RWM101: Foundations of Real World Math (2022.A.01)

A first step toward understanding fractions is to review their different parts – like we identified the exponent and base in an exponential expression in section 3.4.

For example, the fraction $\frac{8}{11}$  uses three symbols: two integers (eight and 11) and a horizontal line called the fraction bar. We call the number on top of the bar the numerator and the number beneath the bar the denominator. You can read our example in several ways: 8 over 11, 8 out of 11, 8 parts per 11, or 8 divided by 11.

Just as different words can represent the same concept (the synonyms big and large), different mathematical expressions can represent the same quantity. For example, imagine a pizza cut into four slices. Each slice represents $\frac{1}{4}$ of the pizza. If you eat two of the four slices, you can say you ate $\frac{2}{4}$ of the entire pizza. As this image indicates, our two $\frac{1}{4}$ slices make up an entire $\frac{1}{2}$ of the pizza.

This is all to say that for our pizza example, $\frac{2}{4}=\frac{1}{2}$. Each of these fractions has a different numerator and denominator, but they both represent the same quantity or value. We say these two fractions are equivalent. Equivalent fractions may be written to look different, but they represent the same value.

To practice, can you determine equivalent ways to write the fraction $\frac{2}{6}$? Are $\frac{3}{4}$ and $\frac{75}{100}$ equivalent? How can you tell?

We call a fraction improper when the numerator is larger than its denominator. For example, $\frac{10}{3}$. There is nothing wrong or bad about improper fractions; they occur all around us, like regular fractions. Imagine, for example, sharing 10 candy bars among three people. Everyone will receive $\frac{10}{3}$ of a candy bar.

Let's say you just bought a full candy bar. Then you discover you already had a half-eaten candy bar in your pocket. How much do you have altogether? You might write that you have $1+\frac{1}{2}$ candy bars, which some math teachers would call a mixed number. It contains a regular whole number and a new fraction. In fact, some math teachers rewrite this expression without the addition sign altogether as:

$1+\frac{1}{2}=1\frac{1}{2}$

While this notation for mixed numbers is common in early math courses, mathematicians and math students largely abandon it once they move on to other courses, and they do so for good reason. It is very easy to confuse mixed number notation for multiplication, reading the above incorrectly as 1 times $\frac{1}{2}$. It is useful to keep in mind that mixed number notation hides a + sign, one that connects a whole number and a fraction. Nevertheless, it can be useful to understand and apply such notation.

But it is more important for you to understand that every mixed number is simply another way of writing an improper fraction. In our example above, you can also think about the one whole candy bar as two halves, so that $1=\frac{1}{2}+\frac{1}{2}$. This allows you to count how much candy you have in a different way. You can combine your two halves with the half candy bar in your pocket, so you have a total of three halves.

In purely math terms, $1+\frac{1}{2}=1\frac{1}{2}=\frac{3}{2}$, which is an improper fraction.

While it is rare for a math text to take a bold stance on almost anything, we are going to make an exception here! No one should ever use mixed number notation, especially in mathematical situations. This being said, we often see "mixed numbers" in our daily lives, most notably at gas stations. The image below shows a price of $\displaystyle 327\frac{9}{10}$ pennies, which is, of course, a mixed number.

One benefit of mixed number notation is it makes it easier to compare the quantity to whole numbers. For instance, we know that the $327\frac{9}{10}$ price is in between $327$ and $328$. When we convert this mixed number into an improper fraction, we obtain:

$327\frac{9}{10} = 327 + \frac{9}{10} = \frac{3270 + 9}{10} = \frac{3279}{10}$.

The second video link below describes how to convert mixed numbers into an improper fraction. Note that while many prefer to use proper and improper fractions to notate rational numbers, these expressions make it hard to determine which improper fraction lies between consecutive whole numbers. As much as we hate to admit it, this one point supports the use of mixed numbers.

Of course, you can disfigure a lovely, beautiful, and clearly-expressed improper fraction to create an ugly, inferior mixed number – there is a downgrading conversion process in this direction, too. For example, consider the improper fraction $13/3$ which we can rewrite as:

$\frac{13}{3} = \frac{12+1}{3} = \frac{12}{3} + \frac{1}{3} = 4+\frac{1}{3} = 4\frac{1}{3}$

Converting a mixed number into an improper fraction is an example of how to add fractions. Before we discuss how to add and subtract fractions more generally in section 4.8, we need to address a related issue that will help you add and subtract fractions more easily.

Let's return to your aunt's coffee shop for a delicious and motivating example. Your aunt recently added a second breakfast sandwich to her menu, the Vegan Supreme. Like her original Bacon Breakfast Sandwich, her customers love her new addition.

