What is a Proper Fraction?

By definition, a proper fraction is a part of a whole. It represents a value that is less than one full unit. In a proper fraction, the top number (numerator) is always smaller than the bottom number (denominator).

Let’s look at some visual examples to see how this works in practice.

Example 1: One-Half ($\frac{1}{2}$)

Imagine a rectangle divided into two equal parts. If you shade one of those parts, you have represented one-half of the whole.

• Notation: $\frac{1}{2}$

• How to read it: “1 by 2” or “one-half”

• Visual: A rectangle with 1 out of 2 sections shaded.

Example 2: One-Third ($\frac{1}{3}$)

Now, imagine a rectangle divided into three equal parts. If only one section is shaded, it represents one-third.

• Notation: $\frac{1}{3}$

• How to read it: “1 by 3” or “one-third”

• Visual: A rectangle with 1 out of 3 sections shaded.

Example 3: One-Quarter ($\frac{1}{4}$)

Fractions aren’t just for rectangles; they apply to circles too! If you divide a circle into four equal slices and take one, you have a quarter.

• Notation: $\frac{1}{4}$

• How to read it: “1 by 4” or “a quarter”

• Visual: A circle divided into four quadrants with 1 section shaded.

Adding Parts Together

What happens when you have more than one part shaded? You simply add the fractions together.

Take a rectangle divided into three equal parts. If you shade two of those parts:

• The first shaded part is $\frac{1}{3}$

• The second shaded part is $\frac{1}{3}$

• Total shaded region: $\frac{1}{3} + \frac{1}{3} = \frac{2}{3}$

In this case, $\frac{2}{3}$ is still a proper fraction because the shaded region is still less than the entire “whole” rectangle.

Summary Table

1. The Power of Addition: Finding the Shaded Region

In our first example, we have a circle divided into four equal quadrants. Each of these quadrants represents $\frac{1}{4}$ of the total circle.

• The Setup: Three out of the four sections are shaded with horizontal lines.

• The Math: To find the total shaded area, we simply add the parts together:
  $$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}$$

• The Result: We have three-fourths of a circle. This tells us that while we have most of the circle, one piece is still missing!

2. Completing the Puzzle: What is a “Whole”?

The second example shows us what happens when every single piece is accounted for. Again, the circle is divided into four equal parts ($\frac{1}{4}$ each).

• The Setup: This time, all four sections are shaded.

• The Math: When we add all four quarters together, something interesting happens:
  $$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{4}{4}$$

• The Result: Whenever the top number (numerator) and the bottom number (denominator) are the same, the fraction equals 1. This represents the Whole.