2.4 Adding and Subtracting Fractions

Learning Outcomes

  • Model fraction addition
  • Add and subtract fractions with a common denominator
  • Find equivalent fractions with the common denominator
  • Add and subtract fractions with different denominators

Key words

  • Common denominator: a common multiple of all denominators
  • Least common denominator: the smallest common multiple of all denominators

Model Fraction Addition

How many quarters are pictured? One quarter plus [latex]2[/latex] quarters equals [latex]3[/latex] quarters.

Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.
Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

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

[latex]\text{one quarter }+\text{ two quarters }=\text{ three quarters} [/latex]

Let’s use fraction circles to model the same example, [latex]\frac{1}{4}+\frac{2}{4}[/latex].

Start with one [latex]\frac{1}{4}[/latex] piece. . [latex]\frac{1}{4}[/latex]
Add two more [latex]\frac{1}{4}[/latex] pieces. . [latex]+\frac{2}{4}[/latex]
The result is [latex]\frac{3}{4}[/latex] . . [latex]\frac{3}{4}[/latex]

So again, we see that [latex]\frac{1}{4}+\frac{2}{4}=\frac{3}{4}[/latex].

try it

Use a model to find each sum. Show a diagram to illustrate your model.

[latex]\frac{1}{8}+\frac{4}{8}[/latex]

 

Use a model to find each sum. Show a diagram to illustrate your model.
[latex]\frac{1}{6}+\frac{4}{6}[/latex]

 

The following video shows more examples of how to use models to add fractions with like denominators.

Add Fractions with a Common Denominator

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

Fraction Addition

If [latex]a,b,\text{ and }c[/latex] are integers where [latex]c\ne 0[/latex], then [latex]\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}[/latex]

To add fractions with a common denominator, add the numerators and place the sum over the common denominator.

Example

Find the sum: [latex]\frac{3}{5}+\frac{1}{5}[/latex]

Try It

Technically, a negative sign on a fraction can be written in the following locations: by the numerator, by the denominator or out in front of the fraction bar.

[latex]-\frac{2}{3}=\frac{-2}{3}=\frac{2}{-3}[/latex]

This is because [latex]\frac{(-)}{(+)}=(-)[/latex] and [latex]\frac{(+)}{(-)}=(-)[/latex] due to division of integers. Usually, we avoid putting the negative sign on the denominator and keep it either out front or on the numerator.

Example

Find the sum: [latex]-\frac{3}{12}+\left(-\frac{5}{12}\right)[/latex]

Try It

Adding and Subtracting Fractions with Different Denominators

Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals [latex]25[/latex] cents and one dime equals [latex]10[/latex] cents, so the sum is [latex]35[/latex] cents. See the image below.

Together, a quarter and a dime are worth [latex]35[/latex] cents, or [latex]{\Large\frac{35}{100}}[/latex] of a dollar.

A quarter and a dime are shown. Below them, it reads 25 cents plus 10 cents. Below that, it reads 35 cents.
Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is [latex]100[/latex]. Since there are [latex]100[/latex] cents in one dollar, [latex]25[/latex] cents is [latex]\frac{25}{100}[/latex] and [latex]10[/latex] cents is [latex]\frac{10}{100}[/latex]. So we add [latex]\frac{25}{100}+\frac{10}{100}[/latex] to get [latex]\frac{35}{100}[/latex], which is [latex]35[/latex] cents.

We have practiced adding and subtracting fractions with common denominators. Now let’s see what we need to do with fractions that have different denominators.

To add fractions, they must have the same denominator. If they have different denominators, we must build equivalent fractions so that they have the same denominator. The new denominator must be a multiple of each of the fractions denominators.

For example, [latex]\frac{5}{6}+\frac{3}{4}[/latex] has denominators of [latex]6[/latex] and [latex]4[/latex]. We need a new denominator that has factors of both [latex]6[/latex] and [latex]4[/latex]. One way to achieve this is to simply multiply the two denominators to get [latex]24[/latex].  [latex]6[/latex] and [latex]4[/latex] both divide exactly into [latex]24[/latex]. Now we can build equivalent fractions with denominators of [latex]24[/latex].

