ORCCA Fractions and Fraction Arithmetic

Section 1.2 Fractions and Fraction Arithmetic

The word “fraction” comes from the Latin word fractio, which means “break into pieces.” For thousands of years, cultures from all over the world have used fractions to understand parts of a whole.

Subsection 1.2.1 Visualizing Fractions

Parts of a Whole.

One approach to understanding fractions is to think of them as parts of a whole.

In Figure 1.2.1, we see $1$ whole divided into $7$ parts. Since $3$ parts are shaded, we have an illustration of the fraction $\frac{3}{7}\text{.}$ The denominator $7$ tells us how many parts to cut up the whole; since we have $7$ parts, they're called “sevenths.” The numerator $3$ tells us how many sevenths to consider.

a rectangle that is seven times as wide as it is tall; the entire rectangle is shaded; there is a 1 in the center of the rectangle; to its right, it is labeled 'one whole'; another rectangle of the same size is aligned directly below the first rectangle; it is subdivided equally into seven squares; the first three squares are shaded; there is a 1/7 in the center of each of the first three squares; to its right, it is labeled 'three sevenths' — Figure 1.2.1. Representing $\frac{3}{7}$ as parts of a whole.

Checkpoint 1.2.2. A Fraction as Parts of a Whole.

We can also locate fractions on number lines. When ticks are equally spread apart, as in Figure 1.2.3, each tick represents a fraction.

Figure 1.2.3. Representing $\frac{3}{7}$ on a number line.

Checkpoint 1.2.4. A Fraction on a Number Line.

Division.

Fractions can also be understood through division.

For example, we can view the fraction $\frac{3}{7}$ as $3$ divided into $7$ equal parts, as in Figure 1.2.5. Just one of those parts represents $\frac{3}{7}\text{.}$

Figure 1.2.5. Representing $\frac{3}{7}$ on a number line.

Checkpoint 1.2.6. Seeing a Fraction as Division Arithmetic.

Subsection 1.2.2 Equivalent Fractions

It's common to have two fractions that represent the same amount. Consider $\frac{2}{5}$ and $\frac{6}{15}$ represented in various ways in Figures 1.2.7–1.2.9.

one rectangle that is five times as wide as it is tall; the entire rectangle is shaded; there is a 1 in the center of the rectangle; a second rectangle that is also five times as wide as it is tall; it is subdivided equally into five squares; the first two squares are shaded; there is a 1/5 in the center of each of the first two squares; a third rectangle that is also five times as wide as it is tall; it is subdivided equally into fifteen adjacent rectangles; the first six of these smaller rectangles are shaded; there is a 1/15 in the center of each of the first six smaller rectangles — Figure 1.2.7. $\frac{2}{5}$ and $\frac{6}{15}$ as equal parts of a whole

Figure 1.2.8. $\frac{2}{5}$ and $\frac{6}{15}$ as equal on a number line

Figure 1.2.9. $\frac{2}{5}$ and $\frac{6}{15}$ as equal results from division

Those two fractions, $\frac{2}{5}$ and $\frac{6}{15}$ are equal, as those figures demonstrate. Also, because they each equal $0.4$ as a decimal. If we must work with this number, the fraction that uses smaller numbers, $\frac{2}{5}\text{,}$ is preferable. Working with smaller numbers decreases the likelihood of making a human arithmetic error. And it also increases the chances that you might make useful observations about the nature of that number.

So if you are handed a fraction like $\frac{6}{15}\text{,}$ it is important to try to reduce it to “lowest terms.” The most important skill you can have to help you do this is to know the multiplication table well. If you know it well, you know that $6=2\cdot3$ and $15=3\cdot5\text{,}$ so you can break down the numerator and denominator that way. Both the numerator and denominator are divisible by $3\text{,}$ so they can be “factored out” and then as factors, cancel out.

\begin{align*} \frac{6}{15}\amp=\frac{2\cdot3}{3\cdot5}\\ \amp=\frac{2\cdot\cancelhighlight{3}}{\cancelhighlight{3}\cdot5}\\ \amp=\frac{2\cdot1}{1\cdot5}\\ \amp=\frac{2}{5} \end{align*}

Checkpoint 1.2.10.

Sometimes it is useful to do the opposite of reducing a fraction, and build up the fraction to use larger numbers.

Checkpoint 1.2.11.

Subsection 1.2.3 Multiplying with Fractions

Example 1.2.12.

Suppose a recipe calls for $\frac{2}{3}$ cup of milk, but we’d like to quadruple the recipe (make it four times as big). We'll need four times as much milk, and one way to measure this out is to fill a measuring cup to $\frac{2}{3}$ full, four times:

four measuring cups, each filled two-thirds with liquid

When you count up the shaded thirds, there are eight of them. So multiplying $\frac{2}{3}$ by the whole number $4\text{,}$ the result is $\frac{8}{3}\text{.}$ Mathematically:

\begin{align*} 4\cdot\frac{2}{3}\amp=\frac{4\cdot2}{3}\\ \amp=\frac{8}{3} \end{align*}

Fact 1.2.13. Multiplying a Fraction and a Whole Number.

When you multiply a whole number by a fraction, you may just multiply the whole number by the numerator and leave the denominator alone. In other words, as long as $d$ is not $0\text{,}$ then a whole number and a fraction multiply this way:

\begin{equation*} a \cdot \frac{c}{d} = \frac{a\cdot c}{d} \end{equation*}

Example 1.2.14.

We could also use multiplication to decrease amounts. Suppose we needed to cut the recipe down to just one fifth. Instead of four of the $\frac{2}{3}$ cup milk, we need one fifth of the $\frac{2}{3}$ cup milk. So instead of multiplying by $4\text{,}$ we multiply by $\frac{1}{5}\text{.}$ But how much is $\frac{1}{5}$ of $\frac{2}{3}$ cup?

