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Adding and Subtracting Fractions and Mixed Numbers

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Lesson Priority: VIP Knowledge

Pre-Algebra $\longrightarrow$
Decimals and Fractions $\longrightarrow$
Numerical Fractions
 
Objectives
  • Know how to add and subtract fractions when the denominators are equal
  • Know how to add and subtract fractions when the denominators are not equal
  • Know the most efficient way to add and subtract mixed numbers
Lesson Description

Adding and subtracting fractions is not very different than adding and subtracting integers. The only major difference is that it's only possible to combine fractions simply when the denominators are the same. Here we will look at the method of adding and subtracting fractions when the denominator is the same, and then look at how to proceed when the denominators are not the same. We'll also look at how to add and subtract mixed numbers.

 
Practice Problems

Practice problems and worksheet coming soon!

 

Less Friendly Numbers

When we reviewed integer arithmetic earlier in the Pre-Algebra course, we had the boon of knowing our underlying arithmetic facts from grammar school. Essentially, once we digested and practiced the positive and negative number properties, we were back to basic facts that we had already been drilling since grade 1. When it comes to adding and subtracting fractions, we have to follow more of a "process".

Preparation is the Key

If the situation is right, adding or subtracting two fractions is essentially identical to adding and subtracting integers. If the situation is not right, we need to make it right first.The key to adding and subtracting fractions is to make sure the denominators are the same before you add or subtract. If they are the same, you're ready to rock. If they are not the same, you need to make them be the same. Let's jump right into each case and see how it works.

Equal Denominators

We'll begin by looking at the case when the denominators are already the same. In this case, we will combine the numerators appropriately, keeping the same denominator.
Define: Fraction Addition and SubtractionTwo fractions with the same denominator can be added or subtracted, resulting in a single fraction with the same denominator, and a numerator that is the result of adding or subtracting the original numerators, respectively. Symbolically:$$\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}$$$$\frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}$$
Let's see how this works. In the following three examples, add or subtract as indicated.
 
Example 1$$\frac{3}{11} + \frac{4}{11}$$$$\blacktriangleright \,\, \frac{3}{11} + \frac{4}{11} = \frac{7}{11}$$
 
Example 2$$\frac{2}{3} - \frac{5}{3}$$$$\blacktriangleright \frac{2-5}{3} = \frac{-3}{3}$$$$=-1$$Notice how we had to simplify in this case. We must always simplify our answers if we can!
 
Example 3$$-\frac{5}{4} - \frac{9}{4}$$$\blacktriangleright$ Once again following along with the rule,$$ \frac{-5-9}{4} = \frac{-14}{4}$$$$=-\frac{7}{2}$$Again, we must simplify to ensure full credit in an exam setting.
You Should Know
Unless the instructions tell you to express your answer as a mixed number, you can assume that expressing your answers as improper fractions is the right choice. Improper fractions are generally easier to work with in Algebra and beyond. If you know your teacher is keen on having you frequently switch back and forth between mixed numbers and improper fractions, remember to review the lesson on Mixed Numbers and Improper Fractions ».

Different Denominators

The process of adding and subtracting fractions seems harmless enough for fractions that have the same denominator - but what do we do if we're combining two fractions with different denominators? Ultimately, we are going to change each fraction to an equivalent fraction so that the denominators match, and then proceed with the same-denominator process we just saw above.Recall from the recent lesson on fraction properties » that multiplying both the numerator and denominator of a fraction is the way we obtain equivalent fractions. But what equivalent fraction should we change to? In the case of adding and subtracting fractions, we not only need to change each fraction to a common denominator, but also identify what that common denominator is in the first place. We can quickly accomplish this goal using recently learned methods about finding least common multiples ».Since there are a few more things going on here than we saw with combining same-denominator fractions, let's break it down one step at a time.Step-by-StepStep 1 - Identify LCDThe Least Common Multiple of the denominators is the smallest, simplest denominator that we can convert each fraction to so that they equal one another, and it is the one we should use. We often call this the Least Common Denominator (akin to the phrase Least Common Multiple). The first step is to identify what that ideal denominator is.Step 2 - Convert FractionsOnce we know what denominator we should be converting each fraction to, we need to execute on that plan. Change each fraction to the desired denominator using equivalent fraction techniques.Step 3 - CombineAt this point, you have two same-denominator fractions that you want to combine. In other words, we're in the original simpler situation that we discussed in the first learning objective of this lesson. Proceed the way we previously defined, by combining the numerators and keeping the same denominator.Step 4 - Clean UpAlso similar to the process of combining same-denominator fractions, we must always check that our final answer is simplified, if it can be.Let's try an example.
 
