

Step 4: Simplify. The GCD of 21 and 12 is 3: $\frac{21}{12} = \frac{21 \div 3}{12 \div 3} = \frac{7}{4}$

Convert to a mixed number: $\frac{7}{4} = 1\frac{3}{4}$

Answer: $\dfrac{7}{4}$ or $1\dfrac{3}{4}$

Common Mistakes

Mistake 1: Adding numerators and denominators separately

❌ $\frac{1}{3} + \frac{1}{4} = \frac{2}{7}$

✅ $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Why this matters: This is the most common error students make with fractions. A fraction is a single number, not two separate numbers. Adding the tops and bottoms separately changes the meaning entirely and produces an incorrect value.

Mistake 2: Finding a common denominator but forgetting to adjust the numerators

❌ $\frac{2}{5} + \frac{1}{3} = \frac{2}{15} + \frac{1}{15} = \frac{3}{15}$

✅ $\frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}$

Why this matters: When you multiply the denominator by a number, you must multiply the numerator by the same number to create an equivalent fraction. Changing only the denominator changes the value of the fraction.

Mistake 3: Using any common multiple instead of the LCD

❌ Using 60 as the common denominator for $\frac{1}{4} + \frac{1}{6}$ instead of 12.

✅ The LCD of 4 and 6 is 12, not 60.

Why this matters: While using a larger common multiple (like the product of the denominators) will still give a correct answer, it forces you to work with bigger numbers. This increases the chance of arithmetic errors and often requires extra simplification at the end. The LCD keeps numbers as manageable as possible.

Practice Problems

Try these on your own before checking the answers:

1. $\dfrac{2}{7} + \dfrac{1}{3}$
2. $\dfrac{3}{8} + \dfrac{5}{12}$
3. $\dfrac{4}{9} + \dfrac{7}{15}$
4. $1\dfrac{1}{4} + 2\dfrac{2}{3}$
5. $\dfrac{3}{5} + \dfrac{1}{4} + \dfrac{2}{3}$

Click to see answers

1. LCD is 21. $\frac{6}{21} + \frac{7}{21} = \frac{13}{21}$
2. LCD is 24. $\frac{9}{24} + \frac{10}{24} = \frac{19}{24}$
3. LCD is 45. $\frac{20}{45} + \frac{21}{45} = \frac{41}{45}$
4. Convert to improper fractions: $\frac{5}{4} + \frac{8}{3}$. LCD is 12. $\frac{15}{12} + \frac{32}{12} = \frac{47}{12} = 3\frac{11}{12}$
5. LCD is 60. $\frac{36}{60} + \frac{15}{60} + \frac{40}{60} = \frac{91}{60} = 1\frac{31}{60}$

Summary

• You can only add fractions when they share the same denominator, because the denominator defines the size of each part.
• The Least Common Denominator (LCD) is the smallest number both denominators divide into evenly. Find it by listing multiples or using prime factorization.
• After finding the LCD, convert each fraction by multiplying both its numerator and denominator by the same factor.
• Add the numerators and keep the LCD as the denominator, then simplify by dividing by the GCD.
• For mixed numbers, convert to improper fractions first, then follow the same four steps.

Related Topics

• Fractions -- Complete Guide
• How to Simplify Fractions to Lowest Terms
• How to Multiply and Divide Fractions Step by Step

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