10.2.3.2: Subtracting Fractions and Mixed Numbers
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- Subtract fractions with like and unlike denominators.
- Subtract mixed numbers without regrouping.
- Subtract mixed numbers with regrouping.
- Solve application problems that require the subtraction of fractions or mixed numbers.
Introduction
Sometimes subtraction, rather than addition, is required to solve problems that involve fractions. Suppose you are making pancakes and need 412 cups of flour but you only have 234 cups. How many additional cups will you have to get to make the pancakes? You can solve this problem by subtracting the mixed numbers.
Subtracting Fractions
The most simple fraction subtraction problems are those that have two proper fractions with a common denominator. That is, each denominator is the same. The process is just as it is for addition of fractions with like denominators, except you subtract! You subtract the second numerator from the first and keep the denominator the same.
Imagine that you have a cake with equal-sized pieces. Some of the cake has already been eaten, so you have a fraction of the cake remaining. You could represent the cake pieces with the picture below.
The cake is cut into 12 equal pieces to start. Two are eaten, so the remaining cake can be represented with the fraction 1012. If 3 more pieces of cake are eaten, what fraction of the cake is left? You can represent that problem with the expression 1012−312.
If you subtract 3 pieces, you can see below that 712 of the cake remains.
You can solve this problem without the picture by subtracting the numerators and keeping the denominator the same:
1012−312=712
If the denominators (bottoms) of the fractions are the same, subtract the numerators (tops) and keep the denominator the same. Remember to simplify the resulting fraction, if possible.
67−17 | Subtract. |
Solution
6−17=57 | Both fractions have a denominator of 7, so subtract the numerators and keep the same denominator. |
67−17=57
59−29 | Subtract. Simplify the answer. |
Solution
59−29=39 | The fractions have a like denominator, also known as a common denominator, so subtract the numerators. |
3÷39÷3=13 | Simplify the fraction. |
59−29=13
If the denominators are not the same (they have unlike denominators), you must first rewrite the fractions with a common denominator. The least common denominator, which is the least common multiple of the denominators, is the most efficient choice, but any common denominator will do. Be sure to check your answer to be sure that it is in simplest form. You can use prime factorization to find the least common multiple (LCM), which will be the least common denominator (LCD). See the example below.
15−16 | Subtract. Simplify the answer. |
Solution
5⋅6=30 | The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together. |
15⋅66=630 16⋅55=530 |
Rewrite each fraction as an equivalent fraction with a denominator of 30. |
630−530=130 | Subtract the numerators. Simplify the answer if needed. |
15−16=130
The example below shows using multiples to find the least common multiple, which will be the least common denominator.
56−14 | Subtract. Simplify the answer. |
Solution
Multiples of 6 include 6, 12, 18, 24 Multiples of 4 include 4, 8, 12, 16, 20 |
Find the least common multiple of the denominators. |
12 is the least common multiple of 6 and 4. | This is the least common denominator. |
56⋅22=101214⋅33=312 | Rewrite each fraction with a denominator of 12. |
1012−312=712 | Subtract the fractions. Simplify the answer if needed. |
56−14=712
23−16 Subtract and simplify the answer.
- 13
- 36
- 56
- 12
- Answer
-
- Incorrect. Find a least common denominator and subtract; then simplify. The correct answer is 12.
- Incorrect. Simplify the fraction. The correct answer is 12.
- Incorrect. Subtract, don’t add, the fractions. The correct answer is 12
- Correct. 46−16=36=12
Subtracting Mixed Numbers
Subtracting mixed numbers works much the same way as adding mixed numbers. To subtract mixed numbers, subtract the whole number parts of the mixed numbers and then subtract the fraction parts in the mixed numbers. Finally, combine the whole number answer and the fraction answer to express the answer as a mixed number.