If we cut the measuring cup into five equal vertical strips along with the three equal horizontal strips, then in total there are $3\cdot5=15$ subdivisions of the cup. Two of those sections represent $\frac{1}{5}$ of the $\frac{2}{3}$ cup.

a measuring cup divided evenly into three horizontal sections, but also evenly into five vertical sections; in total, there are fifteen sections; the two that correspond to the lower two-thirds and leftmost one-fifth are shaded

In the end, we have $\frac{2}{15}$ of a cup. The denominator $15$ came from multiplying $5$ and $3\text{,}$ the denominators of the fractions we had to multiply. The numerator $2$ came from multiplying $1$ and $2\text{,}$ the numerators of the fractions we had to multiply.

\begin{align*} \frac{1}{5}\cdot\frac{2}{3}\amp=\frac{1\cdot2}{5\cdot3}\\ \amp=\frac{2}{15} \end{align*}

Fact 1.2.15. Multiplication with Fractions.

As long as $b$ and $d$ are not $0\text{,}$ then fractions multiply this way:

\begin{equation*} \frac{a}{b} \cdot \frac{c}{d} = \frac{a\cdot c}{b \cdot d} \end{equation*}

Checkpoint 1.2.16. Fraction Multiplication Practice.

Subsection 1.2.4 Division with Fractions

How does division with fractions work? Are we able to compute/simplify each of these examples?

$\displaystyle 3\div\frac{2}{7}$
$\displaystyle \frac{18}{19}\div5$
$\displaystyle \frac{14}{3}\div\frac{8}{9}$
$\displaystyle \frac{\ \frac{2}{5}\ }{\frac{5}{2}}$

We know that when we divide something by $2\text{,}$ this is the same as multiplying it by $\frac{1}{2}\text{.}$ Conversely, dividing a number or expression by $\frac{1}{2}$ is the same as multiplying by $\frac{2}{1}\text{,}$ or just $2\text{.}$ The more general property is that when we divide a number or expression by $\frac{a}{b}\text{,}$ this is equivalent to multiplying by the reciprocal $\frac{b}{a}\text{.}$

Fact 1.2.17. Division with Fractions.

As long as $b\text{,}$ $c$ and $d$ are not $0\text{,}$ then division with fractions works this way:

\begin{equation*} \frac{a}{b} \div \frac{c}{d} = \frac{a}{b}\cdot\frac{d}{c} \end{equation*}

Example 1.2.18.

With our examples from the beginning of this subsection:

$\displaystyle \begin{aligned}[t] 3\div\frac{2}{7}\amp=3\multiplyright{\frac{7}{2}}\\ \amp=\divideunder{3}{1}\cdot\frac{7}{2}\\ \amp=\frac{21}{2}\\ \ \end{aligned} $
$\displaystyle \begin{aligned}[t] \frac{18}{19}\div5\amp=\frac{18}{19}\div\divideunder{5}{1}\\ \amp=\frac{18}{19}\multiplyright{\frac{1}{5}}\\ \amp=\frac{18}{95}\\ \ \end{aligned} $
$\displaystyle \begin{aligned}[t] \frac{14}{3}\div\frac{8}{9}\amp=\frac{14}{3}\multiplyright{\frac{9}{8}}\\ \amp=\frac{14}{\highlight{1}}\cdot\frac{\highlight{3}}{8}\\ \amp=\frac{\highlight{7}}{1}\cdot\frac{3}{\highlight{4}}\\ \amp=\frac{21}{4} \end{aligned} $
$\displaystyle \begin{aligned}[t] \frac{\ \frac{2}{5}\ }{\frac{5}{2}}\amp=\frac{2}{5}\highlight{{}\div{}}\frac{5}{2}\\ \amp=\frac{2}{5}\multiplyright{\frac{2}{5}}\\ \amp=\frac{4}{25}\\ \ \end{aligned} $

Checkpoint 1.2.19. Fraction Division Practice.

Subsection 1.2.5 Adding and Subtracting Fractions

In this section, we will review how to add and subtract numerical fractions, both in the case of having the same denominator and in the case of having different denominators.

With whole numbers and integers, operations of addition and subtraction are relatively straightforward. The situation is almost as straightforward with fractions if the two fractions have the same denominator. Consider

\begin{equation*} \frac{7}{2} + \frac{3}{2} = 7\text{ halves}+3\text{ halves} \end{equation*}

In the same way that $7$ tacos and $3$ tacos make $10$ tacos, we have:

\begin{equation*} \begin{array}{ccccc} 7\text{ halves}{}\amp+{}\amp 3\text{ halves}\amp=\amp 10\text{ halves} \\ \frac{7}{2}{}\amp+{}\amp\frac{3}{2}\amp=\amp\frac{10}{2} \\ \amp \amp \amp=\amp5 \\ \end{array} \end{equation*}

Fact 1.2.20. Adding/Subtracting with Fractions Having the Same Denominator.

To add or subtract two fractions having the same denominator, keep that denominator, and add or subtract the numerators.

\begin{align*} \frac{a}{b}+\frac{c}{b}\amp=\frac{a+c}{b}\amp\frac{a}{b}-\frac{c}{b}\amp=\frac{a-c}{b} \end{align*}

If it's possible, useful, or required of you, simplify the result by reducing to lowest terms.

Checkpoint 1.2.21. Fraction Addition/Subtraction Practice.

Whenever we'd like to combine fractional amounts that don't represent the same number of parts of a whole (that is, when the denominators are different), finding sums and differences is more complicated.

Example 1.2.22. Quarters and Dimes.

Find the sum $\frac{3}{4}+\frac{2}{10}\text{.}$ Does this seem intimidating? Consider this:

$\frac{1}{4}$ of a dollar is a quarter, and so $\frac{3}{4}$ of a dollar is $75$ cents.
$\frac{1}{10}$ of a dollar is a dime, and so $\frac{2}{10}$ of a dollar is $20$ cents.

