maths.free › Arithmetic › 4. Fractions › Add and Subtract Fractions with Common Denominators

Add and Subtract Fractions with Common Denominators

Model fraction addition

Model Fraction Addition

How many quarters are pictured? One quarter plus \(2\) quarters equals \(3\) quarters.

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

\[\begin{array}{lllll}\frac{1}{4} & & \frac{2}{4} & & \frac{3}{4} \\ \text{one quarter} & + & \text{two quarters} & = & \text{three quarters}\end{array}\]

Let’s use fraction circles to model the same example, \(\frac{1}{4}+\frac{2}{4}.\)

Start with one \(\frac{1}{4}\) piece.
Add two more \(\frac{1}{4}\)pieces.
The result is \(\frac{3}{4}\).

So again, we see that

\[\frac{1}{4}+\frac{2}{4}=\frac{3}{4}\]

Example

Try it.

Use a model to find the sum \(\frac{3}{8}+\frac{2}{8}.\)

Solution

Start with three \(\frac{1}{8}\) pieces.
Add two \(\frac{1}{8}\)pieces.
How many \(\frac{1}{8}\)pieces are there?

There are five \(\frac{1}{8}\) pieces, or five-eighths. The model shows that \(\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.\)

Add Fractions with a Common Denominator

shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.

Example

Try it.

Find the sum: \(\frac{3}{5}+\frac{1}{5}.\)

Solution

	\(\frac{3}{5}+\frac{1}{5}\)
Add the numerators and place the sum over the common denominator.	\(\frac{3+1}{5}\)
Simplify.	\(\frac{4}{5}\)

Example

Try it.

Find the sum: \(\frac{x}{3}+\frac{2}{3}.\)

Solution

	\(\frac{x}{3}+\frac{2}{3}\)
Add the numerators and place the sum over the common denominator.	\(\frac{x+2}{3}\)

Note that we cannot simplify this fraction any more. Since \(x\) and \(2\) are not like terms, we cannot combine them.

Example

Try it.

Find the sum: \(-\frac{9}{d}+\frac{3}{d}.\)

Solution

We will begin by rewriting the first fraction with the negative sign in the numerator.

\(-\frac{a}{b}=\frac{-a}{b}\)

	\(-\frac{9}{d}+\frac{3}{d}\)
Rewrite the first fraction with the negative in the numerator.	\(\frac{-9}{d}+\frac{3}{d}\)
Add the numerators and place the sum over the common denominator.	\(\frac{-9+3}{d}\)
Simplify the numerator.	\(\frac{-6}{d}\)
Rewrite with negative sign in front of the fraction.	\(-\frac{6}{d}\)

Example

Try it.

Find the sum: \(\frac{2n}{11}+\frac{5n}{11}.\)

Solution

	\(\frac{2n}{11}+\frac{5n}{11}\)
Add the numerators and place the sum over the common denominator.	\(\frac{2n+5n}{11}\)
Combine like terms.	\(\frac{7n}{11}\)

Example

Try it.

Find the sum: \(-\frac{3}{12}+(-\frac{5}{12}).\)

Solution

	\(-\frac{3}{12}+(-\frac{5}{12})\)
Add the numerators and place the sum over the common denominator.	\(\frac{-3+(-5)}{12}\)
Add.	\(\frac{-8}{12}\)
Simplify the fraction.	\(-\frac{2}{3}\)

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into \(12\) slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or \(\frac{7}{12}\) of the pizza) left in the box. If Leonardo eats \(2\) of these remaining pieces (or \(\frac{2}{12}\) of the pizza), how much is left? There would be \(5\) pieces left (or \(\frac{5}{12}\) of the pizza).

\[\frac{7}{12}-\frac{2}{12}=\frac{5}{12}\]

Let’s use fraction circles to model the same example, \(\frac{7}{12}-\frac{2}{12}.\)

Start with seven \(\frac{1}{12}\) pieces. Take away two \(\frac{1}{12}\) pieces. How many twelfths are left?

Again, we have five twelfths, \(\frac{5}{12}.\)

Example

Try it.

