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# Subtraction of Fractions

## Subtract fractions and mixed numbers with like and unlike denominators.

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Subtraction of Fractions

Have you ever picked fresh peaches? Have you ever made a peach pie?

Jesse has decided to make peach pie for the bake sale. His recipe calls for 212\begin{align*}2 \frac{1}{2}\end{align*} pounds of peaches. Jesse’s older brother Jeff drives him to the farmer’s market to pick up his peaches. When Jesse gets there, he is amazed at how fast paced the market is. There are several people working behind the counter and they seem to be adding up all of the figures in their heads. Jesse is amazed. He loves math but he can’t even imagine adding up so many numbers in his head at one time.

Jesse is fascinated. So much so that he loses his focus and puts many, many peaches in his cloth bag. When the girl weighs it, Jesse watches her add up the math in her head. She tells Jesse how much and he pays her. Then she hands Jesse back the bag.

“How many pounds is this?” Jesse asks.

“You bought 614\begin{align*}6 \frac{1}{4}\end{align*} pounds of peaches,” She says focusing on another customer.

Jesse is surprised. He knows that he wasn’t paying very good attention, but he has a lot more peaches than he needs. How much more does he have? After Jesse makes his pie, how many pounds of peaches will be left over?

To figure this out, you will need to understand how to subtract fractions and mixed numbers. Pay close attention because this Concept will cover all that you need to know. You will see this problem again at the end of the Concept and you will be ready to solve it!

### Guidance

In real life, we use fractions all the time. Let’s say you are cutting a piece of wood 334\begin{align*}3 \frac{3}{4}\end{align*} feet long and you need to cut 12\begin{align*}\frac{1}{2}\end{align*} foot off of the piece of wood. What do you need to do to figure out how much wood you have left, after you make the cut? You guessed it. Subtraction is the key. Subtracting fractions and mixed numbers is a skill that you are going to need to use all the time.

How do we subtract fractions and mixed numbers?

If you know how to add fractions, then you already know how to subtract them. The key is to make sure that the fractions that you are subtracting have the same denominator. If the fractions have the same denominator, then subtract the numerators just like you subtract whole numbers and keep the denominator the same in your answer.

6929\begin{align*}\frac{6}{9}-\frac{2}{9}\end{align*}

Notice that the denominators are the same, so we can simply subtract the numerators.

62=4\begin{align*}6 - 2 = 4\end{align*}

Our answer is 49\begin{align*}\frac{4}{9}\end{align*}.

If the denominators are not the same, make sure to find the lowest or least common denominator first and then do your subtracting. Think about the first example with sawing wood. If you subtract 12\begin{align*}\frac{1}{2}\end{align*} foot from a piece of wood that is 334\begin{align*}3 \frac{3}{4}\end{align*} feet long, you have to find a common denominator first.

We can choose 4 as the least common denominator and rename each fraction in terms of fourths. To do this, we create equivalent fractions. If you use the equivalent fraction 24\begin{align*}\frac{2}{4}\end{align*} for 12\begin{align*}\frac{1}{2}\end{align*}, then you have the same denominator as the fraction in 334\begin{align*}3 \frac{3}{4}\end{align*}.

33424=314\begin{align*}3 \frac{3}{4}-\frac{2}{4} = 3 \frac{1}{4}\end{align*}

This problem actually uses a mixed number and a fraction. We can also subtract two mixed numbers. We do this in the same way. We subtract the fractions and then subtract the whole numbers.

456146\begin{align*}4 \frac{5}{6}-1 \frac{4}{6}\end{align*}

First, we subtract the fraction parts. These fractions have the same denominator, so we can simply subtract the numerators.

54=1\begin{align*}5 - 4 = 1\end{align*} the fraction here is 16\begin{align*}\frac{1}{6}\end{align*}

Next, we subtract the whole numbers.

41\begin{align*}4 - 1\end{align*}

Our final answer is 316\begin{align*}3 \frac{1}{6}\end{align*}.

Sometimes, when you subtract mixed numbers, you will have to do an extra step. Think about this one.

Imagine you are cutting, or subtracting, 134\begin{align*}1 \frac{3}{4}\end{align*} feet of wood from a piece of wood that is 312\begin{align*}3 \frac{1}{2}\end{align*} feet long. Your subtraction problem looks like this: 312134\begin{align*}3 \frac{1}{2} - 1 \frac{3}{4}\end{align*}.