So if you know what to look for, the expression $\frac{3}{4}+\frac{2}{10}$ is like adding $75$ cents and $20$ cents, which gives you $95$ cents. As a fraction of one dollar, that is $\frac{95}{100}\text{.}$ So we can report

\begin{equation*} \frac{3}{4}+\frac{2}{10}=\frac{95}{100}\text{.} \end{equation*}

(Although we should probably reduce that last fraction to $\frac{19}{20}\text{.}$)

This example was not something you can apply to other fraction addition situations, because the denominators here worked especially well with money amounts. But there is something we can learn here. The fraction $\frac{3}{4}$ was equivalent to $\frac{75}{100}\text{,}$ and the other fraction $\frac{2}{10}$ was equivalent to $\frac{20}{100}\text{.}$ These equivalent fractions have the same denominator and are therefore “easy” to add. What we saw happen was:

\begin{align*} \frac{3}{4}+\frac{2}{10}\amp=\frac{75}{100}+\frac{20}{100}\\ \amp=\frac{95}{100} \end{align*}

This realization gives us a strategy for adding (or subtracting) fractions. However, first let's review how to find the least common denominator.

Finding the Least Common Denominator.

When adding or subtracting fractions with different denominators, it is necessary to first find a common denominator. Usually, we want to find the least common denominator (LCD), which is the smallest common multiple of each denominator.

It is common to refer to the process of finding the LCD as "building" the LCD. The reason for this is that a common method for finding the LCD, which will be the most useful method to use when we get to more complicated fractions later in the course, is to use the prime factors of each denominator as building blocks to build the LCD. These building blocks will eventually be multiplied together to form the LCD.

Definition 1.2.23. Prime Factor.

A prime factor of a number is a prime number that divides into that number evenly. To be a prime number means that, when the number is factored using only positive integers, it can only be factored into 1 times itself.

Example 1.2.24.

Consider the previous example Example 1.2.22, in which we added the fractions $\frac{3}{4}$ and $\frac{2}{10}\text{.}$ Notice that the common denominator used in that example was $100\text{.}$ However, that is not the least common denominator. Find the least common denominator (LCD) of these fractions, using the prime factors to build the LCD.

Explanation

To find the least common denominator with this method, we must first find the prime factors of each denominator, $4$ and $10\text{.}$ This can be done by first factoring each number into any two factors. Then, if any of those factors are not prime (we call them composite in that case), split each of those composite factors into two factors. Keep going with this process until all factors are prime.

We can factor $4$ into the factors $2$ and $2\text{.}$ Notice that both of these factors are prime, so $4$ factors into $2\cdot2\text{,}$ when we write it as a product of its prime factors. Note that we call this the prime factorization of $4\text{.}$

We can factor $10$ into the factors $2$ and $5\text{.}$ Notice that both of these factors are prime, so $10$ factors into $2\cdot5\text{,}$ when we write it as a product of its prime factors.

Now we have the building blocks we need to build the LCD of the fractions $\frac{3}{4}$ and $\frac{2}{10}\text{.}$ One way to do this is to first include all the prime factors of the first denominator, which in this case will be $2$ and $2\text{,}$ and then take only those prime factors of the second denominator that we don't already have as a result of the previous step.

Thus, we would only need to take the factor of $5$ from $10\text{,}$ since we already have a factor of $2$ from the first denominator. The last step is to multiply all of those prime factors together.

Therefore, the LCD of the fractions $\frac{3}{4}$ and $\frac{2}{10}$ is $2\cdot2\cdot5=20\text{.}$ Notice that this is smaller than the common denominator of $100$ that was used previously.

Let's look at another example of finding the LCD.

Example 1.2.25.

Find the LCD of the fractions $\frac{7}{18}$ and $\frac{1}{54}\text{.}$

Explanation

To find the LCD, we must first find the prime factors of $18$ and $54\text{.}$ For $18\text{,}$ we can first factor it as $3\cdot6\text{.}$ Notice that $3$ is prime, but $6$ is not. So, we need to factor $6$ further. We can factor $6$ into $2\cdot3$ and both of these factors are prime. Thus, the prime factorization of $18$ is $2\cdot3\cdot3\text{.}$

We can factor $54$ into $6\cdot9$ and neither $6$ nor $9$ are prime. We can factor $6$ into $2\cdot3$ and $9$ into $3\cdot3\text{;}$ all of these factors are prime. Thus, the prime factorization of $54$ is $2\cdot3\cdot3\cdot3\text{.}$

Note that you could have started each factorization differently, like using $2\cdot27$ for $54\text{,}$ but you will eventually end up with the same prime factors in the end.

Now we have the prime factors we need to build our LCD. Using the method mentioned before, we will first take all of the factors from our first denominator of $18\text{,}$ which gives us factors of $2\text{,}$ $3\text{,}$ and $3\text{.}$ Next, we take only the factors of $54$ that we don't already have from our previous step. Notice that we already have the $2$ and two of the three $3$'s. Thus, we need only take one more factor of $3$ to ensure that all the prime factors of $54$ are represented in the LCD.

Therefore, the LCD of $\frac{7}{18}$ and $\frac{1}{54}$ is $2\cdot3\cdot3\cdot3=54\text{.}$

Remark 1.2.26.

You probably noticed in the last example that the LCD turned out to be one of the original denominators. This will always be the case whenever one denominator is a multiple of the other. So, you can save yourself some time if you notice this relationship (or check for it) beforehand.

In summary, here are the steps for finding the LCD of two fractions:

List 1.2.27. Finding the LCD of two fractions

Factor each denominator into its prime factors.
Take all prime factors from the first denominator.
Take only the prime factors from the second denominator that haven't already been taken from the first denominator.
Multiply together all the prime factors which were taken in the last two steps. The result will be the LCD of the two fractions.

