1.2 Working With Fractions
- Page ID
- 152638
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By the end of this section, you will be able to:
- break down a number into prime factors
- add, subtract, multiply, divide, take reciprocals, and simplify fractions
By fraction, we mean an expression that has some guy divided by some other guy. As long as that second guy isn't zero. We're never allowed to divide by zero. If I ever catch you dividing by zero, you're grounded for a month.
A fraction is written \( \dfrac{a}{b} \), where \( b \neq 0 \). The top \(a\) is called the numerator, and the bottom \(b\) is called the denominator.
Clearly, working with fractions is intimately related to worrying about division. Sometimes a fraction is equivalent to a whole number, like how \( \dfrac{12}{3} = 4 \), in which case we say "12 is divisible by 3," or "3 divides evenly into 12." Sometimes it's not, like \( \dfrac{2}{37} \), where we would say "2 is not divisible by 37 and everything is terrible." Sometimes a fraction will even have a negative sign in it, and we already learned what happens when negatives come into division. For example,
\[ \frac{-6}{2} = -3, \quad \frac{6}{-2} = -3, \quad - \left( \frac{6}{2} \right) = -3 \notag \]
all turn out to be the same. In general...
For positive numbers \(a\) and \( b\), we have
\[ \frac{-a}{b} = \frac{a}{-b} = - \frac{a}{b}. \notag \]
Another vocabulary word we will need is reciprocal. The reciprocal of \(\dfrac{a}{b}\) is \(\dfrac{b}{a}\). Just flip him over. We'll come back to that later.
We need to be able to do arithmetic and algebra on fractions, and there are rules about what is legal to do when computing with fractions. Before we get into those, however, let's take a quick detour that will help us later.
Factors of Numbers
When you factor a number or expression, you are splitting it into pieces (also called factors) that you can multiply together to get the original number. For example, \(2 \cdot 12, 3 \cdot 8, 6 \cdot 4,\) and \( 2 \cdot 2 \cdot 2 \cdot 3 \) are all ways to break \(24\) down into factors. In the last part there, I broke it down all the way into pieces that were prime numbers (a prime number is only divisible by 1 and itself, like 2, 3, 5, 7, 11, 13, ...). The ability to recognize what factors are "hiding" inside a particular integer will make your life much easier!
Factor \(48\) all the way down to prime factors.
Solution
You can systematically break \(48\) into pieces and then break those pieces into pieces and then break those pieces...until we find:
\[ 48 = 2 \cdot 24 = 2 \cdot 2 \cdot 12 = 2 \cdot 2 \cdot 2 \cdot 6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \notag \]
- Find two numbers that multiply to give \(-27\).
- Factor \(51\).
- Find three pairs of numbers that multiply to give \(36\).
- Factor \(81\) all the way down to prime factors.
- Answer
-
- \((-3)(9)\) or \(3(-9)\).
- This one is sneaky! It "feels prime," but actually \( 51 = 3 \cdot 17 \). Did I get you? 51 is my favorite number.
- \( 36 = 3 \cdot 12 = 9 \cdot 4 = 6 \cdot 6 \) all work.
- \( 81 = 9 \cdot 9 = 3 \cdot 3 \cdot 3 \cdot 3 \).
Okay now we're ready to get to work...
Multiplying and Dividing Fractions
\[ \dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{a \cdot c}{b \cdot d} \quad \quad \text{ and } \quad \quad \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} = \dfrac{ad}{bc} \notag \]
Translation: To multiply two fractions, just multiply across the top and multiply across the bottom. That's it! To divide fractions, you leave the first as it is and multiply by the divisor's reciprocal instead. A fraction's reciprocal is its "multiplicative inverse," meaning if you multiply a fraction by its reciprocal, you will get 1.
These formulas also work backwards, such as \(\frac{a \cdot c}{b \cdot d} = \frac{a}{b} \cdot \frac{c}{d} \), and now that we know about factoring, we can use this fact to learn how to cancel. Consider the fraction \( \frac{35}{21} \). Using factoring, and the multiplication rule backwards, we can write
\[ \frac{35}{21} = \frac{5 \cdot 7}{3 \cdot 7} = \frac{5}{3} \cdot \frac{7}{7} = \frac{5}{3} \cdot 1 = \frac{5}{3}. \notag \]
Omg that's so much better! We call this fraction fully reduced because the numerator and denominator have no common factors left. Those 7s that 35 and 21 had in common came together to form a 1.
Cancelling is NOT a magic wand that you can wave and make numbers disappear. "I saw a 4 on a billboard once, so I'm gonna cancel this 4 right over here—" NO! Grounded. Straight to jail. You may NOT cancel in a fraction unless you can make the 1 appear as in the process above. I'm watching you.
Simplify the fractions to fully reduced form.
