Lesson Objectives
• Learn how to write a fraction which represents a given situation
• Learn how to identify the numerator and denominator of a fraction
• Learn how to write a division problem using a fraction bar
• Learn how to identify a proper fraction, improper fraction, or mixed number

## What is a Fraction?

At the beginning of our pre-algebra course, we focused on the whole numbers. These numbers are used to describe and perform operations with whole objects. As an example, suppose we purchase a pizza for our family to eat for dinner. At this point, we have a whole pizza. Since no part of the pizza is missing, we can use the whole number 1 to describe the amount of pizza we currently have. Let’s now suppose it’s finally dinner time and we use our pizza cutter to slice our pizza up into 8 equal slices. As people start eating, the amount of pizza that remains is no longer represented with the whole number 1. As long as some amount remains, meaning the amount of pizza left is not zero, we have a fractional amount of pizza left. A fraction is normally used to represent a part of a whole amount. In the case of a proper fraction, we are referring to an amount that is less than 1, but more than 0. Let's suppose our family eats 3 slices out of the 8 slices available. At this point, there are 5 pieces of pizza left, out of a total of 8. We can represent this with the fraction: $$\frac{5}{8}$$ The top number in a fraction is known as the numerator. The numerator shows the number of equal parts of the whole amount under discussion. Since we described the amount of pizza that was left, our numerator is 5. This is because 5 is the number of pieces of pizza left. If we had described a scenario in which we discussed the amount of pizza that was eaten or removed, the numerator would change to 3. $$\frac{5}{8}\hspace{.5em}»\hspace{.5em}pizza \hspace{.5em}remaining$$ $$\frac{3}{8}\hspace{.5em}»\hspace{.5em}eaten \hspace{.5em}pizza$$ So we can see that the numerator can change based on the context of the situation. The bottom number in a fraction is known as the denominator. The denominator shows the number of equal parts in the whole. In our scenario, the denominator was 8 since the whole pizza was split up into 8 equal parts.
Example 1: Identify the numerator and denominator in each fraction: $$\frac{5}{9}$$
• 5 is the numerator or top number
• 9 is the denominator or bottom number
$$\frac{7}{12}$$
• 7 is the numerator or top number
• 12 is the denominator or bottom number
$$\frac{19}{31}$$
• 19 is the numerator or top number
• 31 is the denominator or bottom number
Example 2: Use a fraction to describe the shaded part (yellow) and non-shaded part (blue) of each image: Our rectangle above is split into 3 equal parts. 2 parts are shaded (yellow) and 1 part is not (blue). $$\frac{2}{3}\hspace{.5em}»\hspace{.5em}shaded$$ $$\frac{1}{3}\hspace{.5em}»\hspace{.5em}not\hspace{.5em}shaded$$ Our polygon above is split into 2 equal parts. 1 part is shaded (yellow) and 1 part is not (red). $$\frac{1}{2}\hspace{.5em}»\hspace{.5em}shaded$$ $$\frac{1}{2}\hspace{.5em}»\hspace{.5em}not\hspace{.5em}shaded$$ Before moving forward, we want to take a look at the fraction bar. This lies in the middle of the numerator and the denominator. This horizontal line represents division. We are not at the point where we will divide fractions yet, but we should know that a fraction represents the division of the numerator by the denominator. This means we can rewrite a problem such as 20 ÷ 4 = 5 as: $$\frac{20}{4}=5$$ We must be very careful when dividing with expressions in fractional form. The numerator and denominator must be evaluated separately. This would mean transforming a problem using the division symbol "÷" requires us to place parentheses around the numerator and the denominator. This will ensure that we evaluate the numerator and denominator separately before performing the division.
Example 3: Evaluate each $$\frac{3^2 \cdot 5 + 7}{-4^3 ÷ 2 + 6}$$ To evaluate this problem, we will place parentheses around the numerator and denominator: $$(3^2 \cdot 5 + 7) ÷ (-4^3 ÷ 2 + 6)$$ Now we can simplify the numerator and denominator separately $$(52) ÷ (-26)$$ Now perform the final division $$52 ÷ -26 = -2$$

### Proper Fractions, Improper Fractions, and Mixed Numbers

When we work with fractions, we will have three different scenarios: proper fractions, improper fractions, and mixed numbers. Proper Fractions are fractions used to describe a quantity that is greater than 0, but less than 1. A proper fraction has a numerator that is strictly less than its denominator. An Improper Fraction is used to describe a quantity that is greater than or equal to 1. An improper fraction will have a numerator that is greater than or equal to its denominator. A Mixed Number represents the sum of a whole number and a proper fraction. Let's take a look at a few examples.
Example 4: Determine if each is a proper fraction, improper fraction, or mixed number. $$\frac{9}{2}$$ This is an improper fraction. The numerator 9 is larger than the denominator 2. $$\frac{7}{7}$$ This is an improper fraction. The numerator 7 is the same as the denominator 7. This means we have 7 ÷ 7, which is 1. $$\frac{13}{15}$$ This is a proper fraction. The numerator 13 is smaller than the denominator 15. $$2\frac{8}{9}$$ This is a mixed number or the result of a whole number plus a proper fraction.