645−315 | Subtract. Simplify the answer and write as a mixed number. |
Solution
6−3=3 45−15=35 |
Subtract the whole numbers and subtract the fractions. |
335 | Combine the fraction and the whole number. Make sure the fraction in the mixed number is simplified. |
645−315=335
Sometimes it might be easier to express the mixed number as an improper fraction first and then solve. Consider the example below.
813−423 | Subtract. Simplify the answer and write as a mixed number. |
Solution
813=8⋅3+13=24+13=253 423=4⋅3+23=12+23=143 |
Write each mixed number as an improper fraction. |
253−143=113 | Since the fractions have a like denominator, subtract the numerators. |
113=323 | Write the answer as a mixed number. Divide 11 by 3 to get 3 with a remainder of 2. |
813−423=323
Since addition is the inverse operation of subtraction, you can check your answer to a subtraction problem with addition. In the example above, if you add 423 to your answer of 323, you should get 813.
423+3234+3+23+237+437+113813
Sometimes you have to find a common denominator in order to solve a mixed number subtraction problem.
712−213 | Subtract.Simplify the answer and write as a mixed number. |
Solution
2⋅3=6 | Recall that a common denominator can easily be found by multiplying the denominators together. |
12⋅33=3613⋅22=26 | Rewrite each fraction using the common denominator 6. |
36−26=16 | Subtract the fractions. |
7−2=5 | Subtract the whole numbers. |
516 | Combine the whole number and the fraction. |
712−213=516
945−423
Subtract. Simplify the answer and write it as a mixed number.
- 215
- 5215
- 4715
- 5
- Answer
-
- Incorrect. Subtract the whole numbers, too. The correct answer is 5215.
- Correct. 9−4=5; 45−23=1215−1015=215. Combining them gives 5215.
- Incorrect. Subtract, don’t add, the fractions. The correct answer is 5215.
- Incorrect. Subtract the fractions as well as the whole numbers. The correct answer is 5215.
Subtracting Mixed Numbers with Regrouping
Sometimes when subtracting mixed numbers, the fraction part of the second mixed number is larger than the fraction part of the first number. Consider the problem: 716−356. The standard procedure would be to subtract the fractions, but 16−56 would result in a negative number. You don’t want that! You can regroup one of the whole numbers from the first number, writing the first mixed number in a different way:
716=7+16=6+1+166+66+16=6+76=676
Now, you can write an equivalent problem to the original:
676−356
Then, you just subtract like you normally subtract mixed numbers:
6−3=3
76−56=26=13
So, the answer is 313.
As with many fraction problems, you may need to find a common denominator. Remember that a key part of adding and subtracting fractions and mixed numbers is making sure to have a common denominator as a first step! In the example below, the original fractions do not have a like denominator. You need to find one before proceeding with the next steps.
715−314 | Subtract. Simplify the answer and write as a mixed number. |
Solution
Multiples of 5 include 5, 10, 15, 20, 25 Multiples of 4 include 4, 8, 12, 16, 20, 24 |
Find a least common denominator. 20 is the least common multiple, so use it for the least common denominator. |
15⋅44=42014⋅55=520 | Rewrite each fraction using the common denominator. |
7420−3520 | Write the expression using the mixed numbers with the like denominator. |
7420=6+1+4206+2020+4206+242062420 | Since the second fraction part, 520, is larger than the first fraction part, 420, regroup one of the whole numbers and write it as 2020. |
7420−352062420−3520 | Rewrite the subtraction expression using the equivalent fractions. |
6−3=32420−520=1920 | Subtract the whole numbers, subtract the fractions. |
31920 | Combine the whole number and the fraction. |
715−314=31920
Sometimes a mixed number is subtracted from a whole number. In this case, you can also rewrite the whole number as a mixed number in order to perform the subtraction. You use an equivalent mixed number that has the same denominator as the fraction in the other mixed number.
