Adding fractions

Here you’ll learn about how to add fractions with like denominators and with unlike denominators. You will also learn how to add mixed numbers.

Students will first learn about adding fractions as part of number and operations in fractions in elementary school.

What is adding fractions?

Adding fractions is when you combine two or more fractions together to find the total.

To do this, fractions need a common denominator (bottom number). Then you can add the fractions by adding the numerators (top numbers).

For example,

Since the denominators are the same, you need to add the numerators 3+7=10.

There are 10 parts. But what size are the parts? They are still eighths, so the denominator stays the same and the answer is 10 eighths, or 1 and 2 eighths

You can also write this answer as the equivalent fraction 1\cfrac{1}{4}.

If the denominators are not the same, you create equivalent fractions before adding.

For example,

Since the denominators are NOT the same, the parts are NOT the same size. Use equivalent fractions to create a common denominator of 9. Multiply the numerator and denominator of \cfrac{2}{3} by 3.

You add to see how many parts there are in total: 6+5=11.

There are 11 parts. But what size are the parts? They are still ninths, so the denominator stays the same and the answer is 11 ninths, or 1 and 2 ninths

What is adding fractions?

Common Core State Standards

How does this relate to 4th grade math and 5th grade math?

• Grade 4 – Number and Operations – Fractions (4.NF.B.3c)
  Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
  
• Grade 5 – Number and Operations – Fractions (5.NF.A.1)
  Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
  For example, \cfrac{2}{3}+\cfrac{5}{4}=\cfrac{8}{12}+\cfrac{15}{12}=\cfrac{23}{12}. (In general, \cfrac{a}{b}+\cfrac{c}{d}=\cfrac{(ad+bc)}{bd}. )

How to add fractions

In order to add fractions with like denominators:

1. Add the numerators (top numbers).
2. Write your answer as a fraction.

In order to add mixed numbers with like denominators:

1. Add the whole numbers.
2. Add the fractions.
3. Write your answer as a mixed number.

In order to add fractions with unlike denominators:

1. Create common denominators (bottom numbers).
2. Add the numerators (top numbers).
3. Write your answer as a fraction.

In order to add mixed numbers with unlike denominators:

1. Add the whole numbers.
2. Create common denominators (bottom numbers).
3. Add the fractions.
4. Write your answer as a mixed number.

Adding fractions examples

Example 1: adding fractions with like denominators

Solve \cfrac{3}{5}+\cfrac{4}{5}.

1. Add the numerators (top numbers).

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 3+4=7.

2Write your answer as a fraction.

There are 7 parts. But what size are the parts? They are still fifths, so the denominator stays the same.

\cfrac{3}{5}+\cfrac{4}{5}=\cfrac{7}{5} \ or \ 1 \cfrac{2}{5}

Example 2: adding mixed numbers with like denominators

Solve 4\cfrac{5}{6}+1\cfrac{2}{6}.

Add the whole numbers.

Add the fractions.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 5+2=7.

There are 7 parts. But what size are the parts? They are still sixths, so the denominator stays the same.

\cfrac{5}{6}+\cfrac{2}{6}=\cfrac{7}{6} \ or \ 1 \cfrac{1}{6}

Write your answer as a mixed number.

Add the whole numbers and fraction together.

Example 3: adding mixed numbers with like denominators

Solve 3 \cfrac{7}{10}+3 \cfrac{5}{10}.

Add the whole numbers.

Add the fractions.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 5+7=12.

There are 12 parts. But what size are the parts? They are still tenths, so the denominator stays the same.

\cfrac{7}{10}+\cfrac{5}{10}=\cfrac{12}{10} \ or \ 1 \cfrac{2}{10}

Write your answer as a mixed number.

Add the whole numbers and fraction together.

You can also write this answer as the equivalent fraction 7 \cfrac{1}{5}.

Example 4: adding fractions with unlike denominators

Solve \cfrac{4}{10}+\cfrac{1}{3}.

Create common denominators (bottom numbers).

Since \cfrac{4}{10} and \cfrac{1}{3} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator. Multiply each fraction by the opposite denominator.

Add the numerators (top numbers).

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 12+10=22.

Write your answer as a fraction.

There are 22 parts. But what size are the parts? They are still thirtieths, so the denominator stays the same.

You can also write this answer as the equivalent fraction \cfrac{11}{15}.

Example 5: adding mixed numbers with unlike denominators

Solve 2 \cfrac{3}{8}+1 \cfrac{7}{12}.

Add the whole numbers.

Create common denominators (bottom numbers).

Since \cfrac{3}{8} and \cfrac{7}{12} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator.

A common denominator of 24 can be used. Multiply the numerator and denominator of each fraction by the factor that will create a denominator of 24.

\cfrac{3}{8}=\cfrac{3 \times 3}{8 \times 3}=\cfrac{9}{24} \ and \ \cfrac{7}{12}=\cfrac{7 \times 2}{12 \times 2}=\cfrac{14}{24}

Add the fractions.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 9+14=23.

There are 23 parts. But what size are the parts? They are still twenty-fourths, so the denominator stays the same.

\cfrac{9}{24}+\cfrac{14}{24}=\cfrac{23}{24}

Write your answer as a mixed number.

Add the whole number and fraction together.

Example 6: adding mixed numbers with unlike denominators

Solve 4 \cfrac{2}{3}+5 \cfrac{5}{6}.

Add the whole numbers.

Create common denominators (bottom numbers).

Since \cfrac{2}{3} and \cfrac{5}{6} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator.

A common denominator of 6 can be used. Multiply the numerator and denominator of \cfrac{2}{3} by the factor that will create a denominator of 6.

\cfrac{2}{3}=\cfrac{2 \times 2}{3 \times 2}=\cfrac{4}{6} \ and \ \cfrac{5}{6}

Add the fractions.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 4+5=9.

There are 9 parts. But what size are the parts? They are still sixths, so the denominator stays the same.

\cfrac{4}{6}+\cfrac{5}{6}=\cfrac{9}{6} or 1 \cfrac{3}{6}

Write your answer as a mixed number.

Add the whole numbers and fraction together.

You can also write this answer as the equivalent mixed number 10 \cfrac{1}{2}.

Teaching tips for adding fractions

• Fraction work in 3rd grade centers around understanding through models, particularly area models and number lines. To build on this in 4th grade and 5th grade, always have physical or digital models available for students to use when necessary.

• To introduce this topic, give students a fraction addition problem and give them time to solve in a way that makes sense to them. Then go over the different solving strategies as a whole group.

• In the beginning, you may want to stick to smaller fractions or problems that do not involve regrouping. However, at some point it is important to transition to showing students all kinds of fraction addition problems mixed together. This gives them an opportunity to identify which solving strategy to use.

• Fraction worksheets may have their place when students are still developing understanding around addition, but once students have a successful strategy and can flexibly operate, incorporate math games or real world projects that involve addition of fractions.

Our favorite mistakes

• Adding without like denominators
  When fractions are added together, they must have like denominators. Before adding fractions with unlike denominators, use equivalent fractions to create a common denominator.

• When adding fractions, only the numerators are added
  The denominator tells us how big the parts are. You add the number of parts (numerators) and keep the denominator (the size of the parts) the same.

Related fractions operations lessons

• Fractions operations
• Multiplying and dividing fractions
• Subtracting fractions
• Adding and subtracting fractions
• Multiplying fractions
• Dividing fractions
• Reciprocal math
• Fraction word problems
• Multiplicative inverse
• Interpret a fraction as division

The next lessons are

• Decimals
• Algebraic expression
• Exponents
• Percent
• Converting fractions, decimals, and percentages

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