Now let's use this skill as we review how to add or subtract fractions with different denominators.

Fact 1.2.28. Adding/Subtracting Fractions with Different Denominators.

To add (or subtract) fractions, use their denominators to find a common denominator; usually the least common denominator is preferred. Then, rewrite the fractions as equivalent fractions that use this common denominator. Write the result keeping that denominator and adding (or subtracting) the numerators. Reduce the fraction, if that is useful or required.

Example 1.2.29.

Let's add $\frac{2}{3}+\frac{2}{5}\text{.}$ The denominators are $3$ and $5\text{,}$ so the number $15$ would be a good common denominator. In fact, since both denominators are prime, their product is the LCD.

\begin{align*} \frac{2}{3}+\frac{2}{5}\amp=\frac{2\multiplyright{5}}{3\multiplyright{5}}+\frac{2\multiplyright{3}}{5\multiplyright{3}}\\ \amp=\frac{10}{15}+\frac{6}{15}\\ \amp=\frac{16}{15} \end{align*}

Notice how we found equivalent fractions. Both fractions needed to have a denominator of $15\text{,}$ so we multiplied the numerator and denominator of each fraction by whatever factor was necessary for the resulting denominator to be $15\text{.}$

Checkpoint 1.2.30. Using Some Flour.

Subsection 1.2.6 Mixed Numbers and Improper Fractions

A simple recipe for bread contains only a few ingredients:

$1\,\sfrac{1}{2}$	tablespoons yeast
$1\,\sfrac{1}{2}$	tablespoons kosher salt
$6\,\sfrac{1}{2}$	cups unbleached, all-purpose flour (more for dusting)

Each ingredient is listed as a mixed number that quickly communicates how many whole amounts and how many parts are needed. It's useful for quickly communicating a practical amount of something you are cooking with, measuring on a ruler, purchasing at the grocery store, etc. But it causes trouble in an algebra class. The number $1\,\sfrac{1}{2}$ means “one and one half.” So really,

\begin{equation*} 1\,\frac{1}{2}=1+\frac{1}{2} \end{equation*}

The trouble is that with $1\,\sfrac{1}{2}\text{,}$ you have two numbers written right next to each other. Normally with two math expressions written right next to each other, they should be multiplied, not added. But with a mixed number, they should be added.

Fortunately we just reviewed how to add fractions. If we need to do any arithmetic with a mixed number like $1\,\sfrac{1}{2}\text{,}$ we can treat it as $1+\frac{1}{2}$ and simplify to get a “nice” fraction instead: $\frac{3}{2}\text{.}$ A fraction like $\frac{3}{2}$ is called an improper fraction because it's actually larger than $1\text{.}$ A proper fraction represents a number that is smaller than one, so its numerator is smaller than its denominator.

\begin{align*} 1\,\frac{1}{2}\amp=1+\frac{1}{2}\\ \amp=\frac{1}{1}+\frac{1}{2}\\ \amp=\frac{2}{2}+\frac{1}{2}\\ \amp=\frac{3}{2} \end{align*}

Subsection 1.2.7 More on Operations with Mixed Numbers

To multiply and divide with mixed numbers,

Convert each mixed number to an improper fraction.
Perform the multiplication or division, using the rules previously discussed.
Simplify, if possible, which may include rewriting the result as a mixed number.

Example 1.2.31. Multiplication and Division with Mixed Numbers.

Here are some examples of multiplicaiton and division involving mixed numbers.

$-3\,\frac{2}{5}\cdot4\,\frac{1}{3}$

First, we convert each mixed number to an improper fraction:

\begin{equation*} -3\,\frac{2}{5}\cdot4\,\frac{1}{3}=-\frac{17}{5}\cdot \frac{13}{3} \end{equation*}

Then, we multiply. Note that these numbers, have opposite signs, so the final result will be negative.

\begin{equation*} -\frac{17}{5}\cdot \frac{13}{3}=-\frac{221}{15} \end{equation*}

Note that this fraction cannot be reduced, but it is an improper fraction, and since the original numbers were given as mixed numbers, it would be appropriate to now convert the final answer into a mixed number.

\begin{equation*} -\frac{221}{15}=-14\,\frac{11}{15} \end{equation*}

Therefore, $-3\,\frac{2}{5}\cdot4\,\frac{1}{3}=-14\,\frac{11}{15}$
$-5\div-3\,\frac{4}{7}$

This time, the numbers are both negative, so the result will be positive. We start by converting each number to fraction form:

\begin{equation*} -5\div-3\,\frac{4}{7}=-\frac{5}{1}\div-\frac{25}{7} \end{equation*}

Next, we rewrite the division as multiplication by the reciprocal of $-\frac{25}{7}\text{:}$

\begin{equation*} -\frac{5}{1}\div-\frac{25}{7}=-\frac{5}{1}\cdot-\frac{7}{25} \end{equation*}

Now we perform the multiplication, which keep in mind will give us a positive result, and then simplify:

\begin{align*} -\frac{5}{1}\cdot-\frac{7}{25}\amp=\frac{7}{5}\\ \amp=1\,\frac{2}{5} \end{align*}

Therefore, $-5\div-3\,\frac{4}{7}=1\,\frac{2}{5}\text{.}$

Checkpoint 1.2.32.

To add and subtract mixed numbers, it is not necessary to first convert the numbers to improper fractions. However, as with fractions, one must have common denominators before adding or subtracting mixed numbers.

To add mixed numbers with common denominators:

Add the whole numbers.
Add the fractions.
Simplify, if possible.

Example 1.2.33.

