GED Mathematical Reasoning: Adding and Subtracting Fractions

To add fractions having common denominators, add or subtract the numerators and keep the common denominator. Then, reduce to lowest terms if possible.

\frac{4}{7} + \frac{1}{7} = \frac{5}{7}

\frac{4}{5} - \frac{1}{5} = \frac{3}{5}

Steps for adding or subtracting fractions that don’t have common denominators

  • Step 1: Find the Least Common Multiple of the denominators.
  • Step 2: Create an equivalent fraction for each fraction being added using the Least Common Multiple as the common denominator.
  • Step 3: Add or subtract the numerators and keep the common denominator.
  • Step 4: Reduce if possible. If the answer is an improper fraction, change it into a mixed number when needed.

 

Example 1

If Elsa has 2\frac{1}{4} yard of green fabric, \frac{1}{3} yard of blue fabric, and \frac{1}{2} yard of yellow fabric, how many total yards of fabric does she have?

  1. \frac{11}{12} yards
  2. 3\frac{1}{6} yards
  3. 3 yards
  4. 3\frac{1}{12} yards

 

For this problem, we’ll need to add the fractions: 2\frac{1}{4}\frac{1}{3}, \frac{1}{2} to find the total amount of fabric. Let’s first change that mixed number into an improper fraction.

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To find the total amount of fabric, then, we’ll add: \frac{9}{4} + \frac{1}{3} + \frac{1}{2}

Step 1: Find the Least Common Multiple of the denominators.

To do this, we’ll list the first several multiples of each denominator and look for the smallest multiple common to all three lists.

The first several multiples of 4 are: 4, 8, 12, 16, 20, and 24

The first several multiples of 3 are: 3, 6, 9, 12, 15, and 18

The first several multiples of 2 are: 2, 4, 6, 8, 10, and 12

Can you spot the Least Common Multiple? It is 12! The smallest number common to all three lists is 12.

Step 2 is to create an equivalent fraction for each fraction being added using the Least Common Multiple as the common denominator. Let’s take this one fraction at a time.

To rewrite \frac{9}{4} with a denominator of 12, create a new fraction with a 12 in the denominator.

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To go from a denominator of 4 to a denominator of 12, we multiply by 3 so we must multiply the numerator by 3, as well. 9 times 3 is 27, which will be the new numerator.

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Next, to rewrite \frac{1}{3} with a denominator of 12, create another new fraction with a 12 in the denominator.

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To go from a denominator of 3 to a denominator of 12, we multiply by 4 so we must multiply the numerator by 4, as well. 1 times 4 is 4, which will be the new numerator.

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Now for our third fraction. To rewrite \frac{1}{2} with a denominator of 12, create one more new fraction with a 12 in the denominator.

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To go from a denominator of 2 to a denominator of 12, we multiply by 6 so we must multiply the numerator by 6, as well. 1 times 6 is 6, which will be the new numerator.

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Many students prefer to write out the work vertically like this:

fractions15

Now we’re ready to add! For Step 3, add the numerators and keep the common denominator.

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27 plus 4 plus 6 equals 37, which will be the numerator of our sum. Keeping the common denominator of 12 yields a result of \frac{37}{12}

Finally… Step 4: Reduce if possible. If the answer is an improper fraction, change it into a mixed number when needed. In looking at the result of \frac{37}{12}, it is already written in lowest terms since there are no factors common to both 37 and 12. We can, however, turn this improper fraction into a mixed number, which makes sense since we are dealing with a total length of fabric. To do so, we’ll divide 12 into 37.

12 goes into 37 three times with a remainder of 1. Therefore,\frac{37}{12} written as a mixed number is: 3\frac{1}{12}. The answer, then, is answer (4): 3\frac{1}{12} yards

Going back to Step 4, it is important to know that reducing a fraction is different than turning an improper fraction into a mixed number. Reducing a fraction means writing it in lowest terms by dividing the numerator and denominator by a common factor. Changing an improper fraction to a mixed number is simply re-writing it in a different form.

 

Example 2

If Brett has\frac{8}{9} pound of butter and uses \frac{1}{3} pound for a recipe. How much butter does he have left?

  1. \frac{7}{9} pound
  2. \frac{7}{6} pound
  3. \frac{5}{9} pound
  4. \frac{1}{2} pound

 

To solve this problem, we’ll need to subtract \frac{8}{9} - \frac{1}{3}. Notice that the fractions we need to subtract do NOT have common denominators. To subtract fractions without common denominators, we’ll follow the same strategy as we did to add. It’s just that we’ll subtract instead.

Step 1: Find the Least Common Multiple of the denominators.

The first few multiples of 9 are: 9, 18, and  27

The first few multiples of 3 are: 3, 6, and 9

The smallest number common to both lists is 9. Therefore, our Least Common Multiple is 9.

Step 2 requires us to create an equivalent fraction for each fraction being subtracted using the Least Common Multiple as the common denominator.

Since \frac{8}{9}  already has a denominator of 9, we may leave this fraction as is.

To rewrite \frac{1}{3} with a denominator of 9, create a new fraction with a 9 in the denominator.

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To go from a denominator of 3 to a denominator of 9, we multiply by 3 so we must multiply the numerator by 3, as well. 1 times 3 is 3, which will be the new numerator.

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Step 3 is to subtract the numerators and keep the common denominator.

8 minus 3 equals 5, which will be the numerator of our difference. Keeping the common denominator of 9 yields a result of \frac{5}{9}.

Step 4 is to reduce if possible.

In this case, the fraction \frac{5}{9} is already written in lowest terms since there are no factors common to both 5 and 9. Our final answer, then, is:  answer (3), \frac{5}{9} pound butter

 

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