- \( \frac{16}{8} \)
- \( \frac{10}{15} \)
- \( \frac{-42}{48} \)
- \( \frac{26}{13} \)
- \( \frac{50}{-125} \)
- Answer
-
- \( 2 \)
- \( \frac{2}{3} \)
- \( - \frac{7}{8} \)
- \( 2 \)
- \( - \frac{2}{5} \)
Canceling as much as possible first can be very helpful before you do any real multiplying or dividing.
Compute \( \dfrac{12}{16} \cdot \dfrac{8}{18} \) and simplify.
Solution
I don't want to multiply \( 12 \cdot 8 \) in my head... Let's clean it up first.
\[\dfrac{12}{16} \cdot \dfrac{8}{18} = \dfrac{3\cdot \textcolor{red}{4}}{4 \cdot \textcolor{red}{4}} \cdot \dfrac{\textcolor{red}{2} \cdot 4}{\textcolor{red}{2} \cdot 9} = \dfrac{3}{4}\cdot \dfrac{4}{9} = \dfrac{3\cdot \textcolor{red}{4}}{\textcolor{red}{4}\cdot 9} = \dfrac{3}{9} = \dfrac{1}{3}. \notag \]
I've written out a lot of steps here to try to make clear what is happening, but we're going to start taking these skills for granted eventually, so you don't have to write this much detail if you don't need it!
Division involving fractions is more likely to appear in the form of compound fractions, or fractions within fractions, or Evil Condominiums of Fractions. They look like this:
\[ \frac{4}{\frac{3}{2}}, \quad \frac{\frac{1}{2}}{3}, \quad \frac{\frac{2}{25}}{\frac{4}{3}} \notag \]
To deal with these, remember it's just a big ol' division and multiply the top of the big fraction by the reciprocal of the bottom. Like this:
\[ \frac{4}{\textcolor{red}{\frac{3}{2}}} = 4 \cdot \textcolor{red}{\frac{2}{3}} = \frac{4}{1}\cdot {\frac{2}{3}} = \frac{8}{3}, \notag \]
\[ \frac{\frac{1}{2}}{-\textcolor{red}{3}} = -\frac{\frac{1}{2}}{\textcolor{red}{\frac{3}{1}}} = -\frac{1}{2} \cdot \textcolor{red}{\frac{1}{3}} = -\frac{1}{6}, \notag \]
\[ \frac{\frac{2}{25}}{\textcolor{red}{\frac{4}{3}}} = \frac{2}{25} \cdot \textcolor{red}{\frac{3}{4}} = \frac{2 \cdot 3}{25 \cdot 2 \cdot 2} = \frac{3}{50} \notag \]
Bring it all together with some exercises...
Compute and simplify.
1. \( \frac{3}{-4} \cdot \frac{-12}{27} \)
2. \( \dfrac{ \frac{2}{5}}{10} \)
3. \( \frac{15}{30} \cdot \frac{9}{12} \)
4. \( \dfrac{ \frac{1}{2}}{-\frac{3}{8}} \)
- Answer
-
- \( \frac{1}{3} \)
- \( \frac{1}{25} \)
- \( \frac{3}{8} \)
- \(- \frac{4}{3} \)
Adding and Subtracting Fractions
\[ \dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a+b}{c} \quad \quad \text{ and } \quad \quad \dfrac{a}{c} - \dfrac{b}{c} = \dfrac{a-b}{c} \notag \]
Translation: If you have the same denominator, you can combine the fractions by just adding or subtracting the numerators! These rules also work backwards, by the way. That is, you can split up a fraction over a plus in the numerator like so: \( \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} \). Watch out! This only works when the plus or minus is in the numerator!
Great, I hear you saying, fine and dandy and all, but what if I don't have the same denominator? Then you must bake one from scratch! Let's demonstrate with an example.
Add \( \dfrac{4}{9} + \dfrac{2}{12} \).
Solution
The denominators currently don't match! Here's the process:
- Factor the denominators into their prime factors: \(9 = 3 \cdot 3 \) and \( 12 = 2 \cdot 2 \cdot 3 \). These two numbers share a factor 3, but the 9 has another 3, while the 12 has two 2s.
- "Merge" the collections of factors into a set \(3, 3, 2, 2 \) that has all the factors they need. Notice that the number \( 2 \cdot 2 \cdot 3 \cdot 3 = 36 \) is divisible by both 9 and 12. This is called the least common multiple (LCM) of 9 and 12. It's the smallest number that both 9 and 12 divide cleanly into, which we like because
- Now that we have the LCM, we multiply top and bottom on each fraction by whatever is necessary to make the denominator match the LCM:
\[ \dfrac{4 \cdot 4}{9 \cdot 4} + \dfrac{2 \cdot 3}{12 \cdot 3} = \dfrac{16}{36} + \dfrac{6}{36} \notag \] - Now that the denominators do match, we just use the addition rule and simplify:
\[ \dfrac{16}{36} + \dfrac{6}{36} = \frac{16+6}{36} = \frac{22}{36} = \frac{2 \cdot 11}{2 \cdot 18} = \frac{11}{18}. \notag \]
Here's the summary and one more example, and then you can try some challenge exercises.