Add: $3\,\frac{7}{9}+2\,\frac{5}{9}\text{.}$

Explanation

First, we add the whole numbers: $3+2=5\text{.}$ Then, we add the fractions: $\frac{7}{9}+\frac{5}{9}=\frac{12}{9}\text{.}$

The last step is to simplify, which means we need to reduce the fraction, if possible. However, it also means that if the fraction turns out to be improper, we must convert it to a mixed number and then combine the whole number part to the result of adding the original whole numbers.

The simplification steps, including the whole numbers, can be shown as follows:

\begin{align*} 5\,\frac{12}{9}\amp=5\,\frac{4}{3}\\ \amp=5+1\,\frac{1}{3}\\ \amp=6\,\frac{1}{3}\text{.} \end{align*}

Therefore, $3\,\frac{7}{9}+2\,\frac{5}{9}= 6\,\frac{1}{3}\text{.}$

Checkpoint 1.2.34.

To subtract mixed numbers with a common denominator:

It's usually easiest to rewrite the problem in vertical form.
Compare the two fractions. If the top fraction is larger than the bottom fraction, then skip the next step and go on to the third step. Otherwise, borrow $1$ from the whole number in the top mixed number and add it to the fraction part, making it an improper fraction part.
Subtract the bottom fraction from the new top fraction.
Subtract the whole numbers.
Simplify, if possible.

Example 1.2.35.

Subtract: $8\,\frac{2}{7}-3\,\frac{4}{7}\text{.}$

Explanation

We'll start by writing the problem in vertical form:

\begin{equation*} \begin{array}{r} 8{\textstyle{2 \over 7}}\\ \underline { - 3{\textstyle{4 \over 7}}} \end{array} \end{equation*}

Notice that the top fraction is smaller than the bottom fraction. This means that we will need to borrow $1$ from the top whole number, $8\text{,}$ and add it to the top fraction, making it an improper fraction. Then, we subtract the bottom fraction from the new top fraction, as well subtract the whole numbers, as shown below:

\begin{equation*} \underline{ \begin{array}{r} 7 + 1 + {\textstyle{2 \over 7}}\\ - 3{\textstyle{4 \over 7}} \end{array} } \end{equation*}

\begin{equation*} \underline{ \begin{array}{r} 7 + {\textstyle{7 \over 7}} + {\textstyle{2 \over 7}}\\ - 3{\textstyle{4 \over 7}} \end{array} } \end{equation*}

\begin{equation*} \begin{array}{r} 7{\textstyle{9 \over 7}}\\ \underline { - 3{\textstyle{4 \over 7}}} \\ 4{\textstyle{5 \over 7}} \end{array} \end{equation*}

Since the fraction is already finished, we are finished. Therefore, $8\,\frac{2}{7}-3\,\frac{4}{7}=4\,\frac{5}{7}\text{.}$

Checkpoint 1.2.36.

To add or subtract mixed numbers that have fraction parts with different denominators, here is one approach:

Convert each mixed number to an improper fraction.
Find the least common denominator.
Rewrite the fractions as equivalent fractions that use this common denominator.
Add or subtract the new fractions by adding or subtracting the two numerators to obtain the numerator of the result and keeping the common denominator as the denominator of the result.
Simplify, if possible, including rewriting the result as a mixed number whenever appropriate.

Example 1.2.37.

Here are some examples involving addition or subtraction of mixed numbers with different denominators:

\begin{align*} 3\,\frac{1}{2}+5\,\frac{2}{3}\amp=\frac{7}{2}+\frac{17}{3}\\ \amp=\multiplyleft{\frac{3}{3}}\frac{7}{2}+\frac{17}{3}\multiplyright{\frac{2}{2}}\\ \amp=\frac{21}{6}+\frac{34}{6}\\ \amp=\frac{55}{6}\\ \amp=9\,\frac{1}{6} \end{align*}
\begin{align*} 2\,\frac{5}{6}-5\,\frac{3}{4}\amp=\frac{17}{6}-\frac{23}{4}\\ \amp=\multiplyleft{\frac{2}{2}}\frac{17}{6}-\frac{23}{4}\multiplyright{\frac{3}{3}}\\ \amp=\frac{34}{12}-\frac{69}{12}\\ \amp=-\frac{35}{12}\\ \amp=-2\,\frac{11}{12} \end{align*}
\begin{align*} -4\,\frac{1}{9}-7\,\frac{2}{3}\amp=-\frac{37}{9}-\frac{23}{3}\\ \amp=-\frac{37}{9}-\frac{23}{3}\multiplyright{\frac{3}{3}}\\ \amp=-\frac{37}{9}-\frac{69}{9}\\ \amp=-\frac{106}{9}\\ \amp=-11\,\frac{7}{9} \end{align*}
\begin{align*} -8\,\frac{1}{5}+2\,\frac{2}{7}\amp=-\frac{41}{5}+\frac{16}{7}\\ \amp=\multiplyleft{\frac{7}{7}}-\frac{41}{5}+\frac{16}{7}\multiplyright{\frac{5}{5}}\\ \amp=-\frac{287}{35}+\frac{80}{35}\\ \amp=-\frac{207}{35}\\ \amp=-5\,\frac{32}{35} \end{align*}

Exercises 1.2.8 Exercises

Review and Warmup

1.

Which letter is -21/4 on the number line?

2.

Which letter is 27/4 on the number line?

3.

The dot in the graph can be represented by what fraction?

4.

The dot in the graph can be represented by what fraction?

5.

The dot in the graph can be represented by what fraction?

6.

The dot in the graph can be represented by what fraction?

Reducing Fractions

7.

Reduce the fraction $\displaystyle{ \frac{2}{10} }\text{.}$

8.

Reduce the fraction $\displaystyle{ \frac{7}{42} }\text{.}$

9.

Reduce the fraction $\displaystyle{ \frac{21}{56} }\text{.}$

10.