Use fraction circles to find the difference: \(\frac{4}{5}-\frac{1}{5}.\)

Solution

Start with four \(\frac{1}{5}\) pieces. Take away one \(\frac{1}{5}\) piece. Count how many fifths are left. There are three \(\frac{1}{5}\) pieces left.

Subtract Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

Example

Try it.

Find the difference: \(\frac{23}{24}-\frac{14}{24}.\)

Solution

	\(\frac{23}{24}-\frac{14}{24}\)
Subtract the numerators and place the difference over the common denominator.	\(\frac{23-14}{24}\)
Simplify the numerator.	\(\frac{9}{24}\)
Simplify the fraction by removing common factors.	\(\frac{3}{8}\)

Example

Try it.

Find the difference: \(\frac{y}{6}-\frac{1}{6}.\)

Solution

	\(\frac{y}{6}-\frac{1}{6}\)
Subtract the numerators and place the difference over the common denominator.	\(\frac{y-1}{6}\)

The fraction is simplified because we cannot combine the terms in the numerator.

Example

Try it.

Find the difference: \(-\frac{10}{x}-\frac{4}{x}.\)

Solution

Remember, the fraction \(-\frac{10}{x}\) can be written as \(\frac{-10}{x}.\)

	\(-\frac{10}{x}-\frac{4}{x}\)
Subtract the numerators.	\(\frac{-10-4}{x}\)
Simplify.	\(\frac{-14}{x}\)
Rewrite with the negative sign in front of the fraction.	\(-\frac{14}{x}\)

Now lets do an example that involves both addition and subtraction.

Example

Try it.

Simplify: \(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}.\)

Solution

	\(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}\)
Combine the numerators over the common denominator.	\(\frac{3+(-5)-1}{8}\)
Simplify the numerator, working left to right.	\(\frac{-2-1}{8}\)
Subtract the terms in the numerator.	\(\frac{-3}{8}\)
Rewrite with the negative sign in front of the fraction.	\(-\frac{3}{8}\)

Key Concepts

Fraction Addition
- If \(a,b,\) and \(c\) are numbers where \(c\ne 0\), then \(\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\).
- To add fractions, add the numerators and place the sum over the common denominator.
Fraction Subtraction
- If \(a,b,\) and \(c\) are numbers where \(c\ne 0\), then \(\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}\).
- To subtract fractions, subtract the numerators and place the difference over the common denominator.

Add and Subtract Fractions with Common Denominators

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

Try it.

\(\frac{2}{5}+\frac{1}{5}\)

Try it.

\(\frac{3}{10}+\frac{4}{10}\)

Solution

\(\frac{7}{10}\)

Try it.

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

Try it.

\(\frac{3}{8}+\frac{3}{8}\)

Solution

\(\frac{3}{4}\)

Add Fractions with a Common Denominator

In the following exercises, find each sum.

Try it.

\(\frac{4}{9}+\frac{1}{9}\)

Try it.

\(\frac{2}{9}+\frac{5}{9}\)

Solution

\(\frac{7}{9}\)

Try it.

\(\frac{6}{13}+\frac{7}{13}\)

Try it.

\(\frac{9}{15}+\frac{7}{15}\)

Solution

\(\frac{16}{15}\)

Try it.

\(\frac{x}{4}+\frac{3}{4}\)

Try it.

\(\frac{y}{3}+\frac{2}{3}\)

Solution

\(\frac{y+2}{3}\)

Try it.

\(\frac{7}{p}+\frac{9}{p}\)

Try it.

\(\frac{8}{q}+\frac{6}{q}\)

Solution

\(\frac{14}{q}\)

Try it.

\(\frac{8b}{9}+\frac{3b}{9}\)

Try it.

\(\frac{5a}{7}+\frac{4a}{7}\)

Solution

\(\frac{9a}{7}\)

Try it.

\(\frac{-12y}{8}+\frac{3y}{8}\)

Try it.

\(\frac{-11x}{5}+\frac{7x}{5}\)

Solution

\(\frac{-4x}{5}\)

Try it.

\(-\frac{1}{8}+(-\frac{3}{8})\)

Try it.

\(-\frac{1}{8}+(-\frac{5}{8})\)

Solution

\(-\frac{3}{4}\)

Try it.