After you find a common denominator, your subtraction problem now looks like this.

324134\begin{align*}3 \frac{2}{4} - 1 \frac{3}{4}\end{align*}

Take a deep breath and don’t panic! This is where you use your expertise at converting mixed numbers to improper fractions. After you have a common denominator for the fractions, multiply the whole number of the mixed number by the denominator of the fraction. Add this product to the numerator of the fraction.

324134=(3×4)+24=144=(4×1)+34=74

Your new subtraction problem for the example looks like this.

14474\begin{align*}\frac{14}{4} -\frac{7}{4}\end{align*}

Now you simply subtract the numerators and you get 74\begin{align*}\frac{7}{4}\end{align*}. Now you convert this back into a mixed number. Do you remember how to do this?

Don’t forget to rewrite the difference as a mixed number and keep the fraction in lowest terms.

Here are the steps for subtracting mixed numbers.

Subtracting Mixed Numbers:

1. Make sure the fractions have a common denominator.
2. If the fraction to the left of the minus sign is smaller than the fraction to the right of the minus sign – convert both mixed numbers into improper fractions
3. Subtract the improper fractions
4. Rewrite the difference as a mixed number.

Take a few minutes to write these steps down in your notebooks.

Subtract the following fractions and mixed numbers. Be sure that your answer is in lowest terms.

#### Example A

1012612\begin{align*}\frac{10}{12}-\frac{6}{12}\end{align*}

Solution: 412=13\begin{align*}\frac{4}{12} = \frac{1}{3}\end{align*}

#### Example B

6734\begin{align*}\frac{6}{7}-\frac{3}{4}\end{align*}

Solution: 328\begin{align*}\frac{3}{28}\end{align*}

#### Example C

41434\begin{align*}4 \frac{1}{4}-\frac{3}{4}\end{align*}

Solution:24=12\begin{align*}\frac{2}{4} = \frac{1}{2}\end{align*}

Here is the original problem once again.

Jesse has decided to make peach pie for the bake sale. His recipe calls for 212\begin{align*}2 \frac{1}{2}\end{align*} pounds of peaches. Jesse’s older brother Jeff drives him to the farmer’s market to pick up his peaches. When Jesse gets there, he is amazed at how fast paced the market is. There are several people working behind the counter and they seem to be adding up all of the figures in their heads. Jesse is amazed. He loves math but he can’t even imagine adding up so many numbers in his head at one time.

Jesse is fascinated. So much so that he loses his focus and puts many, many peaches in his cloth bag. When the girl weighs it, Jesse watches her add up the math in her head. She tells Jesse how much and he pays her. Then she hands Jesse back the bag.

“How many pounds is this?” Jesse asks.

“You bought 614\begin{align*}6 \frac{1}{4}\end{align*} pounds of peaches,” She says focusing on another customer.

Jesse is surprised. He knows that he wasn’t paying very good attention, but he has a lot more peaches than he needs. How much does he have? After Jesse makes his pie, how many pounds of peaches will be left over?

To solve this problem, we will need to subtract the number of pounds that Jesse needs for his recipe from the number of pounds that he purchases.

614212\begin{align*}6 \frac{1}{4}-2 \frac{1}{2}\end{align*}

These fractions have different denominators, so we need to rename them in terms of a common denominator. The lowest common denominator here is 4.

614\begin{align*}6 \frac{1}{4}\end{align*} is all set.

212=224\begin{align*}2 \frac{1}{2}=2 \frac{2}{4}\end{align*}

We can rewrite the problem.

614224\begin{align*}6 \frac{1}{4}-2 \frac{2}{4}\end{align*}

Next, we have to convert these to improper fractions because we can’t subtract two-fourths from one-fourth.

254104=154\begin{align*}\frac{25}{4}-\frac{10}{4}=\frac{15}{4}\end{align*}

To write it in simplest form, we convert this improper fraction to a mixed number.

Our answer is 334\begin{align*}3 \frac{3}{4}\end{align*} pounds of peaches left over.

### Vocabulary

Lowest Common Denominator
when two fractions have different denominators, we use the lowest common denominator to rename each fraction in terms of that common number. The lowest common denominator is also a least common multiple of the denominators.
Equivalent Fractions
equal fractions
Improper Fractions
when the numerator of a fraction is larger than the denominator

### Guided Practice

Here is one for you to try on your own.