Reduce the fraction $\displaystyle{ \frac{28}{49} }\text{.}$

11.

Reduce the fraction $\displaystyle{ \frac{140}{315} }\text{.}$

12.

Reduce the fraction $\displaystyle{ \frac{24}{42} }\text{.}$

13.

Reduce the fraction $\displaystyle{ {{\frac{22}{39}}} }\text{.}$

14.

Reduce the fraction $\displaystyle{ {{\frac{22}{39}}} }\text{.}$

15.

Reduce the fraction $\displaystyle{ \frac{120}{75} }\text{.}$

Building Fractions

16.

Find an equivalent fraction to $\frac{2}{3}$ with denominator $15\text{.}$

17.

Find an equivalent fraction to $\frac{1}{4}$ with denominator $20\text{.}$

18.

Find an equivalent fraction to $\frac{5}{9}$ with denominator $18\text{.}$

19.

Find an equivalent fraction to $\frac{10}{11}$ with denominator $22\text{.}$

Multiplying/Dividing Fractions

20.

Multiply: $\displaystyle{ \frac{1}{9} \cdot \frac{1}{9} }$

21.

Multiply: $\displaystyle{ \frac{3}{10} \cdot \frac{3}{7} }$

22.

Multiply: $\displaystyle{ \frac{13}{11} \cdot \frac{10}{13} }$

23.

Multiply: $\displaystyle{ \frac{15}{7} \cdot \frac{5}{9} }$

24.

Multiply: $\displaystyle{10\cdot \frac{1}{3} }$

25.

Multiply: $\displaystyle{7\cdot \frac{1}{9} }$

26.

Multiply: $\displaystyle{-\frac{2}{5} \cdot \frac{3}{14}}$

27.

Multiply: $\displaystyle{-\frac{10}{7} \cdot \frac{5}{6}}$

28.

Multiply: $\displaystyle{45\cdot\left( -{\frac{5}{9}} \right)}$

29.

Multiply: $\displaystyle{21\cdot\left( -{\frac{6}{7}} \right)}$

30.

Multiply: $\displaystyle{ {{\frac{6}{25}}} \cdot {{\frac{7}{4}}} \cdot {{\frac{5}{49}}} }$

31.

Multiply: $\displaystyle{ {{\frac{15}{49}}} \cdot {{\frac{2}{25}}} \cdot {{\frac{7}{4}}} }$

32.

Multiply: $\displaystyle{ {{\frac{3}{7}}} \cdot {{\frac{2}{9}}} \cdot {35} }$

33.

Multiply: $\displaystyle{ {{\frac{15}{2}}} \cdot {{\frac{1}{9}}} \cdot {14} }$

34.

Divide: $\displaystyle{ \frac{1}{8} \div \frac{5}{3} }$

35.

Divide: $\displaystyle{ \frac{2}{9} \div \frac{7}{5} }$

36.

Divide: $\displaystyle{ \frac{1}{6} \div \left(-\frac{7}{4}\right) }$

37.

Divide: $\displaystyle{ \frac{1}{12} \div \left(-\frac{4}{9}\right) }$

38.

Divide: $\displaystyle{-\frac{8}{5} \div (-6) }$

39.

Divide: $\displaystyle{-\frac{16}{9} \div (-12) }$

40.

Divide: $\displaystyle{35 \div \frac{7}{5} }$

41.

Divide: $\displaystyle{8 \div \frac{4}{5} }$

Adding/Subtracting Fractions

42.

Add: $\displaystyle{\frac{19}{24} + \frac{1}{24}}$

43.

Add: $\displaystyle{\frac{1}{12} + \frac{1}{12}}$

44.

Add: $\displaystyle{\frac{2}{7} + \frac{1}{28}}$

45.

Add: $\displaystyle{\frac{2}{7} + \frac{3}{28}}$

46.

Add: $\displaystyle{\frac{3}{8} + \frac{5}{24}}$

47.

Add: $\displaystyle{\frac{3}{10} + \frac{19}{20}}$

48.

Add: $\displaystyle{\frac{4}{9} + \frac{1}{6}}$

49.

Add: $\displaystyle{\frac{3}{10} + \frac{1}{8}}$

50.

Add: $\displaystyle{\frac{1}{6} + \frac{1}{10}}$

51.

Add: $\displaystyle{\frac{3}{10} + \frac{1}{6}}$

52.

Add: $\displaystyle{\frac{5}{6} + \frac{9}{10}}$

53.

Add: $\displaystyle{\frac{4}{5} + \frac{7}{10}}$

54.

Add: $\displaystyle{-\frac{3}{7} + \frac{6}{7}}$

55.

Add: $\displaystyle{-\frac{3}{11} + \frac{4}{11}}$

56.

Add: $\displaystyle{-\frac{2}{9} + \frac{25}{36}}$

57.

Add: $\displaystyle{-\frac{4}{9} + \frac{1}{54}}$

58.

Add: $\displaystyle{-\frac{5}{6} + \frac{1}{9}}$

59.

Add: $\displaystyle{-\frac{1}{10} + \frac{1}{9}}$

60.

Add: $\displaystyle{ -5 + \frac{1}{8}}$

61.

Add: $\displaystyle{ -4 + \frac{3}{5}}$

62.

Add: $\displaystyle{ {{\frac{1}{4}}} + {{\frac{1}{3}}} + {{\frac{1}{6}}} }$

63.

Add: $\displaystyle{ {{\frac{1}{3}}} + {{\frac{1}{6}}} + {{\frac{3}{8}}} }$

64.

Add: $\displaystyle{ {{\frac{7}{10}}} + {{\frac{4}{9}}} + {{\frac{1}{6}}} }$

65.

Add: $\displaystyle{ {{\frac{5}{6}}} + {{\frac{3}{4}}} + {{\frac{2}{3}}} }$

66.