\(-\frac{3}{16}+(-\frac{7}{16})\)

Try it.

\(-\frac{5}{16}+(-\frac{9}{16})\)

Solution

\(-\frac{7}{8}\)

Try it.

\(-\frac{8}{17}+\frac{15}{17}\)

Try it.

\(-\frac{9}{19}+\frac{17}{19}\)

Solution

\(\frac{8}{19}\)

Try it.

\(\frac{6}{13}+(-\frac{10}{13})+(-\frac{12}{13})\)

Try it.

\(\frac{5}{12}+(-\frac{7}{12})+(-\frac{11}{12})\)

Solution

\(-\frac{13}{12}\)

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

Try it.

\(\frac{5}{8}-\frac{2}{8}\)

Try it.

\(\frac{5}{6}-\frac{2}{6}\)

Solution

\(\frac{1}{2}\)

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

Try it.

\(\frac{4}{5}-\frac{1}{5}\)

Try it.

\(\frac{4}{5}-\frac{3}{5}\)

Solution

\(\frac{1}{5}\)

Try it.

\(\frac{11}{15}-\frac{7}{15}\)

Try it.

\(\frac{9}{13}-\frac{4}{13}\)

Solution

\(\frac{5}{13}\)

Try it.

\(\frac{11}{12}-\frac{5}{12}\)

Try it.

\(\frac{7}{12}-\frac{5}{12}\)

Solution

\(\frac{1}{6}\)

Try it.

\(\frac{4}{21}-\frac{19}{21}\)

Try it.

\(-\frac{8}{9}-\frac{16}{9}\)

Solution

\(-\frac{8}{3}\)

Try it.

\(\frac{y}{17}-\frac{9}{17}\)

Try it.

\(\frac{x}{19}-\frac{8}{19}\)

Solution

\(\frac{x-8}{19}\)

Try it.

\(\frac{5y}{8}-\frac{7}{8}\)

Try it.

\(\frac{11z}{13}-\frac{8}{13}\)

Solution

\(\frac{11z-8}{13}\)

Try it.

\(-\frac{8}{d}-\frac{3}{d}\)

Try it.

\(-\frac{7}{c}-\frac{7}{c}\)

Solution

\(-\frac{14}{c}\)

Try it.

\(-\frac{23}{u}-\frac{15}{u}\)

Try it.

\(-\frac{29}{v}-\frac{26}{v}\)

Solution

\(-\frac{55}{v}\)

Try it.

\(\frac{6c}{7}-\frac{5c}{7}\)

Try it.

\(\frac{12d}{11}-\frac{9d}{11}\)

Solution

\(\frac{3d}{11}\)

Try it.

\(\frac{-4r}{13}-\frac{5r}{13}\)

Try it.

\(\frac{-7s}{3}-\frac{7s}{3}\)

Solution

\(-\frac{14s}{3}\)

Try it.

\(-\frac{3}{5}-(-\frac{4}{5})\)

Try it.

\(-\frac{3}{7}-(-\frac{5}{7})\)

Solution

\(\frac{2}{7}\)

Try it.

\(-\frac{7}{9}-(-\frac{5}{9})\)

Try it.

\(-\frac{8}{11}-(-\frac{5}{11})\)

Solution

\(-\frac{3}{11}\)

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

Try it.

\(-\frac{5}{18}\cdot \frac{9}{10}\)

Try it.

\(-\frac{3}{14}\cdot \frac{7}{12}\)

Solution

\(-\frac{1}{8}\)

Try it.

\(\frac{n}{5}-\frac{4}{5}\)

Try it.

\(\frac{6}{11}-\frac{s}{11}\)

Solution

\(\frac{6-s}{11}\)

Try it.

\(-\frac{7}{24}+\frac{2}{24}\)

Try it.

\(-\frac{5}{18}+\frac{1}{18}\)

Solution

\(-\frac{2}{9}\)

Try it.

\(\frac{8}{15}\div \frac{12}{5}\)

Try it.

\(\frac{7}{12}\div \frac{9}{28}\)

Solution

\(\frac{49}{27}\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: \(2x+9+3x-4.\)
If you missed this problem, review .

