Modeling subtraction of mixed numbers

Let’s think of pizzas again to model subtraction of mixed numbers with a common denominator. Suppose you just baked a whole pizza and want to give your brother half of the pizza. What do you have to do to the pizza to give him half? You have to cut it into at least two pieces. Then you can give him half.

We will use fraction circles (pizzas!) to help us visualize the process.

Start with one whole.

Algebraically, you would write:

What if we start with more than one whole? Let’s find out.

In the next example, we’ll subtract more than one whole.

What if you start with a mixed number and need to subtract a fraction? Think about this situation: You need to put three quarters in a parking meter, but you have only a [latex]$1[/latex] bill and one quarter. What could you do? You could change the dollar bill into [latex]4[/latex] quarters. The value of [latex]4[/latex] quarters is the same as one dollar bill, but the [latex]4[/latex] quarters are more useful for the parking meter. Now, instead of having a [latex]$1[/latex] bill and one quarter, you have [latex]5[/latex] quarters and can put [latex]3[/latex] quarters in the meter.

This models what happens when we subtract a fraction from a mixed number. We subtracted three quarters from one dollar and one quarter.

We can also model this using fraction circles, much like we did for addition of mixed numbers.

Example

Use a model to subtract: [latex]1\Large\frac{1}{4}-\Large\frac{3}{4}[/latex]

Show Solution

Solution:

Rewrite vertically. Start with one whole and one fourth.		[latex]\color{red}{1 \Large\frac{1}{4}}[/latex] [latex]-- \Large\frac{3}{4}[/latex]
Since the fractions have denominator 4, cut the whole into 4 pieces. You now have [latex]\Large\frac{4}{4}[/latex] and [latex]\Large\frac{1}{4}[/latex] which is [latex]\Large\frac{5}{4}[/latex] .		[latex]\color{red}{ \Large\frac{5}{4}}[/latex] [latex]-- \Large\frac{3}{4}[/latex]
Take away [latex]\Large\frac{3}{4}[/latex] . There is [latex]\Large\frac{1}{2}[/latex] left.		[latex]\Large\frac{5}{4}[/latex] [latex]\Large\frac{\color{red}{-- \LARGE\frac{3}{4}}}{ \LARGE\frac{2}{4}= \LARGE\frac{1}{2}}[/latex]

Subtract mixed numbers with a common denominator

Now we will subtract mixed numbers without using a model. But it may help to picture the model in your mind as you read the steps.

Example

Find the difference: [latex]5\Large\frac{3}{5}-\normalsize2\Large\frac{4}{5}[/latex]

Solution:

	[latex]5\Large\frac{3}{5}-\normalsize2\Large\frac{4}{5}[/latex]
Rewrite the problem in vertical form.
Since [latex]\Large\frac{3}{5}[/latex] is less than [latex]\Large\frac{4}{5}[/latex] , take 1 from the 5 and add it to the [latex]\Large\frac{3}{5}:\left(\Large\frac{5}{5}+\Large\frac{3}{5}=\Large\frac{8}{5}\right)[/latex]
Subtract the fractions.
Subtract the whole parts. The result is in simplest form.

Since the problem was given with mixed numbers, we leave the result as mixed numbers.

Watch the following video to see two more examples of how to subtract mixed numbers with like denominators.

Just as we did with addition, we could subtract mixed numbers by converting them first to improper fractions. We should write the answer in the form it was given, so if we are given mixed numbers to subtract we will write the answer as a mixed number.