Subtract: $\displaystyle{\frac{23}{36} - \frac{7}{36}}$

67.

Subtract: $\displaystyle{\frac{21}{20} - \frac{19}{20}}$

68.

Subtract: $\displaystyle{\frac{4}{7} - \frac{3}{28}}$

69.

Subtract: $\displaystyle{\frac{1}{8} - \frac{43}{48}}$

70.

Subtract: $\displaystyle{\frac{23}{24} - \frac{5}{6}}$

71.

Subtract: $\displaystyle{\frac{17}{18} - \frac{4}{9}}$

72.

Subtract: $\displaystyle{-\frac{5}{6}-\frac{7}{10}}$

73.

Subtract: $\displaystyle{-\frac{5}{6}-\frac{7}{10}}$

74.

Subtract: $\displaystyle{-\frac{3}{10} - \left(-\frac{5}{6}\right)}$

75.

Subtract: $\displaystyle{-\frac{5}{6} - \left(-\frac{3}{10}\right)}$

76.

Subtract: $\displaystyle{ 4 - \frac{20}{3}}$

77.

Subtract: $\displaystyle{ 5 - \frac{16}{9}}$

Operations with Mixed Numbers

78.

Multiply: $\displaystyle{{2 {\textstyle\frac{2}{9}}} \cdot {2 {\textstyle\frac{5}{8}}} }$

79.

Multiply: $\displaystyle{{5 {\textstyle\frac{5}{9}}} \cdot {1 {\textstyle\frac{2}{25}}} }$

80.

Divide the following: $\displaystyle{{2 {\textstyle\frac{7}{10}}} \div {2 {\textstyle\frac{1}{4}}} }$

81.

Divide the following: $\displaystyle{{5 {\textstyle\frac{1}{4}}} \div {1 {\textstyle\frac{7}{8}}} }$

82.

Add the following: $\displaystyle{5{{\frac{1}{18}}} + 3{{\frac{7}{18}}} }$

83.

Add the following: $\displaystyle{5{{\frac{3}{8}}} + 2{{\frac{3}{5}}} }$

84.

Add the following: $\displaystyle{3{{\frac{5}{9}}} + 2{{\frac{1}{2}}} }$

85.

Add the following: $\displaystyle{3{{\frac{2}{5}}} + 5{{\frac{3}{4}}} }$

86.

Subtract the following: $\displaystyle{ 7{{\frac{33}{35}}} -4{{\frac{13}{35}}} }$

87.

Subtract the following: $\displaystyle{ 9{{\frac{3}{14}}} -5{{\frac{1}{14}}} }$

88.

Subtract the following: $\displaystyle{ 9{{\frac{2}{3}}} -4{{\frac{3}{10}}} }$

89.

Subtract the following: $\displaystyle{-1{{\frac{3}{8}}} -2{{\frac{1}{2}}} }$

Applications

90.

Stephanie walked ${{\frac{1}{7}}}$ of a mile in the morning, and then walked ${{\frac{3}{10}}}$ of a mile in the afternoon. How far did Stephanie walk altogether?

Stephanie walked a total of of a mile.

91.

Janieve walked ${{\frac{3}{8}}}$ of a mile in the morning, and then walked ${{\frac{1}{5}}}$ of a mile in the afternoon. How far did Janieve walk altogether?

Janieve walked a total of of a mile.

92.

Gosheven and Diane are sharing a pizza. Gosheven ate ${{\frac{1}{8}}}$ of the pizza, and Diane ate ${{\frac{1}{10}}}$ of the pizza. How much of the pizza was eaten in total?

They ate of the pizza.

93.

Randi and Stephen are sharing a pizza. Randi ate ${{\frac{1}{9}}}$ of the pizza, and Stephen ate ${{\frac{3}{10}}}$ of the pizza. How much of the pizza was eaten in total?

They ate of the pizza.

94.

A trail’s total length is ${{\frac{43}{72}}}$ of a mile. It has two legs. The first leg is ${{\frac{3}{8}}}$ of a mile long. How long is the second leg?

The second leg is of a mile in length.

95.

A trail’s total length is ${{\frac{37}{90}}}$ of a mile. It has two legs. The first leg is ${{\frac{1}{9}}}$ of a mile long. How long is the second leg?

The second leg is of a mile in length.

96.

The pie chart represents a school’s student population.

Together, white and black students make up of the school’s population.

97.

Izabelle is participating in a running event. In the first hour, she completed ${{\frac{3}{8}}}$ of the total distance. After another hour, in total she had completed ${{\frac{23}{40}}}$ of the total distance.

What fraction of the total distance did Izabelle complete during the second hour?

Izabelle completed of the distance during the second hour.

98.

The pie chart represents a school’s student population.

more of the school is white students than black students.

99.

Each page of a book is ${7 {\textstyle\frac{1}{2}}}$ inches in height, and consists of a header (a top margin), a footer (a bottom margin), and the middle part (the body). The header is ${{\frac{2}{5}}}$ of an inch thick and the middle part is ${6 {\textstyle\frac{2}{5}}}$ inches from top to bottom.

What is the thickness of the footer?

The footer is of an inch thick.

100.

Kayla and Dawn are sharing a pizza. Kayla ate ${{\frac{1}{5}}}$ of the pizza, and Dawn ate ${{\frac{1}{6}}}$ of the pizza. How much more pizza did Kayla eat than Dawn?

Kayla ate more of the pizza than Dawn ate.

101.

Page and Carmen are sharing a pizza. Page ate ${{\frac{1}{5}}}$ of the pizza, and Carmen ate ${{\frac{1}{6}}}$ of the pizza. How much more pizza did Page eat than Carmen?

Page ate more of the pizza than Carmen ate.

102.

