3.3: Subtracting Fractions
Introduction
Plenty of Peaches
Jesse has decided to make peach pie for the bake sale. His recipe calls for \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 do the math in her head. She tells Jesse how much he owes and he pays her. Then she hands Jesse back the bag.
“How many pounds is this?” Jesse asks.
“You bought \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 lesson will cover all that you need to know. You will see this problem again at the end of the lesson and you will be ready to solve it!
What You Will Learn
By the end of this lesson you will know how to complete the following:
- Subtract fractions and mixed numbers.
- Estimate differences of fractions and mixed numbers.
- Identify and apply the inverse property of addition in fraction operations, using numerical and variable expressions.
- Model and solve real-world problems using simple equations involving sums and differences of fractions or mixed numbers.
Teaching Time
I. Subtract Fractions and Mixed Numbers
In real life, we use fractions all the time. Let’s say you are cutting a piece of wood that is \begin{align*}3 \frac{3}{4}\end{align*} feet long and you need to cut \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 will 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.
Example
\begin{align*}\frac{6}{9}-\frac{2}{9}\end{align*}
Notice that the denominators are the same, so we can simply subtract the numerators.
\begin{align*}6 - 2 = 4\end{align*}
Our answer is \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 want to subtract \begin{align*}\frac{1}{2}\end{align*} foot from a piece of wood that is \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 \begin{align*}\frac{2}{4}\end{align*} for \begin{align*}\frac{1}{2}\end{align*}, then you have the same denominator as the fraction in \begin{align*}3 \frac{3}{4}\end{align*}.
\begin{align*}3 \frac{3}{4}-\frac{2}{4} = 3 \frac{1}{4}\end{align*}
This example 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.
Example
\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.
\begin{align*}5 - 4 = 1\end{align*} the fraction here is \begin{align*}\frac{1}{6}\end{align*}
Next, we subtract the whole numbers.
\begin{align*}4 - 1\end{align*}
Our final answer is \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 example.
Imagine you are cutting, or subtracting, \begin{align*}1 \frac{3}{4}\end{align*} feet of wood from a piece of wood that is \begin{align*}3 \frac{1}{2}\end{align*} feet long. Your subtraction problem looks like this: \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.
\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.
\begin{align*}3 \frac{2}{4} &= \frac{(3 \times 4)+2}{4}=\frac{14}{4}\\ 1 \frac{3}{4} &= \frac{(4 \times 1)+3}{4}=\frac{7}{4}\end{align*}
Your new subtraction problem for the example looks like this.
\begin{align*}\frac{14}{4} -\frac{7}{4}\end{align*}
Now you simply subtract the numerators and you get \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:
- Make sure the fractions have a common denominator.
- 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
- Subtract the improper fractions
- Rewrite the difference as a mixed number.
Take a few minutes to write these steps down in your notebooks.
3G. Lesson Exercises
Subtract the following fractions and mixed numbers. Be sure that your answer is in lowest terms.
- \begin{align*}\frac{10}{12}-\frac{6}{12}\end{align*}
- \begin{align*}\frac{6}{7}-\frac{3}{4}\end{align*}
- \begin{align*}4 \frac{1}{4}-\frac{3}{4}\end{align*}
Take a few minutes to check your answers with a partner. Is your work accurate?
II. Estimate Differences of Fractions and Mixed Numbers
Estimation is a method for finding an approximate solution to a problem. For example, \begin{align*}52 - 21\end{align*} is exactly 31. We can estimate the difference by rounding to the tens place and subtracting \begin{align*}50 - 20\end{align*}. Then we can say that \begin{align*}52 - 21\end{align*} is “about 30.”
To approximate the values of fractions, we compare the fractions in relation to three benchmarks, 0, \begin{align*}\frac{1}{2}\end{align*} and 1. Is the fraction closer to 0, \begin{align*}\frac{1}{2}\end{align*} or 1? If it’s closer to 1, we say that the value of the fraction is “about 1.”
We can use these approximate values of fractions to estimate the differences of fractions and mixed numbers.
First, we approximate the value of each fraction or mixed number using the benchmarks 0, \begin{align*}\frac{1}{2}\end{align*} and 1.
Next, we find the difference between the approximate values.
Even when you are asked to find an exact answer, estimation is a useful way to get an idea of a reasonable solution to a problem. Once you have finished solving for an exact answer of a problem, you can check your answer against the estimate. Refer back to the following steps if necessary.
