When students first encounter the instruction to state if each pair of ratios forms a proportion, they often wonder what separates a simple comparison from a true mathematical balance. Worth adding: learning how to state if each pair of ratios forms a proportion is a foundational skill in algebra, geometry, and everyday problem solving. A ratio compares two quantities, while a proportion declares that two ratios are equal. Whether you are scaling a recipe, reading a map, or calculating discounts, understanding proportions helps you make accurate decisions based on proportional relationships.
What Is a Ratio?
A ratio is a way to compare two numbers by showing how many times one value contains or is contained within the other. Ratios can be written in several forms: as a fraction such as 3/4, with a colon like 3:4, or in words as “3 to 4.” The order matters in a ratio; 3:4 is not the same as 4:3 because the relationship between the two quantities changes Less friction, more output..
What Is a Proportion?
A proportion is an equation stating that two ratios are equivalent. Still, for example, if you have the ratios 1/2 and 2/4, you can say they form a proportion because both simplify to the same value. Even so, when written as 1/2 = 2/4, this equality is a proportion. The ability to state if each pair of ratios forms a proportion depends on verifying that the two ratios represent the same relationship, even if the numbers look different at first glance.
Short version: it depends. Long version — keep reading.
The Core Rule: Cross-Multiplication
The most reliable method to state if each pair of ratios forms a proportion is cross-multiplication. If you have two ratios written as fractions, a/b and c/d, you multiply the numerator of the first ratio by the denominator of the second ratio, and the denominator of the first ratio by the numerator of the second ratio. If the two products are equal, the ratios form a proportion Worth knowing..
Mathematically, a/b = c/d is a proportion if and only if:
a × d = b × c
These products are often called the cross products. In a proportion, the product of the means equals the product of the extremes. The “means” are the inner terms (b and c), while the “extremes” are the outer terms (a and d).
Not obvious, but once you see it — you'll see it everywhere.
Step-by-Step: How to State If Each Pair of Ratios Forms a Proportion
Follow these steps every time you need to determine whether two ratios form a proportion:
- Write both ratios in fraction form if they are not already.
- Identify the numerators and denominators of each fraction.
- Multiply the numerator of the first fraction by the denominator of the second fraction.
- Multiply the denominator of the first fraction by the numerator of the second fraction.
- Compare the two products.
- If the products are equal, the pair forms a proportion. If they are not equal, the pair does not form a proportion.
This method works regardless of whether the ratios are written with colons, fractions, or words, as long as you convert them into fractional form first.
Worked Examples
Example 1: Do 3/4 and 6/8 form a proportion?
Cross-multiply:
- 3 × 8 = 24
- 4 × 6 = 24
Since 24 = 24, the ratios form a proportion.
Example 2: Do 5/7 and 10/25 form a proportion?
Cross-multiply:
- 5 × 25 = 125
- 7 × 10 = 70
Since 125 ≠ 70, the ratios do not form a proportion Worth keeping that in mind. Still holds up..
Example 3: Do 2:5 and 8:20 form a proportion?
Rewrite as fractions: 2/5 and 8/20. Cross-multiply:
- 2 × 20 = 40
- 5 × 8 = 40
Since 40 = 40, the ratios form a proportion Small thing, real impact..
Example 4: Do 9/12 and 3/4 form a proportion?
You can simplify 9/12 by dividing both the numerator and denominator by 3, which gives 3/4. Because both ratios are identical after simplification, they form a proportion. Cross-multiplication confirms this:
- 9 × 4 = 36
- 12 × 3 =
