6.1 Introduction and Key Insights

What is a ratio anyway? Mathematically speaking, a ratio is nothing more than a fraction. We use the word "ratio" to signal how we will use this fraction. When we say the fraction $10/3$ is a ratio, we are emphasizing how the number 10 compares to the number three. Some read this ratio as "ten compared to three". We use lots of different notation and language to express ratios.

These expressions all express the same ratio:

$10 \text{ to } 3 \hspace{0.5cm} 10:3 \hspace{0.5cm} \frac{10}{3}$

When we use ratios, we are usually less interested in the exact numerical value. For example, we probably would not write $10/3 = 3.\overline{3} \approx 3.33$. We treat the fraction as a ratio and are more interested in how the whole number values compare.

Consider the square shapes below for a visual example. We can use various ratios to describe different aspects of this picture. For example, we can observe there are three dark blue boxes compared to two light blue ones. We can say the dark blue and light blue boxes are in a ratio of 3:2. The fraction $3/2$ also expresses this comparison.

We can use the ratio $\frac{2}{3}$ to compare the number of dark blue to light blue boxes, but instead of only comparing "parts to parts", we can also compare "parts to whole". In this image, we have five squares, so the ratios $3:5$, $3/5$, and $\frac{3}{5}$ all express how many dark blue boxes exist compared to the total number of boxes.

This idea of ratios is also present in computing basic probabilities. For example, if someone jumbled all of our five squares in a bag and asked you to choose one at random (that is, you could not see which one you were choosing), you would use the ratio $2/5$ to express your chance (or probability) of choosing a light blue square.