Example 4$$\frac{6}{5} + \frac{3}{2}$$Step 1 - Identify LCDThe denominators in this problem are $5$ and $2$, and the LCM of $5$ and $2$ is $10$. Therefore, the least common denominator we want is $10$.Step 2 - Convert FractionsEach fraction needs to be converted to an equivalent form that has a denominator of $10$. Recall from what we've learned in the past about equivalent fractions » that we can convert a fraction to an equivalent one of a different denominator by multiplying by a form of the number $1$. Here, we will multiply the first fraction by $2/2$ and we will multiply the second fraction by $5/5$.$$\frac{6}{5} \cdot \frac{2}{2} = \frac{12}{10}$$$$\frac{3}{2} \cdot \frac{5}{5} = \frac{15}{10}$$Step 3 - CombineNow, having changed each fraction to have the same denominators, we can proceed like we did at the beginning of this lesson for combining fractions that have the same denominator.$$\frac{6}{5} + \frac{3}{2}$$$$\longrightarrow \frac{12}{10} + \frac{15}{10}$$$$ = \frac{12+15}{10}$$$$ = \boxed{\frac{27}{10}}$$Step 4 - Clean UpMany teachers will not give you full credit for your work unless you simplify your answer. It often happens that the result you get after combining in Step 3 can then be simplified. For this particular problem, it looks like $27/10$ cannot be simplified, since $27$ and $10$ do not have any common factors.Try this next example which, unlike the one we just looked at step-by-step, will require you to simplify the final answer.
 
Example 4$$\frac{1}{2} - \frac{1}{6}$$
Show solution
$\blacktriangleright$ Step 1The LCM of $2$ and $6$ is $6$, so $6$ is the least common denominator.Step 2$$\frac{1}{2} \cdot \frac{3}{3} = \frac{3}{6}$$$1/6$ already has the needed denominator.Step 3$$\frac{3}{6} - \frac{1}{6} = \frac{2}{6}$$Step 4$$\frac{2}{6} = \frac{1}{3}$$
You Should Know
It may seem odd and potentially wasteful to spend time changing fractions to equivalent fractions in order to add / subtract, only to end up again changing to equivalent forms when we're asked to simplify our final answer. However, aside from using a calculator that simplifies fractions for you, there is no magic shortcut to avoid this.

Combining Mixed Numbers

Finally, we need to be ready to add and subtract mixed numbers. Just like adding and subtracting fractions with differing denominators became manageable when we turned them into same-denominator fractions, so too will mixed numbers when we turn them into pure fractions.Recall from the recent lesson on Mixed Numbers and Improper Fractions » that we can quickly change between mixed number and improper fraction forms. In order to add or subtract mixed numbers, you should turn them into improper fractions first, and then proceed as before (depending on whether the denominators are the same).We will learn by example with the following two problems, by performing the indicated operations.
 
Example 5$$2 \, \frac{3}{8} + 3 \, \frac{7}{8}$$$\blacktriangleright$ While it is tempting with addition to try and keep the result in mixed number form, we'll always guarantee that we do the least amount of work if we convert to improper fractions. It also keeps your process consistent.$$2 \, \frac{3}{8} \longrightarrow \frac{19}{8}$$$$3 \, \frac{7}{8} \longrightarrow \frac{31}{8}$$$$\frac{19}{8} + \frac{31}{8} = \frac{50}{8}$$$$= \frac{25}{4}$$It's also acceptable to leave it in improper form for your final answer unless instructed otherwise.
 