Example
Estimate the difference between \begin{align*}\frac{7}{8}-\frac{4}{9}\end{align*}
First, we need to rewrite each fraction in terms of its benchmark. Seven-eighths is almost one, so we can say that the best benchmark is 1. Four - ninths is a little less than one-half, so we can use one-half as the benchmark.
Next, to find the best estimate, we subtract the benchmarks.
\begin{align*}1 - \frac{1}{2}=\frac{1}{2}\end{align*}
Our estimate is one-half.
We can also estimate differences of mixed numbers.
Example
Estimate the difference between \begin{align*}5 \frac{14}{16}-5 \frac{1}{16}\end{align*}
First, we need to rewrite each of these in terms of a benchmark. Now we use the whole numbers of 5 and 6 as our starting points. Five and fourteen-sixteenths is close to 6. We use 6 as our benchmark for the first mixed number.
Five and one-sixteenth is closest to five. We use 5 as our second benchmark.
Now we subtract the benchmarks to find an accurate estimate.
\begin{align*}6 - 5 = 1\end{align*}
Our estimate is 1.
3H. Lesson Exercises
Use benchmarks to estimate the differences.
- \begin{align*}6 \frac{5}{6}-\frac{1}{10}\end{align*}
- \begin{align*}\frac{8}{9}-\frac{3}{7}\end{align*}
- \begin{align*}4 \frac{4}{5}-1 \frac{3}{5}\end{align*}
Take a few minutes to check your work with a partner.
III. Identify and Apply the Inverse Property of Addition in Fraction Operations, using Numerical and Variable Expressions
Think back to the last lesson where Teri was making the blueberry pie. Imagine that the recipe called for \begin{align*}3 \frac{3}{4}\end{align*} cups of blueberries. You looked in the refrigerator and you find \begin{align*}1 \frac{1}{2}\end{align*} cups of blueberries. You needed to go to the store to buy the rest of the blueberries you need for the recipe.
Identify the information that you know and the information that you don’t know. You know: the amount of blueberries required by the recipe \begin{align*}= 3 \frac{3}{4}\end{align*} cups; the amount of blueberries in your refrigerator \begin{align*}= 1 \frac{1}{2}\end{align*} cups. You don’t know the amount of blueberries you need to buy at the store. Let’s call that \begin{align*}x\end{align*}, since it’s an unknown. If we write an equation to model the relationship between these amounts, it looks like this.
blueberries in refrigerator + blueberries you buy at the store = blueberries required for the pie recipe
When you insert the values (numerical and unknown) into the equation, it looks like this.
\begin{align*}1 \frac{1}{2} + x = 3 \frac{3}{4}\end{align*}
This is how we can write an equation to solve a real-world problem.
How do we solve this equation and figure out the value of the variable?
We can use the Inverse Property of Addition to help us in solving equations.
The Inverse Property of Addition states that the sum of a number and its opposite are equal to 0. Let’s test it out.
\begin{align*}5 + (-5) = 0\end{align*}
Even if you can’t remember how to work with negative numbers, you can still think this through and it will make sense. We have five and we take away five. Well, that answer is 0.
What does this have to do with solving equations?
To solve an equation, we need to get the variable by itself on one side of the equals. In the example above, the \begin{align*}x\end{align*} is on one side with one and one-half. We need to move one and one-half to the other side of the equals. We can do this by using the Inverse Property of Addition. To get the \begin{align*}x\end{align*} alone, we need to have zero on the same side of the equation as the \begin{align*}x\end{align*}. Here is how we use the Inverse Property of Addition.
\begin{align*}1 \ \frac{1}{2} + x &= 3 \ \frac{3}{4}\\ \left(-1 \ \frac{1}{2}\right)+1 \ \frac{1}{2} + x &= 3 \ \frac{3}{4}-1 \frac{1}{2}\\ 0+x &= 3 \ \frac{3}{4}-1 \ \frac{1}{2}\\ x &= 3 \ \frac{3}{4}-1 \ \frac{1}{2}\end{align*}
Now we can simply subtract the mixed numbers. First, we need to rename each fraction in terms of fourths. Three and three-fourths is all set. We convert one and one-half to one and two-fourths and subtract.
Our answer is \begin{align*}2 \frac{1}{4}\end{align*}.
3I. Lesson Exercises
Solve each equation using the Inverse Property of Addition. Be sure that your answer is in simplest form.
- \begin{align*}\frac{1}{6}+x=\frac{5}{6}\end{align*}
- \begin{align*}\frac{4}{5}+ y=2 \frac{1}{2}\end{align*}
Take a few minutes to check your answers with a partner. Question 2 had a lot of steps to it. Did you get all of them? Is your answer in simplest form?
