A University-Level Understanding of Number with an eye towards elementary education

Ratios, proportions, and rates are ideas that connect numbers to the real world. They help us compare quantities, scale figures up or down, and describe how two measurements change together. For example, when we say that a recipe calls for \(2\) cups of flour for every \(3\) cups of sugar, we are describing a ratio. When we say that two such comparisons are equal, we are working with a proportion. And when we describe how one quantity changes compared to another with different units—such as kilometers per hour or dollars per item—we are describing a rate.

These ideas are important in many areas of mathematics and in everyday life. Students first meet them in elementary school when comparing objects ("there are twice as many squares as circles") or when learning about fractions. In middle school, they are used in scaling drawings, converting units, understanding percentages, and working with similar figures in geometry. Later, proportional reasoning becomes a foundation for algebra, functions, and modeling real-world problems.

Ratios, proportions, and rates also help us reason more flexibly about numbers. Instead of focusing only on exact counts, they let us focus on relationships between quantities. This relational thinking is central to mathematics: whether comparing sides of a triangle, predicting outcomes in probability, or interpreting slopes of lines in algebra, proportional reasoning is the bridge that connects different ideas.

A ratio is a way of comparing two quantities. Ratios tell us how much of one thing there is compared to another. For example, we might compare the number of apples to the number of oranges in a basket, or the number of boys to the number of girls in a class. Ratios are often written in one of three ways: with a colon (e.g. \(3:4\)), with the word “to” (e.g. “3 to 4”), or as a fraction (e.g. \(\tfrac{3}{4}\)).

Just as fractions can be simplified or scaled up, ratios can also be rewritten in different but equivalent forms. If we have a ratio \(a:b\text{,}\) then for any nonzero number \(k\) we can multiply both parts of the ratio by \(k\) and obtain \((ka):(kb)\text{.}\) Since the fraction form of the ratio is \(\tfrac{a}{b}\text{,}\) and

both \(a:b\) and \((ka):(kb)\) describe the same relationship between \(a\) and \(b\text{.}\) In the same way, dividing both parts of a ratio by a common factor gives a simpler equivalent ratio. This is why we often reduce ratios to their lowest terms, just as we do with fractions.

For example, the ratio \(6:4\) can be simplified by dividing both terms by 2, giving \(3:2\text{.}\) Both ratios represent the same relationship, because in each case the first quantity is one and a half times the second. The numbers may look different, but the comparison is unchanged.

Ratios can be used in two different but related ways: to compare one part of a group to another part, or to compare one part of a group to the whole. These are called part-to-part and part-to-whole comparisons. Understanding the difference is important, since both are common in real-world situations.

Ratios are closely connected to fractions. A part-to-whole ratio can always be written directly as a fraction, because it compares one subset to the entire set. For example, if 12 of the 30 students in a class are boys, the ratio of boys to the whole class is \(12:30\text{,}\) which corresponds to the fraction \(\tfrac{12}{30} = \tfrac{2}{5}\text{.}\) This fraction can also be interpreted as “two-fifths of the class are boys.” A part-to-part ratio, on the other hand, can be turned into a fraction by placing one part over the other. For instance, if the same class has 12 boys and 18 girls, the part-to-part ratio is \(12:18\text{,}\) or as a fraction \(\tfrac{12}{18} = \tfrac{2}{3}\text{,}\) meaning that there are two boys for every three girls. In both cases, fractions and ratios capture the same idea of comparing quantities, but part-to-whole ratios align naturally with the notion of fractions of a whole, while part-to-part ratios highlight the relative size of two categories within the whole.

A proportion is a statement that two ratios are equal. Proportions are especially useful because they allow us to find missing values when two quantities change in the same way. This is one of the most common uses of ratios in problem solving: scaling up or down while keeping the same relationship.

In the examples above we used cross multiplication to solve proportions. Let’s explain why this method works.

Suppose we have the proportion

with \(b \neq 0\) and \(d \neq 0\text{.}\) By the property of equality, we can multiply both sides of the equation by \(bd\) without changing the truth of the statement. Doing so gives:

Simplifying both sides:

or, written in the more familiar order,

This is exactly the rule of cross multiplication: in any true proportion, the product of the means equals the product of the extremes.

Let’s check this with numbers. Suppose

Cross multiplication says that \(2 \times 15 = 3 \times 10\text{.}\) Both sides equal 30, so the proportion is indeed true.

This shows why cross multiplication is not just a “trick,” but a direct consequence of the properties of equality.

Proportional reasoning shows up throughout the school curriculum, often in ways that may not seem connected at first. Recognizing these connections helps students see mathematics as an integrated subject rather than as separate topics. Below we highlight some important areas where proportional reasoning is central, with explicit examples.

These examples show that proportional reasoning connects many topics that students study across grade levels. Whether converting units, interpreting percentages, graphing lines, working with similar figures, or calculating probabilities, the underlying structure is the same: one quantity changes in a constant ratio with another.