Example 6$$5 \, \frac{5}{6} - 3 \, \frac{4}{9}$$$\blacktriangleright$ Beyond the need to rewrite this problem using improper fractions, we also have to obtain equivalent fractions with the least common denominator.$$5 \, \frac{5}{6} \longrightarrow \frac{35}{6}$$$$3 \, \frac{4}{9} \longrightarrow \frac{31}{9}$$So now we can write our problem as$$\frac{35}{6} - \frac{31}{9}$$A quick inspection leads us to the fact that our LCD is $18$.$$\frac{35}{6} \cdot \frac{3}{3} - \frac{31}{9} \cdot \frac{2}{2}$$$$\frac{105}{18} - \frac{62}{18}$$$$=\frac{49}{18}$$
Remember!
Unlike the optional choice to express your final answer as a mixed number or improper fraction when possible, the consistent arithmetic processes for mixed numbers are non-negotiable in their requirement to work with improper fractions. Regardless of your teacher's preference for expressing your final answers, you should work with improper fractions if you are going to add or subtract mixed numbers, and as we'll see in the next lesson », it's not just easier, but an absolute necessity with multiplication and division.

Navigating Negatives

Just as we often saw the need to subtract bigger numbers from smaller ones and get a negative results, so too will we need to be able to do this flawlessly with fractions. Fortunately, since fraction addition and subtraction boil down to adding and subtracting integer numerators, it really isn't any different than what we already know about adding and subtracting integers » from earlier in the course.
 
Example 7Subtract. Simplify if possible.$$\frac{4}{9} - \frac{8}{11}$$$\blacktriangleright$ Follow the steps: find a common denominator, convert each fraction to it, subtract, and simplify.Common denominator: $\mathrm{LCM(}9, \, 11\mathrm{)} = 99$$$\frac{4}{9} \cdot \frac{11}{11} = \frac{44}{99}$$$$\frac{8}{11} \cdot \frac{9}{9} = \frac{72}{99}$$$$\frac{44}{99} - \frac{72}{99} = \frac{44-72}{99}$$$$=\frac{-28}{99}$$No further simplifying is possible. The answer can be expressed as $\frac{-28}{99}$ or $-\frac{28}{99}$.
 
Example 8Subtract. State your answer as a mixed number.$$2 \, \frac{1}{6} - 4 \, \frac{1}{9}$$$\blacktriangleright$ Again, follow the steps. As tempting as it is to try and work with the integers and fractions separately, you will absolutely end up doing more work that way. Turn them both into improper fractions.$$2 \, \frac{1}{6} = \frac{13}{6}$$$$4 \, \frac{1}{9} = \frac{37}{9}$$Common denominator: $\mathrm{LCM(}6, \, 9\mathrm{)} = 18$$$\frac{13}{6} \cdot \frac{3}{3} = \frac{39}{18}$$$$\frac{37}{9} \cdot \frac{2}{2} = \frac{74}{18}$$$$\frac{39}{18} - \frac{74}{18} = \frac{39-74}{18}$$$$=-\frac{35}{18}$$Obtain our final answer by converting to a mixed number. This is only a necessary step because the instructions specifically asked us to do so.$$-\frac{35}{18} = -1 \, \frac{17}{18}$$And since $\frac{17}{18}$ cannot be simplified, $-1 \, \frac{17}{18}$ is our final answer.

Put It To The Test

For each of the following examples, perform the indicated arithmetic. Leave answers in improper fraction form for convenience.
 
Example 9$$\frac{4}{11} + \frac{23}{11}$$
Show solution
$\blacktriangleright$ Since this problem is looking at two fractions with the same denominator, we need only combine the numerators and check for potential simplification.$$\frac{4}{11} + \frac{23}{11} = \frac{4 + 23}{11}$$$$=\frac{27}{11}$$$27$ and $11$ have no common factors, so the answer cannot be simplified.
 
Example 10$$\frac{27}{4} - \frac{13}{4}$$
Show solution
$\blacktriangleright$ Once again, equal denominators mean we can get right to the answer by performing the arithmetic on the numerator.$$\frac{27}{4} - \frac{13}{4} = \frac{27-13}{4}$$$$=\frac{14}{4} = \frac{7}{2}$$Note that it was possible and necessary to simplify. You'll lose points if you don't!
 