However, your aunt can only prepare one type of breakfast sandwich next weekend due to some scheduled kitchen maintenance. She decides to conduct a survey to help choose which one she should offer and learns that 600 out of 1,000 customers prefer the Vegan Supreme. We can record this figure as a fraction, namely $\frac{600}{1000}$, but it turns out that this fraction is equivalent to a smaller, easier-to-read-and-understand expression. Namely:

$\frac{600}{1000} = \frac{3}{5}$

It is easier to interpret this smaller-looking-but-equivalent fraction than to work with the original figure $600/1000$. When we reduce the complex appearance of the original fraction – a process we call reducing the fraction or expressing it in lowest terms – it is clear that your aunt should offer the Vegan Supreme. Three-fifths of the customers she surveyed preferred it!

Reducing fractions not only helps us make sandwich decisions. It also makes arithmetic computations easier. Reducing fractions to their lowest terms simplifies and clarifies how we combine these numbers via our standard operations.

Now that you are comfortable reducing fractions into their lowest terms, we want to build some familiarity with rewriting fractions. Specifically, we want to rewrite two fractions so that they have the same denominator.

Consider the fractions $3/12$ and $4/12$. These two expressions have a common denominator (12). Now practice your fraction-reducing skills from the previous section to note that:

$\frac{3}{12} = \frac{1}{4}$ and $\frac{4}{12} = \frac{1}{3}$

The fractions $1/3$ and $1/4$ can each be re-expressed so that they have a common denominator of 12. Something similar is also true for the fractions $20/60$ and $24/60$; they have a common denominator of $60$, and each of these fractions reduces to $1/3$ and $2/5$.

The process of finding a single common denominator for two fractions is, in some sense, the reverse of what we have just discussed: start with two different-denominator fractions and then rewrite them so they have the same denominator.

The video you will watch next discusses how to find "the least common denominator". For our two previous examples, the least common denominator of $1/4$ and $1/3$ is 12, but the least common denominator of $1/3$ and $2/5$ is 15 (not 60, the larger denominator we used above). Indeed, instead of writing:

$\frac{1}{3} = \frac{20}{60}$ and $\frac{2}{5} = \frac{24}{60}$,

we could have written:

$\frac{1}{3} = \frac{5}{15}$ and $\frac{2}{5} = \frac{6}{15}$.

To find the least common denominator, you will use the least common multiple of the denominators, a nice callback to section 2.2.

When we add or subtract fractions, it is helpful to consider two different cases: when the fractions have the same denominator and when the fractions have different denominators. We first focus on adding and subtracting fractions that have the same or common denominator. It is a relatively short process, as we simply combine the numerators.

For instance, consider adding $9/15 + 2/15$. Because the denominators match, we are able to compute:

$\frac{9}{15} + \frac{2}{15} = \frac{9+2}{15} = \frac{11}{15}$.

$\frac{9}{15} - \frac{2}{15} = \frac{9-2}{15} = \frac{7}{15}$.

$\frac{a}{b} \pm \frac{c}{b} = \frac{a\pm c}{b}$.

Adding and subtracting fractions and mixed numbers comes up in a variety of real-world applications. Let's return to your aunt's coffee shop. Your aunt has placed some online ads to reach even more customers, and several of the ads include coupons. One fine Saturday morning, one of your regular patrons presents two coupons. One reads, "half off". The second reads, "one-third off".

Assuming your aunt allows her customers to combine the coupons, how much of a reduction will they enjoy? Try to see if you can figure this out!

We need to add $\frac{1}{2}$ and $\frac{1}{3}$, which first requires a common denominator. We can write:

$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Our frugal customer only has to pay one-sixth of the original price; after you take off five-sixths!

One last thought about our coupon-combining diner: if their total bill was $30.36, how much do they owe after they apply both coupons? The answer is in the next section, since this calculation requires multiplying fractions.

Great news! Your aunt's coffee shop is so successful that she is opening up a second shop in a different neighborhood. While everyone is excited, you and your aunt are busy arranging the details for her grand opening. You and your aunt will be cutting a ribbon to commemorate the event, and you have already purchased 10 feet of ribbon to use. Only now, you realize you only need $1/2$ feet of fabric for the event, and you wonder: how many $1/2$ foot-long sections of fabric are in your 10-foot ribbon?

This simple question amounts to trying to divide 10 by $1/2$. You may be able to puzzle out the answer (there are 20 $1/2$-foot-long sections in your 10-foot-long ribbon!).