Adding and Subtracting Fractions

Learning OUTCOMES

  • Use addition and subtraction when evaluating expressions with fractions

Addition and Subtraction of Fractions with Common Denominators

How many quarters are pictured below?

Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.

You can quickly count three, but if you look carefully the image represents 1 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

14+24=34

one quarter + two quarters = three quarters

Let’s use fraction circles to model the same example, 14+24.

Start with one 14 piece. one quarter circle 14
Add two more 14 pieces. plus 2 quarter circles +24
The result is 34 . 3 quarter circles 34

So again, we see that

14+24=34

example

Use a model to find the sum 38+28.

Solution:

Start with three 18 pieces. 3 eighths circles 38
Add two 18 pieces. plus 2 eighths circles +28
How many 18 pieces are there? 5 eighths circles 58

There are five 18 pieces, or five-eighths. The model shows that 38+28=58.

try it

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

18+48

Use a model to find each sum. Show a diagram to illustrate your model.
16+46

 

The following video shows more examples of how to use models to add fractions with like denominators (the value in the lower part of a fraction that represents how many equal parts a whole has been divided into).

Subtracting two fractions with common denominators follows the same process as adding fractions with common denominators. 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 712 of the pizza) left in the box. If Leonardo eats 2 of these remaining pieces (or 212 of the pizza), how much is left? There would be 5 pieces left (or 512 of the pizza).

712212=512

Let’s use fraction circles to model the same example, 712212.

Start with seven 112 pieces. Take away two 112 pieces. How many twelfths are left?

The bottom reads 7 twelfths minus 2 twelfths equals 5 twelfths. Above 7 twelfths, there is a circle divided into 12 equal pieces, with 7 pieces shaded in orange. Above 2 twelfths, the same circle is shown, but 2 of the 7 pieces are shaded in grey. Above 5 twelfths, the 2 grey pieces are no longer shaded, so there is a circle divided into 12 pieces with 5 of the pieces shaded in orange.
Again, we have five twelfths, 512.

Example

Use fraction circles to find the difference: 4515

Solution:
Start with four 15 pieces. Take away one 15 piece. Count how many fifths are left. There are three 15 pieces left, or 35 of the circle left.

The bottom reads 4 fifths minus 1 fifth equals 3 fifths. Above 4 fifths, there is a circle divided into 5 equal pieces, with 4 pieces shaded in orange. Above 1 fifth, the same circle is shown, but 1 of the 4 shaded pieces is shaded in grey. Above 3 fifths, the 1 grey piece is no longer shaded, so there is a circle divided into 5 pieces with 3 of the pieces shaded in orange.

Try It

The examples above show that to add or subtract the same-size pieces—meaning that the fractions have the same denominator—we just add or subtract the number of pieces.

Fraction Addition

If a,b, and c are numbers where c0, then

ac+bc=a+bc

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

Fraction Subtraction

If a,b, and c are numbers where c0, then

acbc=abc

To subtract fractions with a common denominators,  subtract the numerators and place the difference over the common denominator.

Example

Find the difference: 23241424

Try It

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

Example

Simplify: 38+(78)58

Try It

Addition and Subtraction of Fractions with Different Denominators

We just reviewed how to add and subtract fractions with common denominators. But how can we add and subtract fractions with unlike denominators?

Let’s think about coins again. 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 25 cents and one dime equals 10 cents, so the sum is 35 cents. See the image below.

Together, a quarter and a dime are worth 35 cents, or 35100 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 100. Since there are 100 cents in one dollar, 25 cents is 25100 and 10 cents is 10100. So we add 25100+10100 to get 35100, which is 35 cents.

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

First, we will use fraction tiles to model finding the common denominator of 12 and 13.

We’ll start with one 12 tile and 13 tile. We want to find a common fraction tile that we can use to match both 12 and 13 exactly.
If we try the 14 pieces, 2 of them exactly match the 12 piece, but they do not exactly match the 13 piece.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into two pieces, each labeled 1 fourth. Underneath the second rectangle are two pieces, each labeled 1 fourth. These rectangles together are longer than the rectangle labeled as 1 third.
If we try the 15 pieces, they do not exactly cover the 12 piece or the 13 piece.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into three pieces, each labeled 1 sixth. Underneath the second rectangle is an equally sized rectangle split vertically into 2 pieces, each labeled 1 sixth.
If we try the 16 pieces, we see that exactly 3 of them cover the 12 piece, and exactly 2 of them cover the 13 piece.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle are three smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 half rectangle. Below the 1 third rectangle are two smaller rectangles, each labeled 1 fifth. Together, these rectangles are longer than the 1 third rectangle.
If we were to try the 112 pieces, they would also work.

