Model Fraction Addition

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

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].

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.

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.

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.

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.

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

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.

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

Example

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

Show Solution

Solution:

	[latex]\frac{7}{12}+\frac{5}{18}[/latex]
Find the LCD of [latex]12[/latex] and [latex]18[/latex].
Rewrite as equivalent fractions with the LCD.	[latex]\frac{7\cdot\color{red}{3}}{12\cdot\color{red}{3}} +\frac{5\cdot\color{red}{2}}{18\cdot\color{red}{2}}[/latex]
Simplify the numerators and denominators.	[latex]\frac{21}{36}+\frac{10}{36}[/latex]
Add.	[latex]\frac{31}{36}[/latex]

Because [latex]31[/latex] is a prime number, it has no factors in common with [latex]36[/latex]. The answer is simplified.

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.

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]

Show Solution

Solution:

	[latex]\frac{7}{15}-\frac{19}{24}[/latex]
Find the LCD.	[latex]15[/latex] is ‘missing’ three factors of [latex]2[/latex] [latex]24[/latex] is ‘missing’ a factor of [latex]5[/latex]
Rewrite as equivalent fractions with the LCD.	[latex]\frac{7\cdot\color{red}{8}}{15\cdot\color{red}{8}} -\frac{19\cdot\color{red}{5}}{24\cdot\color{red}{5}}[/latex]
Simplify each numerator and denominator.	[latex]\frac{56}{120}-\frac{95}{120}[/latex]
Subtract.	[latex]-\frac{39}{120}[/latex]
Rewrite showing the common factor of [latex]3[/latex].	[latex]-\frac{13\cdot 3}{40\cdot 3}[/latex]
Remove the common factor to simplify.	[latex]-\frac{13}{40}[/latex]

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

Example

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

Show Solution

Solution:

	[latex]\frac{1}{2}-\left(-\frac{1}{4}\right)[/latex]
Find the LCD of [latex]2[/latex] and [latex]4[/latex].
Rewrite as equivalent fractions using the LCD [latex]4[/latex].	[latex]\frac{1\cdot\color{red}{2}}{2\cdot\color{red}{2}} - (--\frac{1}{4})[/latex]
Simplify the first fraction.	[latex]\frac{2}{4}-\left(-\frac{1}{4}\right)[/latex]
Subtract.	[latex]\frac{2-\left(-1\right)}{4}[/latex]
Simplify.	[latex]\frac{3}{4}[/latex]

One of the fractions already had the least common denominator, so we only had to convert the other fraction.