Giải đáp

\(5x+5\)
Draw a model of the fraction \(\frac{3}{4}.\)
If you missed this problem, review .

Giải đáp
Simplify: \(\frac{3+2}{6}.\)
If you missed this problem, review .

Giải đáp

\(\frac{5}{6}\)
Use a model to find the sum \(\frac{3}{8}+\frac{2}{8}.\)

Giải đáp

Start with three \(\frac{1}{8}\) pieces.
Add two \(\frac{1}{8}\)pieces.
How many \(\frac{1}{8}\)pieces are there?
There are five \(\frac{1}{8}\) pieces, or five-eighths. The model shows that \(\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.\)
Use a model to find each sum. Show a diagram to illustrate your model.
\(\frac{1}{8}+\frac{4}{8}\)

Giải đáp

\(\frac{5}{8}\)
Use a model to find each sum. Show a diagram to illustrate your model.
\(\frac{1}{6}+\frac{4}{6}\)

Giải đáp

\(\frac{5}{6}\)
Find the sum: \(\frac{3}{5}+\frac{1}{5}.\)

Giải đáp

\(\frac{3}{5}+\frac{1}{5}\)
Add the numerators and place the sum over the common denominator. \(\frac{3+1}{5}\)
Simplify. \(\frac{4}{5}\)
Find each sum: \(\frac{3}{6}+\frac{2}{6}.\)

Giải đáp

\(\frac{5}{6}\)
Find each sum: \(\frac{3}{10}+\frac{7}{10}.\)

Giải đáp

1
Find the sum: \(\frac{x}{3}+\frac{2}{3}.\)

Giải đáp

\(\frac{x}{3}+\frac{2}{3}\)
Add the numerators and place the sum over the common denominator. \(\frac{x+2}{3}\)
Note that we cannot simplify this fraction any more. Since \(x\) and \(2\) are not like terms, we cannot combine them.
Find the sum: \(\frac{x}{4}+\frac{3}{4}.\)

Giải đáp

\(\frac{x+3}{4}\)
Find the sum: \(\frac{y}{8}+\frac{5}{8}.\)

Giải đáp

\(\frac{y+5}{8}\)

Find the sum: \(-\frac{9}{d}+\frac{3}{d}.\)

Giải đáp

We will begin by rewriting the first fraction with the negative sign in the numerator.

\(-\frac{a}{b}=\frac{-a}{b}\)

	\(-\frac{9}{d}+\frac{3}{d}\)
Rewrite the first fraction with the negative in the numerator.	\(\frac{-9}{d}+\frac{3}{d}\)
Add the numerators and place the sum over the common denominator.	\(\frac{-9+3}{d}\)
Simplify the numerator.	\(\frac{-6}{d}\)
Rewrite with negative sign in front of the fraction.	\(-\frac{6}{d}\)

Find the sum: \(-\frac{7}{d}+\frac{8}{d}.\)

Giải đáp

\(\frac{1}{d}\)
Find the sum: \(-\frac{6}{m}+\frac{9}{m}.\)

Giải đáp

\(\frac{3}{m}\)
Find the sum: \(\frac{2n}{11}+\frac{5n}{11}.\)

Giải đáp

\(\frac{2n}{11}+\frac{5n}{11}\)
Add the numerators and place the sum over the common denominator. \(\frac{2n+5n}{11}\)
Combine like terms. \(\frac{7n}{11}\)
Find the sum: \(\frac{3p}{8}+\frac{6p}{8}.\)

Giải đáp

\(\frac{9p}{8}\)
Find the sum: \(\frac{2q}{5}+\frac{7q}{5}.\)

Giải đáp

\(\frac{9q}{5}\)
Find the sum: \(-\frac{3}{12}+(-\frac{5}{12}).\)

Giải đáp

\(-\frac{3}{12}+(-\frac{5}{12})\)
Add the numerators and place the sum over the common denominator. \(\frac{-3+(-5)}{12}\)
Add. \(\frac{-8}{12}\)
Simplify the fraction. \(-\frac{2}{3}\)
Find each sum: \(-\frac{4}{15}+(-\frac{6}{15}).\)

Giải đáp

\(-\frac{2}{3}\)
Find each sum: \(-\frac{5}{21}+(-\frac{9}{21}).\)

