Adding Fractions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Adding Fractions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Adding fractions combines parts of a whole by rewriting both with a common denominator and then adding the numerators.

You can only add like-sized pieces directly — $\frac{1}{4}$ and $\frac{1}{3}$ must be converted to twelfths before adding.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Adding fractions combines parts of a whole, renaming to a common denominator when the pieces differ.

Common stuck point: The procedure for adding fractions is the easy part; the trap is adding numerators and denominators straight across. Asking "Am I combining two fractions into one sum, matching denominators first if they differ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I combining two fractions into one sum, matching denominators first if they differ?

Worked Examples

Example 1

easy

Add

\frac{1}{4} + \frac{3}{4}

Answer

1

First step

Check that the denominators are the same: both fractions have denominator

4

Full solution

2
Add the numerators: $1 + 3 = 4$ , giving $\frac{4}{4}$ .
3
Simplify: $\frac{4}{4} = 1$ (the two fractions together make a whole).

When two fractions with the same denominator sum to a value where numerator equals denominator, the result is exactly 1 whole. Recognising this shortcut avoids unnecessary simplification steps.

Example 2

medium

Add

\frac{2}{3} + \frac{3}{8} + \frac{1}{4}

Example 3

easy

Worked example: add

\frac{3}{10}+\frac{1}{5}

Example 4

medium

Worked example: add

\frac{7}{10}+\frac{2}{15}

Example 5

medium

Worked example:

\frac{7}{12}+\frac{5}{18}

Example 6

hard

Worked example:

\frac{11}{15}+\frac{7}{12}

, in lowest terms.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Add

\frac{5}{12} + \frac{3}{12}

Example 2

hard

Three tanks contain

\frac{5}{8}

\frac{3}{4}

, and

\frac{1}{3}

of their capacity in water. If each tank has a capacity of

120

\frac{3}{4}

cup flour and

\frac{2}{3}

cup sugar. How much total dry ingredient by volume?

Example 18

medium

Compute

\frac{3}{10} + \frac{2}{15}

Example 19

medium

Compute

\frac{5}{12} + \frac{7}{18}

Example 20

challenge

Compute

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}

Example 21

challenge

Find two distinct positive unit fractions (

\frac{1}{a}

and

\frac{1}{b}

, with

a \neq b

) whose sum equals

\frac{1}{2}

Example 22

challenge

Compute

\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5}

Example 23

easy

Compute

\frac{2}{9}+\frac{4}{9}

Example 24

easy

Compute

\frac{1}{6}+\frac{1}{3}

Example 25

easy

Compute

\frac{2}{6}+\frac{1}{6}

\frac{3}{4}

m wide; an extension is

\frac{5}{8}

m wide. Total width?

Example 32

medium

Compute

\frac{1}{2}+\frac{1}{4}+\frac{1}{6}

Example 33

medium

Compute

\frac{5}{9}+\frac{1}{6}

Example 34

hard

Solve for

x

x+\frac{2}{7}=\frac{3}{4}

Example 35

hard

Three pipes fill a pool: pipe A adds

\frac{1}{3}

, pipe B adds

\frac{1}{4}

, pipe C adds

\frac{1}{6}

of the pool. How much of the pool is unfilled?

Example 36

hard

A baker uses

\frac{2}{3}

cup of milk for cookies and

\frac{3}{8}

cup for muffins. If a measuring cup holds

1

cup, how much more milk fits before overflow?

Example 37

hard

Compute

2\frac{3}{4}+1\frac{5}{6}+\frac{2}{3}

Example 38

hard

Find the LCD and compute

\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}

Example 39

challenge

Compute

\sum_{k=1}^{n}\frac{1}{k(k+1)}

in closed form.

Example 40

challenge

Find all pairs of distinct positive unit fractions

\frac{1}{a},\frac{1}{b}

with

a<b

whose sum equals

\frac{2}{3}

← Back to Adding Fractions Formula Explained Practice Mode

Related Concepts

Fractions Equivalent Fractions Least Common Multiple

Background Knowledge

These ideas may be useful before you work through the harder examples.

fractionsequivalent fractionsleast common multiple