Subtracting Mixed Numbers With Regrouping

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Subtracting mixed numbers requires taking the combined difference of two types of numbers. A useful method is called regrouping. This lesson shows when and how to regroup mixed numbers for subtraction.

Subtracting Mixed Numbers with Regrouping

Jamie's aunt and uncle own and operate the ''Regrouping Mixed Numbers'' dressmaking shop in the center of town. They buy their fabric in bolts of 40 yards by 1 yard and often have leftover material. Instead of discarding this material, they give it to Jamie. Last week, Jamie received 8 5/9 yards of material with a pattern for making a dress needing 6 1/9 yards. She needs to know how much material will be left over. This is great for Jamie because she knows how to subtract mixed numbers.

A mixed number is a number that contains a whole number and a fraction. 8 5/9 and 6 1/9 are mixed numbers. In this lesson, we will subtract mixed numbers and, when necessary, we will use the method of regrouping. In regrouping, we write an equivalent mixed number by taking one from the whole number portion and adding it to the fraction part. So, let's see if Jamie will have enough material leftover to make a blouse.

No Regrouping Necessary

If the fraction parts of a mixed number have the same denominator and if we are subtracting a smaller fraction from a larger fraction, then the subtraction is straightforward: subtract the fractions and then subtract the whole numbers.

For example,


The fractions 5/9 and 1/9 have the same denominator, 9, and we are subtracting the smaller fraction, 1/9, from the larger fraction, 5/9. Thus, 5/9 minus 1/9 is 4/9 and 8 - 6 is 2. The answer is 2 4/9 and we are done.

Regrouping First Case

A small change in plans. Jamie is getting 8 1/9 yards of fabric for a dress needing 6 5/9 yards. Still wanting to know how much fabric will be left over, Jamie is presented with the following subtraction problem:


The fractions have the same denominator, but the larger fraction, 5/9, is being subtracted from the smaller fraction, 1/9. What to do? The answer is to regroup the 8 1/9 mixed number. Let's take a look at the regrouping steps.

First, take one away from the whole number 8 to make it a 7. Then, add this 1 to the fraction part. Not just 1, but a 1 written as 9/9. The 'new' upper mixed number still has the same value of the 'original' mixed number because we reduced the whole number by the same amount that we then added to the fraction.


The new fraction for the top mixed number is 1/9 + 9/9 = 10/9. The new whole number for the mixed number is 8 - 1 = 7. Thus, 8 1/9 is equivalent to 7 10/9. Now we can do the subtraction.

10/9 minus 5/9 is 5/9 while 7 minus 6 is 1:

Regrouping Second Case

It doesn't happen often, but sometimes Jamie receives a whole number of yards to work with. Today, she received 8 yards of material and a dress pattern needing 6 1/4 yards.


The 8 does not have a fractional part unless you think of the fraction as being 0/4. We still need a larger fraction as the top number. Once again, we apply the regrouping method.

Take one away from the 8 to get 7 for the whole number part. Add this 1 in the form of 4/4 to the fraction part: 0/4 + 4/4 is 4/4.


So 8 became the equivalent 7 4/4 and now we can subtract the fraction parts: 4/4 minus 1/4 is 3/4, while for the whole number parts, 7 minus 6 is 1. The answer is 1 3/4.


It's the end of the month and Jamie receives one last gift of fabric.

Regrouping Third Case

A blouse pattern specifies 2 2/3 yards of material. The latest gift of fabric is 5 1/4 yards. How much material will be left over?

Jamie now tackles the following subtraction problem:


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