Benito works in a bakery and has baked the world’s longest loaf of cinnamon bread. His loaf measures 1158\begin{align*}11 \frac{5}{8}\end{align*} feet. He cuts a piece of bread 112\begin{align*}1 \frac{1}{2}\end{align*} feet for his friend Pamela and he cuts another piece 223\begin{align*}2 \frac{2}{3}\end{align*} feet for his friend Serena. How much bread does he have left?

Let’s take careful inventory of the information that the problem gives us. We know that the whole loaf of bread is 1158\begin{align*}11 \frac{5}{8}\end{align*} feet. Pamela gets a piece 112\begin{align*}1 \frac{1}{2}\end{align*} feet, that is her piece and Serena gets a piece 223\begin{align*}2 \frac{2}{3}\end{align*} feet long, that is her piece.

This is our given information.

What do we want to find out? We want to know the length of the bread after he cuts Pamela and Serena’s pieces (loaf after cutting =x\begin{align*}= x\end{align*}).

Let’s write an equation to show the relationship between the values:

Whole loaf – Pamela’s piece – Serena’s piece = loaf after cutting

When we substitute the given values, we have the following equation.

1158112223=x\begin{align*}11 \frac{5}{8} - 1 \frac{1}{2}-2 \frac{2}{3} = x\end{align*}

Now, we simply solve from left to right. First, find a common denominator between the fractions in 1158\begin{align*}11 \frac{5}{8}\end{align*} and 112\begin{align*}1 \frac{1}{2}\end{align*}. Let’s use 8, so we solve 1158148=1018\begin{align*}11 \frac{5}{8}-1 \frac{4}{8} = 10 \frac{1}{8}\end{align*}.

Next, we can simplify the problem.

\begin{align*}10 \frac{1}{8} - 2 \frac{2}{3} = x\end{align*}

The lowest common denominator for the fractions is going to be 24. We simplify the problem further.

\begin{align*}10 \frac{3}{24} - 2 \frac{16}{24} = x\end{align*}

You can already see that you will have to convert the mixed numbers to improper fractions. Simplify again.

Next, we just convert the answer to a mixed number and write in simplest terms.

Solution: \begin{align*}7 \frac{11}{24}\end{align*} feet or about \begin{align*}7 \frac{1}{2}\end{align*} feet

### Practice

Directions: Subtract.

1. \begin{align*}\frac{7}{10}-\frac{1}{4}\end{align*}

2. \begin{align*}\frac{4}{8}-\frac{1}{2}\end{align*}

3. \begin{align*}\frac{4}{7}-\frac{1}{21}\end{align*}

4. \begin{align*}\frac{3}{4}-\frac{1}{8}\end{align*}

5. \begin{align*}3 \frac{1}{10}-\frac{1}{5}\end{align*}

6. \begin{align*}4 \frac{3}{8}-3 \frac{1}{4}\end{align*}

7. \begin{align*}2 \frac{1}{6}-1 \frac{1}{3}\end{align*}

8. \begin{align*}9 \frac{1}{2}- 7 \frac{1}{7}\end{align*}

Directions: Estimate the difference

9. \begin{align*}\frac{14}{16}-\frac{1}{11}\end{align*}

10. \begin{align*}\frac{28}{60}-\frac{6}{13}\end{align*}

11. \begin{align*}\frac{6}{7}-\frac{2}{145}\end{align*}

12. \begin{align*}\frac{32}{33}-\frac{5}{12}\end{align*}

13. \begin{align*}14 \frac{20}{21}-3 \frac{18}{19}\end{align*}

14. \begin{align*}2 \frac{1}{49}-1 \frac{13}{14}\end{align*}

15. \begin{align*}3 \frac{2}{21}-2 \frac{6}{11}\end{align*}

16. \begin{align*}4 \frac{8}{17}-\frac{71}{73}\end{align*}

### Vocabulary Language: English

Equivalent Fractions

Equivalent Fractions

Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.
fraction

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
improper fraction

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.
LCD

LCD

The least common denominator or lowest common denominator (LCD) of two fractions is the smallest number that is a multiple of both of the original denominators.
LCM

LCM

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both of the original numbers.
Lowest Common Denominator

Lowest Common Denominator

The lowest common denominator of multiple fractions is the least common multiple of all of the related denominators.