Two rectangles are shown side by side. The first is labeled 1 half. The second is shorter and is labeled 1 third. Underneath the first rectangle is an equally sized rectangle split vertically into 6 pieces, each labeled 1 twelfth. Underneath the second rectangle is an equally sized rectangle split vertically into 4 pieces, each labeled 1 twelfth.
Even smaller tiles, such as 124 and 148, would also exactly cover the 12 piece and the 13 piece.

The denominator of the largest piece that covers both fractions is the least common denominator (LCD) of the two fractions. So, the least common denominator of 12 and 13 is 6.

Notice that all of the tiles that cover 12 and 13 have something in common: Their denominators are common multiples of 2 and 3, the denominators of 12 and 13. The least common multiple (LCM) of the denominators is 6, and so we say that 6 is the least common denominator (LCD) of the fractions 12 and 13.

Least Common Denominator

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

To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD.

Example

Find the LCD for the fractions: 712 and 518

Solution:

Factor each denominator into its primes. Two adjacent factoring trees. The first tree is for 12. 12 factors to 3 and 4. 4 factors to 2 and 2. The second tree is for 18. 18 factors to 3 and 6. 6 factors to 2 and 3.
List the primes of 12 and the primes of 18 lining them up in columns when possible. The prime factorization of 12 written as 12 equals 2 times 2 times 3. Below this, the prime factorization of 18 is written as 18 equals 2 times 3 times 3. The two equations align vertically along the equals sign. The first 2 from the prime factorization of 12 and the 2 in the prime factorization of 18 aling vertically. The second 2 in the prime factorization of 12 does not align vertically with any factor from the prime factorization of 18. The 3 from the prime factorization of 12 aligns with the first 3 from the prime factorization of 18. The second 3 from the prime factorization of 18 does not align with any factor from the prime factorization of 12. Below the two equations, a horizontal line is drawn.
Bring down the columns. Arrows are drawn through each column of factors from the prime factorizations of 12 and 18. Below the horizontal line is the equation LCM equals 2 times 2 times 3 times 3.
Multiply the factors. The product is the LCM. LCM=36
The LCM of 12 and 18 is 36, so the LCD of 712 and 518 is 36. LCD of 712 and 518 is 36.

Try it

To find the LCD of two fractions, find the LCM of their denominators. Notice how the steps shown below are similar to the steps we took to find the LCM.

Find the least common denominator (LCD) of two fractions

  1. Factor each denominator into its primes.
  2. List the primes, matching primes in columns when possible.
  3. Bring down the columns.
  4. Multiply the factors. The product is the LCM of the denominators.
  5. The LCM of the denominators is the LCD of the fractions.

Example

Find the least common denominator for the fractions: 815 and 1124

Try It

Earlier, we used fraction tiles to see that the LCD of 14and16 is 12. We saw that three 112 pieces exactly covered 14 and two 112 pieces exactly covered 16, so

14=312 and 16=212.

On the left is a rectangle labeled 1 fourth. Below it is an identical rectangle split vertically into 3 equal pieces, each labeled 1 twelfth. On the right is a rectangle labeled 1 sixth. Below it is an identical rectangle split vertically into 2 equal pieces, each labeled 1 twelfth.
We say that 14 and 312 are equivalent fractions and also that 16 and 212 are equivalent fractions.

We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference.

Equivalent Fractions Property

If a,b,c are whole numbers where b0,c0,then

ab=acbc and acbc=ab

To add or subtract fractions with different denominators, we will first have to convert each fraction to an equivalent fraction with the LCD. Let’s see how to change 14 and 16 to equivalent fractions with denominator 12 without using models.

Example

Convert 14 and 16 to equivalent fractions with denominator 12, their LCD.

Solution:

Find the LCD. The LCD of 14 and 16 is 12.
Find the number to multiply 4 to get 12. 43=12
Find the number to multiply 6 to get 12. 62=12
Use the Equivalent Fractions Property to convert each fraction to an equivalent fraction with the LCD, multiplying both the numerator and denominator of each fraction by the same number. 14      16

1343      1262

Simplify the numerators and denominators. 312   212

We do not reduce the resulting fractions. If we did, we would get back to our original fractions and lose the common denominator.

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 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.

Example

Convert 815 and 1124 to equivalent fractions with denominator 120, their LCD.

Try it

In our next video we show two more examples of how to use the column method to find the least common denominator of two fractions.

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: 12+13

Solution:

12+13
Find the LCD of 2, 3. The prime factorization of 2 written as 2 equals 2. Below this is the prime factorization of 3 written as 3 equals 3. The two equations align vertically along the equals sign. Both the 2 and the 3 from the prime factorizations do not align vertically with each other. Below the equations is a horizontal line. Below this line is the equation LCD equals 2 times 3, which simplifies to LCD equals 6.
Change into equivalent fractions with the LCD 6. 1323+1232
Simplify the numerators and denominators. 36+26
Add. 56

Remember, always check to see if the answer can be simplified. Since 5 and 6 have no common factors, the fraction 56 cannot be reduced.

Try It

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

Example

Add: 712+518

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

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