A few years back, a car was purchased for ${\$16{,}000}\text{.}$ Today it is worth ${{\frac{1}{5}}}$ of its original value. What is the car’s current value?

The car’s current value is .

103.

A school had a fund-raising event. The revenue came from three resources: ticket sales, auction sales, and donations. Ticket sales account for ${{\frac{2}{9}}}$ of the total revenue; auction sales account for ${{\frac{1}{8}}}$ of the total revenue. What fraction of the revenue came from donations?

of the revenue came from donations.

104.

A few years back, a car was purchased for ${\$22{,}800}\text{.}$ Today it is worth ${{\frac{1}{6}}}$ of its original value. What is the car’s current value?

The car’s current value is .

105.

A town has $150$ residents in total, of which ${{\frac{2}{3}}}$ are African Americans. How many African Americans reside in this town?

There are African Americans residing in this town.

106.

A company received a grant, and decided to spend ${{\frac{18}{25}}}$ of this grant in research and development next year. Out of the money set aside for research and development, ${{\frac{5}{6}}}$ will be used to buy new equipment. What fraction of the grant will be used to buy new equipment?

of the grant will be used to buy new equipment.

107.

A food bank just received $21$ kilograms of emergency food. Each family in need is to receive ${{\frac{3}{8}}}$ kilograms of food. How many families can be served with the $21$ kilograms of food?

families can be served with the $21$ kilograms of food.

108.

A construction team maintains a ${16}$-mile-long sewage pipe. Each day, the team can cover ${{\frac{2}{5}}}$ of a mile. How many days will it take the team to complete the maintenance of the entire sewage pipe?

It will take the team days to complete maintaining the entire sewage pipe.

109.

A child is stacking up tiles. Each tile’s height is ${{\frac{2}{3}}}$ of a centimeter. How many layers of tiles are needed to reach ${12}$ centimeters in total height?

To reach the total height of ${12}$ centimeters, layers of tiles are needed.

110.

A restaurant made $350$ cups of pudding for a festival.

Customers at the festival will be served ${{\frac{1}{6}}}$ of a cup of pudding per serving. How many customers can the restaurant serve at the festival with the $350$ cups of pudding?

The restaurant can serve customers at the festival with the $350$ cups of pudding.

111.

A $2\times4$ piece of lumber in your garage is ${49 {\textstyle\frac{5}{16}}}$ inches long. A second $2\times4$ is ${65 {\textstyle\frac{11}{32}}}$ inches long. If you lay them end to end, what will the total length be?

The total length will be inches.

112.

A $2\times4$ piece of lumber in your garage is ${66 {\textstyle\frac{11}{32}}}$ inches long. A second $2\times4$ is ${54 {\textstyle\frac{13}{16}}}$ inches long. If you lay them end to end, what will the total length be?

The total length will be inches.

113.

Each page of a book consists of a header, a footer and the middle part. The header is ${{\frac{1}{10}}}$ inches in height; the footer is ${{\frac{19}{20}}}$ inches in height; and the middle part is ${6 {\textstyle\frac{7}{10}}}$ inches in height.

What is the total height of each page in this book? Use mixed number in your answer if needed.

Each page in this book is inches in height.

114.

To pave the road on Ellis Street, the crew used $2{{\frac{4}{9}}}$ tons of cement on the first day, and used $4{{\frac{3}{4}}}$ tons on the second day. How many tons of cement were used in all?

tons of cement were used in all.

115.

When driving on a high way, noticed a sign saying exit to Johnstown is $2{{\frac{3}{4}}}$ miles away, while exit to Jerrystown is $4{{\frac{1}{2}}}$ miles away. How far is Johnstown from Jerrystown?

Johnstown and Jerrystown are miles apart.

116.

When driving on a high way, noticed a sign saying exit to Johnstown is $1{{\frac{3}{4}}}$ miles away, while exit to Jerrystown is $4{{\frac{1}{2}}}$ miles away. How far is Johnstown from Jerrystown?

Johnstown and Jerrystown are miles apart.

117.

A cake recipe needs ${2 {\textstyle\frac{1}{4}}}$ cups of flour. Using this recipe, to bake $9$ cakes, how many cups of flour are needed?

To bake $9$ cakes, cups of flour are needed.

118.

A cake recipe needs ${1 {\textstyle\frac{3}{4}}}$ cups of flour. Using this recipe, to bake $5$ cakes, how many cups of flour are needed?

To bake $5$ cakes, cups of flour are needed.

Sketching Fractions

119.

Sketch a number line showing each fraction. (Be sure to carefully indicate the correct number of equal parts of the whole.)

$\displaystyle \dfrac{2}{3}$
$\displaystyle \dfrac{6}{8}$
$\displaystyle \dfrac{5}{4}$
$\displaystyle -\dfrac{4}{5}$

120.

Sketch a number line showing each fraction. (Be sure to carefully indicate the correct number of equal parts of the whole.)

$\displaystyle \dfrac{1}{6}$
$\displaystyle \dfrac{3}{9}$
$\displaystyle \dfrac{7}{6}$
$\displaystyle -\dfrac{8}{5}$

121.

Sketch a picture of the product $\frac{3}{5} \cdot \frac{1}{2}\text{,}$ using a number line or rectangles.

122.

Sketch a picture of the sum $\frac{2}{3} + \frac{1}{8}\text{,}$ using a number line or rectangles.

Challenge

123.

Given that $a\neq0\text{,}$ simplify $\displaystyle{}{\frac{6}{a}+\frac{2}{a}}\text{.}$

124.

Given that $a\neq0\text{,}$ simplify $\displaystyle{}{\frac{7}{a}+\frac{9}{2a}}\text{.}$

125.

Given that $a\neq0\text{,}$ simplify $\displaystyle{}{\frac{8}{a}-\frac{4}{5a}}\text{.}$