Getting a Common Denominator:
- Determine the prime factors of each of the denominators involved.
- Merge the collections of prime factors to find the LCM.
- Make each denominator match the LCM by multiplying the fraction top and bottom by whatever is needed.
- Now that there is a common denominator, add or subtract as usual.
Compute \( \dfrac{1}{3} \cdot \dfrac{9}{4} - \dfrac{5}{12} + \dfrac{3}{\frac{2}{7}} \) and simplify.
Solution
It looks scary, but we just have to start somewhere! Let's do that first term's multiplication and simplify.
\[ \textcolor{red}{ \dfrac{1}{3} \cdot \dfrac{9}{4}} - \dfrac{5}{12} + \dfrac{3}{\frac{2}{7}} = \textcolor{red}{\frac{1\cdot 3 \cdot 3}{3 \cdot 4}} - \dfrac{5}{12} + \dfrac{3}{\frac{2}{7}} = \textcolor{red}{ \frac{3}{4} }- \dfrac{5}{12} + \dfrac{3}{\frac{2}{7}} \notag \]
Now let's clean up that compound fraction too.
\[ = \frac{3}{4} - \dfrac{5}{12} + \textcolor{red}{3\cdot \frac{7}{2}} = \frac{3}{4} - \dfrac{5}{12} + \textcolor{red}{\frac{21}{2} } \notag \]
We need a common denominator now, and the LCM of 4, 12, and 2 is just 12. We make everybody match...
\[= \frac{3\textcolor{red}{\cdot 3}}{4 \textcolor{red}{\cdot 3}} - \dfrac{5}{12} + \frac{21\textcolor{red}{\cdot 6}}{2 \textcolor{red}{\cdot 6}} = \frac{9}{12} - \frac{5}{12} + \frac{126}{12} \notag \]
And finally do the addition and subtraction!
\[ = \frac{9-5+126}{12} = \frac{130}{12} = \frac{2\cdot 65}{2 \cdot 6} = \frac{65}{6} \notag \]
Compute and simplify.
1. \( \dfrac{-5}{15} \cdot \dfrac{3}{2} - \dfrac{1}{2} \)
2. \( \dfrac{\frac{6}{4}}{\frac{3}{12}} + 4 \cdot \dfrac{9}{16} \)
3. \( \dfrac{ \frac{2}{3} + \frac{1}{6}}{4} \)
4. \( \dfrac{1}{2} + \dfrac{1}{3} - \dfrac{1}{5} \)
- Answer
-
1. \( -1 \). Be careful with your negative signs: \( -\frac{1}{2} - \frac{1}{2} = \frac{-1 -1}{2} = ... \)
2. \( \frac{33}{4} \). If you got stuck with what to do with the \(6\), remember that \( 6 = \frac{6}{1} \) and you can get a common denominator.
3. \( \frac{5}{24} \). Clean up the numerator first and then worry about the big division downstairs.
4. \(\frac{19}{30} \).
Fraction Formats
Let's wrap up this section with a note on ways you can write fractions. When the numerator is a larger number than the denominator, such as \( \dfrac{7}{2} \), it's called an improper fraction. This is rude, because this format is generally the easiest to work with if you're going along doing math problems. I pretty much always leave fractions in this form.
You may be familiar with mixed numbers, which is a way of writing an improper fraction by figuring out how many "wholes" are included and pulling them out front. Like this: \(\dfrac{7}{2} = \dfrac{6+1}{2} = \dfrac{6}{2}+\dfrac{1}{2} = 3\frac{1}{2}\). I have "seven halves," but "six halves" is just a whole three, and there's one extra "half" left over. You see these written with a whole number squashed up next to a small fraction, and that displeases me philosophically since squashing things next to each other often means multiplication. I won't use this notation, BUT it's still a good idea to get your head around it, because it will be very helpful when a) working with decimals in about five seconds here and b) estimating numbers with our brains later.
Find the mixed number equivalent of improper fractions, and find the improper fraction equivalent of mixed numbers.
- \( \frac{3}{2} \)
- \( \frac{4}{3} \)
- \( \frac{11}{2} \)
- \( 1\frac{1}{4} \)
- \( 2\frac{1}{3} \)
- \( 5\frac{3}{4} \)
- Answer
-
- \( 1 \frac{1}{2} \)
- \( \frac{3+1}{3} = \frac{3}{3} + \frac{1}{3} = 1\frac{1}{3} \)
- \( 5\frac{1}{2} \)
- \( 1\frac{1}{4} = 1+\frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \)
- \( 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \)
- \( \frac{23}{4} \)