Giải đáp

\(-\frac{2}{3}\)
Use fraction circles to find the difference: \(\frac{4}{5}-\frac{1}{5}.\)

Giải đáp

Start with four \(\frac{1}{5}\) pieces. Take away one \(\frac{1}{5}\) piece. Count how many fifths are left. There are three \(\frac{1}{5}\) pieces left.
Use a model to find each difference. Show a diagram to illustrate your model.
\(\frac{7}{8}-\frac{4}{8}\)

Giải đáp

\(\frac{3}{8}\), models may differ.
Use a model to find each difference. Show a diagram to illustrate your model.
\(\frac{5}{6}-\frac{4}{6}\)

Giải đáp

\(\frac{1}{6}\), models may differ

Find the difference: \(\frac{23}{24}-\frac{14}{24}.\)

Giải đáp

	\(\frac{23}{24}-\frac{14}{24}\)
Subtract the numerators and place the difference over the common denominator.	\(\frac{23-14}{24}\)
Simplify the numerator.	\(\frac{9}{24}\)
Simplify the fraction by removing common factors.	\(\frac{3}{8}\)

Find the difference: \(\frac{19}{28}-\frac{7}{28}.\)

Giải đáp

\(\frac{3}{7}\)
Find the difference: \(\frac{27}{32}-\frac{11}{32}.\)

Giải đáp

\(\frac{1}{2}\)
Find the difference: \(\frac{y}{6}-\frac{1}{6}.\)

Giải đáp

\(\frac{y}{6}-\frac{1}{6}\)
Subtract the numerators and place the difference over the common denominator. \(\frac{y-1}{6}\)
The fraction is simplified because we cannot combine the terms in the numerator.
Find the difference: \(\frac{x}{7}-\frac{2}{7}.\)

Giải đáp

\(\frac{x-2}{7}\)
Find the difference: \(\frac{y}{14}-\frac{13}{14}.\)

Giải đáp

\(\frac{y-13}{14}\)
Find the difference: \(-\frac{10}{x}-\frac{4}{x}.\)

Giải đáp

Remember, the fraction \(-\frac{10}{x}\) can be written as \(\frac{-10}{x}.\)
\(-\frac{10}{x}-\frac{4}{x}\)
Subtract the numerators. \(\frac{-10-4}{x}\)
Simplify. \(\frac{-14}{x}\)
Rewrite with the negative sign in front of the fraction. \(-\frac{14}{x}\)
Find the difference: \(-\frac{9}{x}-\frac{7}{x}.\)

Giải đáp

\(-\frac{16}{x}\)
Find the difference: \(-\frac{17}{a}-\frac{5}{a}.\)

Giải đáp

\(-\frac{22}{a}\)

Simplify: \(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}.\)

Giải đáp

	\(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}\)
Combine the numerators over the common denominator.	\(\frac{3+(-5)-1}{8}\)
Simplify the numerator, working left to right.	\(\frac{-2-1}{8}\)
Subtract the terms in the numerator.	\(\frac{-3}{8}\)
Rewrite with the negative sign in front of the fraction.	\(-\frac{3}{8}\)

Simplify: \(\frac{2}{5}+(-\frac{4}{5})-\frac{3}{5}.\)

Giải đáp

−1
Simplify: \(\frac{5}{9}+(-\frac{4}{9})-\frac{7}{9}.\)

Giải đáp

\(-\frac{2}{3}\)
\(\frac{2}{5}+\frac{1}{5}\)
\(\frac{3}{10}+\frac{4}{10}\)

Giải đáp

\(\frac{7}{10}\)
\(\frac{1}{6}+\frac{3}{6}\)
\(\frac{3}{8}+\frac{3}{8}\)

Giải đáp

\(\frac{3}{4}\)

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Add and Subtract Fractions with Common Denominators

Model fraction addition
Add fractions with a common denominator
Model fraction subtraction
Subtract fractions with a common denominator
If
To add fractions, add the numerators and place the sum over the common denominator.
If
To subtract fractions, subtract the numerators and place the difference over the common denominator.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Thử đi.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

More in Arithmetic

Order of operations Fractions Percentages Powers and roots