By Kevin Beck | Updated Aug 30, 2022
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While the idea of a proportion feels familiar, articulating a precise mathematical definition can be challenging. Think of a 10‑year‑old compared to an average adult, and then an adult compared to a professional basketball player: each pair is related by the same type of size relationship, even though the absolute values differ.
The concept of a ratio works similarly. In a sporting event, knowing that the number of opposing fans far exceeds that of the home team’s supporters might change how loudly you cheer when your favorite club scores.
In mathematics and statistics, questions involving proportions, percentages, and ratios arise frequently. A concise explanation of these concepts, coupled with practical examples, will make you a more confident math student.
A ratio is essentially a comparison expressed as a fraction or quotient, such as 3/4 or 179/2,385. It is a specialized type of fraction used to compare related quantities. For instance, if a room contains 11 boys and 13 girls, the ratio of boys to girls is 11 to 13, which can be written as 11/13 or 11:13.
The term “ratio” comes from the Latin word for “reason.” A rational number is one that can be expressed as a fraction; irrational numbers, like π, cannot.
A proportion is an equation that sets two ratios equal to one another, using different absolute numbers in the fractions. Proportions are written in the same style as ratios, for example, a/b = c/d or a:b = c:d.
Most ratio problems can be solved without a specialized calculator. Consider this scenario: you visit the gym 17 times during a 30‑day month. What is the ratio of gym days to non‑gym days?
Don’t simply divide gym days by total days. Subtract gym days from the total to find non‑gym days: 30 – 17 = 13. The correct ratio is therefore 17:13 (or 17/13).
Sometimes the proportionality between two ratios is obvious. If you and your dog are the only animals in a room, and a nearby gym contains 457 people and 457 dogs, the proportion of people to dogs is identical in both spaces.
Other times you need to check. For example, is 17/52 proportional to 3/9? Use cross‑multiplication: 17 × 9 = 153 and 3 × 52 = 156. Since 153 ≠ 156, the ratios are not equal; 3/9 is slightly larger.
The proportionality constant, k, captures the fixed ratio between two variables. If a is proportional to b, then a = k·b. When a and b are inversely proportional, their product remains constant: a = C/b and b = C/a.
Example: In a coffee shop, the number of archery fans is proportional to the number of baseball fans. Initially, there are 6 archery fans and 9 baseball fans. If the baseball fan count rises to 24, how many archery fans will there be?
First find k: k = 6 ÷ 9 = 2/3 ≈ 0.667. Then solve a = 0.667 × 24, yielding a = 16 archery fans.
