Definition of proportion
Definition of Proportion establishes that a proportion is an equation of two equal ratios or rates: a/b = c/d. The fundamental property — cross products are equal: a · d = b · c — allows students to verify and solve proportions. From OpenStax Prealgebra 2E: '4 is to 9 as 20 is to 45' is written as 4/9 = 20/45. To verify 6/11 = 30/55, check that 6 · 55 = 330 and 11 · 30 = 330. Equal cross products confirm the proportion is true. This concept underpins ratio reasoning, scaling, and percent problems.
Key Concepts
Property A proportion is an equation of the form $\frac{a}{b} = \frac{c}{d}$, where $b \neq 0, d \neq 0$. The proportion states two ratios or rates are equal. For any proportion of this form, its cross products are equal: $a \cdot d = b \cdot c$. Cross products can be used to test whether a proportion is true.
Examples The sentence "4 is to 9 as 20 is to 45" is written as the proportion $\frac{4}{9} = \frac{20}{45}$. To determine if $\frac{6}{11} = \frac{30}{55}$ is a proportion, we check the cross products. Since $6 \cdot 55 = 330$ and $11 \cdot 30 = 330$, the equation is a proportion. To check if $\frac{8}{10} = \frac{30}{40}$ is a proportion, we find the cross products. $8 \cdot 40 = 320$ and $10 \cdot 30 = 300$. Since the products are not equal, it is not a proportion.
Explanation A proportion is a statement that two ratios are equal, like a balanced scale. The cross product rule is a quick check: if the products of the numbers on the diagonal are equal, the ratios form a true proportion.
Common Questions
What is a proportion in math?
A proportion is an equation stating two ratios or rates are equal: a/b = c/d, where b ≠ 0 and d ≠ 0.
What are cross products?
In the proportion a/b = c/d, the cross products are a·d and b·c. If the proportion is true, these products are equal.
How do you verify 6/11 = 30/55 is a proportion?
Check cross products: 6 × 55 = 330 and 11 × 30 = 330. Since they are equal, the equation is a true proportion.
How do you write '4 is to 9 as 20 is to 45' as a proportion?
Write as 4/9 = 20/45.
How is a proportion different from a ratio?
A ratio compares two quantities. A proportion is a statement that two ratios are equal.
How do proportions apply to real-world problems?
Proportions are used in recipe scaling, map reading, similar triangles, and currency conversion — any situation where two equivalent rates or ratios are set equal.