IV. Model and Solve Real-World Problems Using Simple Equations Involving Sums and Differences of Fractions or Mixed Numbers
Have you begun to realize how useful fractions can be in everyday life? Jorge drinks \begin{align*}\frac{2}{3}\end{align*} of a glass of lemonade. Ursula and Andy will meet in \begin{align*}\frac{1}{4}\end{align*} of an hour.
In this section, we’ll look at some real-world problems which involve adding and subtracting fractions. When solving real-world problems, it’s important to first define terms. What information does the problem give us? What information does the problem ask us to find out? Once we know where we are and where we want to go, we can figure out how to get there.
Example
Benito works in a bakery and has baked the world’s longest loaf of cinnamon bread. His loaf measures \begin{align*}11 \frac{5}{8}\end{align*} feet. He cuts a piece that measures \begin{align*}1 \frac{1}{2}\end{align*} feet long, and gives it to his friend Pamela. He then cuts another piece \begin{align*}2 \frac{2}{3}\end{align*} feet long 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 \begin{align*}11 \frac{5}{8}\end{align*} feet long. Pamela gets a piece \begin{align*}1 \frac{1}{2}\end{align*} feet long, that is her piece and Serena gets a piece \begin{align*}2 \frac{2}{3}\end{align*} feet long. This is the 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 \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.
\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 \begin{align*}11 \frac{5}{8}\end{align*} and \begin{align*}1 \frac{1}{2}\end{align*}. Let’s use 8, so we solve \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*}
I can already see that I will have to convert the mixed numbers to improper fractions. Simplify again.
\begin{align*}\frac{243}{24}-\frac{64}{24} &= x\\ x &= \frac{179}{24}\end{align*}
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
Now let’s go back to the introduction problem and help Jesse with his peach problem.
Real Life Example Completed
Plenty of Peaches
Here is the original problem once again. Read it and underline any important information.
Jesse has decided to make peach pie for the bake sale. His recipe calls for \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 do the math in her head. She tells Jesse how much he owes and he pays her. Then she hands Jesse back the bag.
“How many pounds is this?” Jesse asks.
“You bought \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.
\begin{align*}6 \frac{1}{4}-2 \frac{1}{2}\end{align*}
These fractions have different denominators, so we need to rename them using a common denominator. The lowest common denominator here is 4.
\begin{align*}6 \frac{1}{4}\end{align*} is all set.
\begin{align*}2 \frac{1}{2}=2 \frac{2}{4}\end{align*}
We can rewrite the problem.
\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.
\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 \begin{align*}3 \frac{3}{4}\end{align*} pounds of peaches left over.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- 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
- Estimation
- finding an approximate answer
- Inverse Property of Addition
- when you add the inverse of a number, the answer is zero.
Technology Integration
Khan Academy Adding and Subtracting Fractions
James Sousa, Subtracting Mixed Numbers with Like Denominators
James Sousa, Subtract Mixed Numbers Using Improper Fractions
James Sousa, Subtraction of Mixed Numbers
Other Videos:
http://www.teachertube.com/viewVideo.php?video_id=141252 – This is a video on adding and subtracting fractions.
Time to 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*}
Directions: Solve for \begin{align*}x\end{align*}.
17. \begin{align*}x + \frac{2}{3} = \frac{5}{6}\end{align*}
18. \begin{align*}x - 1 \frac{1}{2} = 4\end{align*}
19. \begin{align*}2 \frac{1}{4} + x = 3 \frac{3}{4}\end{align*}
20. \begin{align*}x - \frac{7}{8} = 2 \frac{3}{4}\end{align*}
21. Ludmilla, Brent and Rudy have \begin{align*}8 \frac{5}{6}\end{align*} feet of taffy that they have to sell to raise money for the school drama club. Brent has already sold \begin{align*}3 \frac{2}{3}\end{align*} feet of taffy and Rudy plans to sell exactly \begin{align*}2 \frac{3}{4}\end{align*} feet. How much taffy does Ludmilla have to sell, if they sell all of the taffy?
22. Ron, Jung-Ho and Sarah have a lawn mowing business. Today they are cutting an enormous lawn. Sarah agrees to start and will mow \begin{align*}\frac{3}{8}\end{align*} of the lawn. Ron will only mow \begin{align*}\frac{1}{7}\end{align*} of the lawn, but he’s willing to work during the hottest time of the day. How much of the lawn is Jung-Ho responsible for mowing in order to complete the job?