[latex]\frac{5}{6}=\frac{5 \color{red}{\cdot 4}}{6 \color{red}{\cdot 4}}=\frac{20}{24}[/latex]

[latex]\frac{3}{4}=\frac{3 \color{red}{\cdot 6}}{4 \color{red}{\cdot 6}}=\frac{18}{24}[/latex]

Then we can add:   [latex]\frac{5}{6}+\frac{3}{4}=\frac{20}{24}+\frac{18}{24}=\frac{20+18}{24}=\frac{38}{24}[/latex]

Finally, we simplify the sum:   [latex]\large\frac{38}{24}=\frac{\color{red}{\cancel 2}\cdot 19}{\color{red}{\cancel 2}\cdot 12}=\frac{19}{12}[/latex]

Notice that although multiplying the denominators yields a common denominator, it is not the smallest number that could be used. Any common multiple of the denominators will work, but the least common multiple (LCM) is typically used for efficiency.

let’s take another look at [latex]\frac{5}{6}+\frac{3}{4}[/latex]. The LCM of the denominators [latex]6[/latex] and [latex]4[/latex] is [latex]12[/latex].

We build equivalent fractions with denominator of [latex]12[/latex]:

[latex]\frac{5}{6}=\frac{5 \color{red}{\cdot 2}}{6 \color{red}{\cdot 2}}=\frac{10}{12}[/latex]

[latex]\frac{3}{4}=\frac{3 \color{red}{\cdot 3}}{4 \color{red}{\cdot 3}}=\frac{9}{12}[/latex]

Then we add: [latex]\frac{5}{6}+\frac{3}{4}=\frac{10}{12}+\frac{9}{12}=\frac{10+9}{12}=\frac{19}{12}[/latex]

Either way we get the same answer, but we are using smaller integer values using the LCM.

Least Common Denominator

The lowest common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

Try it

Convert two fractions to equivalent fractions with their LCD as the common denominator

  1. Find the LCD.
  2. For each fraction, determine the natural number needed to multiply the denominator to get the LCD.
  3. Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2.
  4. Simplify the numerator and denominator.

Try it

Try it

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.

Add or subtract fractions with different denominators

  1. Find the LCD.
  2. Convert each fraction to an equivalent form with the LCD as the denominator.
  3. Add or subtract the fractions.
  4. Write the result in simplified form.

Example

Add: [latex]\frac{1}{2}+\frac{1}{3}[/latex]

Solution:

[latex]\frac{1}{2}+\frac{1}{3}[/latex]
Find the LCD of [latex]2[/latex], [latex]3[/latex]. .
Change into equivalent fractions with the LCD [latex]6[/latex]. [latex]\frac{1\cdot\color{red}{3}}{2\cdot\color{red}{3}} +\frac{1\cdot\color{red}{2}}{3\cdot\color{red}{2}}[/latex]
Simplify the numerators and denominators. [latex]\frac{3}{6}+\frac{2}{6}[/latex]
Add. [latex]\frac{5}{6}[/latex]

Remember, always check to see if the answer can be simplified. Since [latex]5[/latex] and [latex]6[/latex] have no common factors (other than 1), the fraction [latex]\frac{5}{6}[/latex] cannot be simplified.

Try It

Watch the following video to see more examples and explanation about how to add two fractions with unlike denominators.

Try It

Example

Add: [latex]\frac{7}{12}+\frac{5}{18}[/latex]

Try It

You can also add more than two fractions as long as you first find a common denominator for all of them. An example of a sum of three fractions is shown below. In this example, you will use the prime factorization method to find the LCM.

Think About It

Add [latex]\frac{3}{4}+\frac{1}{6}+\frac{5}{8}[/latex].  Simplify the answer and write as a mixed number.

What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would add three fractions with different denominators together.

Subtracting Fractions

When you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions. Below are some examples of subtracting fractions whose denominators are not alike.

Example

Subtract: [latex]\frac{7}{15}-\frac{19}{24}[/latex]

Try It

The following video provides two more examples of how to subtract two fractions with unlike denominators.

Example

Add: [latex]-\frac{11}{30}+\frac{23}{42}[/latex]

Try It

Example

Subtract: [latex]\frac{1}{2}-\left(-\frac{1}{4}\right)[/latex]