Example 11$$\frac{7}{10} - \frac{11}{25}$$
Show solution
$\blacktriangleright$ This example requires us to convert our fractions to a common denominator so that we can subtract them. The best common denominator is the LCM of $10$ and $25$.Common denominator: $\mathrm{LCM(}10, \, 25\mathrm{)} = 50$$$\frac{7}{10} \cdot \frac{5}{5} = \frac{35}{50}$$$$\frac{11}{25} \cdot \frac{2}{2} = \frac{22}{50}$$Now we can subtract.$$\frac{35}{50} - \frac{22}{50} = \frac{13}{50}$$
 
Example 12$$5 \, \frac{7}{16} + 6 \, \frac{7}{12}$$
Show solution
$\blacktriangleright$ Again, while it's possible to try and work the addition of the whole numbers and fractions as two separate processes, the one-size-fits-all approach of following the usual steps and working with improper fractions will guarantee a smooth ride for any problem you ever tackle.$$5 \, \frac{7}{16} \longrightarrow \frac{87}{16}$$$$6 \, \frac{7}{12} \longrightarrow \frac{79}{12}$$Now we need a common denominator.Common denominator: $\mathrm{LCM(}16, \, 12\mathrm{)} = 48$$$\frac{87}{16} \cdot \frac{3}{3} = \frac{261}{48}$$$$\frac{79}{12} \cdot \frac{4}{4} = \frac{316}{48}$$Therefore,$$5 \, \frac{7}{16} + 6 \, \frac{7}{12} = \frac{261}{48} + \frac{316}{48}$$$$=\frac{577}{48}$$Obtain the final answer via long division.$$577 \div 48 = 12 \,\, \mathrm{R} \, 1$$$$\longrightarrow 5 \, \frac{7}{16} + 6 \, \frac{7}{12} = 12 \, \frac{1}{48}$$
 
Example 13$$\frac{11}{8} + \frac{15}{7}$$
Show solution
$\blacktriangleright$ Once again, we look to convert these fractions to have the same denominator.Common denominator: $\mathrm{LCM(}8, \, 7\mathrm{)} = 56$$$\frac{11}{8} \cdot \frac{7}{7} = \frac{77}{56}$$$$\frac{15}{7} \cdot \frac{8}{8} = \frac{120}{56}$$$$\frac{77}{56} + \frac{120}{56}$$$$=\frac{197}{56}$$
 
Example 14$$\frac{41}{60} - \frac{17}{24}$$
Show solution
$\blacktriangleright$ Convert the fractions to a common denominator.Common denominator: $\mathrm{LCM(}60, \, 24\mathrm{)} = 120$$$\frac{41}{60} \cdot \frac{2}{2} = \frac{82}{120}$$$$\frac{17}{24} \cdot \frac{5}{5} = \frac{85}{120}$$Now we can subtract.$$\frac{82}{120} - \frac{85}{120}$$$$=-\frac{3}{120}$$Finally, make sure you simplify! Don't risk losing points!$$=-\frac{1}{40}$$
 
Example 15Subtract. Leave your answer as an improper fraction.$$3 \, \frac{3}{4} - 4 \, \frac{13}{18}$$
Show solution
$\blacktriangleright$ Let's work with improper fractions entirely.$$3 \, \frac{3}{4} \longrightarrow \frac{15}{4}$$$$4 \, \frac{13}{18} \longrightarrow \frac{85}{18}$$Now we'll need a common denominator.Common denominator: $\mathrm{LCM(}4, \, 18\mathrm{)} = 36$$$\frac{15}{4} \cdot \frac{9}{9} = \frac{120}{36}$$$$\frac{85}{18} \cdot \frac{2}{2} = \frac{170}{36}$$Now we can subtract.$$\frac{120}{36} - \frac{170}{36}$$$$=-\frac{50}{36}$$Finally, as usual, simplify.$$=-\frac{25}{18}$$
Remember!
You must simplify your answers completely, if you like getting good grades. Teachers are universally adamant about this, and generally deduct a little off from each question in which you fail to do this. Incredibly strict teachers will mark the whole problem wrong (though this is admittedly rare in my travels).
 
Lesson Takeaways
  • Understand the mechanics of adding and subtracting fractions
  • Know what to do if you encounter different denominators or mixed numbers
  • Remember to simplify down every answer every time!
  • Deal with negative answers the same way you do for integers

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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