8−425 | Subtract. Simplify the answer and write as a mixed number. |
Solution
8=7+17+55755 | Regroup one from the whole number and write it as 55. |
755−425 | Rewrite the subtraction expression using the equivalent fractions. |
7−4=355−25=35 | Subtract the whole numbers, subtract the fractions. |
335 | Combine the whole number and the fraction. |
8−425=335
If the fractional part of the mixed number being subtracted is larger than the fractional part of the mixed number from which it is being subtracted, or if a mixed number is being subtracted from a whole number, follow these steps:
- Subtract 1 from the whole number part of the mixed number being subtracted.
- Add that 1 to the fraction part to make an improper fraction. For example: 723=6+33+23=653
- Then, subtract as with any other mixed numbers.
Alternatively, you can change both numbers to improper fractions and then subtract.
15−1314 Subtract. Simplify the answer and write as a mixed number.
- 214
- 2814
- 134
- 234
- Answer
-
- Incorrect. This is the answer to 1514−13. The fraction has to be subtracted from the 15. The correct answer is 134.
- Incorrect. Subtract, don’t add, the quantities. The correct answer is 134.
- Correct. 1444−1314=134
- Incorrect. Subtract 1 from the whole number when rewriting it as a mixed number. The correct answer is 134.
Subtracting Fractions and Mixed Numbers to Solve Problems
Knowing how to subtract fractions and mixed numbers is useful in a variety of situations. When reading problems, look for key words that indicate that the problem can be solved using subtraction.
Sherry loves to quilt, and she frequently buys fabric she likes when she sees it. She had purchased 5 yards of blue print fabric and decided to use 238 yards of it in a quilt. How much of the blue print fabric will she have left over after making the quilt?
Solution
5−238 | Write an expression using subtraction to describe the situation. |
488−238 | Rewrite the whole number as a mixed number. |
488−238=258 | Subtract. Check that the mixed number is simplified. |
Sherry has 258 yards of blue print fabric left over.
Pilar and Farouk are training for a marathon. On a recent Sunday, they both completed a run. Farouk ran 1278 miles and Pilar ran 1434 miles. How many more miles did Pilar run than Farouk?
Solution
1434−1278 | Write an expression using subtraction to describe the situation. |
1468−1278 | Rewrite the mixed numbers using the least common denominator. |
146813+1+6813+88+6813148 | Since the fraction part of the second mixed number is larger than the fraction part of the first mixed number, regroup one as a fraction and rewrite the first mixed number. |
13148−1278 | Write the subtraction expression in its new form. |
178 | Subtract. |
Pilar ran 178 miles more than Farouk.
Mike and Jose are painting a room. Jose used 23 of a can of paint and Mike used 12 of a can of paint. How much more paint did Jose use? Write the answer as a fraction of a can.
Solution
23−12 | Write an expression using subtraction to describe the situation. |
2⋅23⋅2=461⋅32⋅3=36 | Rewrite the fractions using a common denominator. |
46−36=16 | Subtract. Check that the fraction is simplified. |
Jose used 16 of a can more paint than Mike.
Mariah’s sunflower plant grew 1823 inches in one week. Her tulip plant grew 334 inches in one week. How many more inches did the sunflower grow in a week than the tulip?
- 22512 inches
- 15112 inches
- 15 inches
- 141112 inches
- Answer
-
- Incorrect. Subtract, don’t add, the fractions. The correct answer is 141112 inches.
- Incorrect. Subtract 23−34, not 34−23. The correct answer is 141112 inches.
- Incorrect. Subtract the fractions as well as the whole numbers in the mixed numbers. The correct answer is 141112 inches.
- Correct. 172012−3912=141112
Summary
Subtracting fractions and mixed numbers combines some of the same skills as adding whole numbers and adding fractions and mixed numbers. When subtracting fractions and mixed numbers, first find a common denominator if the denominators are not alike, rewrite each fraction using the common denominator, and then subtract the numerators. When subtracting mixed numbers, if the fraction in the second mixed number is larger than the fraction in the first mixed number, rewrite the first mixed number by regrouping one whole as a fraction. Alternatively, rewrite all fractions as improper fractions and then subtract. This process is also used when subtracting a mixed number from a whole number.