Understanding Non-Unit Fractions and Whole Numbers

When you first learn about fractions, you usually start with unit fractions—fractions that have a $1$ on top, like $\frac{1}{4}$ or $\frac{1}{3}$. But what happens when we have numbers larger than $1$ on top? Let's explore how these fractions work and how they relate to whole numbers!

Building Fractions from Unit Fractions

A non-unit fraction is simply a fraction with a numerator (the top number) greater than $1$. You can think of any fraction as being built out of smaller unit fractions.

For example, look at $\frac{3}{4}$. The denominator (the bottom number) tells us the whole is cut into $4$ equal pieces, so each piece is $\frac{1}{4}$. The numerator (the top number) tells us we have $3$ of those pieces.

Therefore, $\frac{3}{4}$ is just three copies of $\frac{1}{4}$: $\frac{3}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4}$

When Fractions Equal 1 Whole

What happens if you have all the pieces that make up a whole? If a pie is cut into $4$ slices, and you have $4$ slices, you have the whole pie!

Whenever the numerator and the denominator are the exact same number, the fraction equals exactly $1$ whole. $\frac{4}{4} = 1$ $\frac{3}{3} = 1$ $\frac{8}{8} = 1$

Fractions Greater Than 1

Sometimes, the numerator is larger than the denominator. This means you have more pieces than it takes to make a single whole, so the overall fraction is greater than $1$.

For example, consider $\frac{5}{3}$. Since it takes $3$ thirds to make a whole ($\frac{3}{3} = 1$), having $5$ thirds means you have one whole and $2$ extra thirds left over.

Sometimes, these "top-heavy" fractions equal larger whole numbers. If you have $\frac{8}{4}$, you can group the fourths to see how many wholes you have: $\frac{8}{4} = \frac{4}{4} + \frac{4}{4} = 1 + 1 = 2$

Example Problems

Example 1: Show that $\frac{3}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4}$ Think of counting items. Just like $3$ apples is $1$ apple + $1$ apple + $1$ apple, having $3$ fourths is exactly $1$ fourth + $1$ fourth + $1$ fourth.

Example 2: What whole number equals $\frac{8}{4}$? Every $4$ pieces of size $\frac{1}{4}$ makes $1$ whole. Since we have $8$ pieces total, we can make exactly two groups of $4$. Therefore, $\frac{8}{4} = 2$.

Example 3: Place $\frac{5}{3}$ on the number line. To place $\frac{5}{3}$ on a number line, count by thirds. Start at $0$, then jump to $\frac{1}{3}, \frac{2}{3}, \frac{3}{3}$ (which is exactly $1$), $\frac{4}{3}$, and finally land on $\frac{5}{3}$. It will sit on the number line between the whole numbers $1$ and $2$.