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Means and Extremes — Definition, Formula & Examples

Means and extremes are the names for the four terms in a proportion. In a:b=c:da : b = c : d, the extremes are the outer terms (aa and dd) and the means are the inner terms (bb and cc).

Given a proportion ab=cd\frac{a}{b} = \frac{c}{d}, the first and fourth terms (aa and dd) are called the extremes, while the second and third terms (bb and cc) are called the means. The cross-product property states that the product of the means equals the product of the extremes.

Key Formula

b×c=a×db \times c = a \times d
Where:
  • aa = First term (extreme)
  • bb = Second term (mean)
  • cc = Third term (mean)
  • dd = Fourth term (extreme)

How It Works

Write the proportion as a:b=c:da : b = c : d. The two values on the outside—aa and dd—are the extremes. The two values on the inside—bb and cc—are the means. Because the proportion is true, the product of the means equals the product of the extremes: b×c=a×db \times c = a \times d. This relationship, often called cross-multiplying, lets you solve for any unknown term.

Worked Example

Problem: In the proportion 3 : 5 = 12 : x, identify the means and extremes, then solve for x.
Identify: The extremes are the outer terms and the means are the inner terms.
Extremes: 3 and xMeans: 5 and 12\text{Extremes: } 3 \text{ and } x \qquad \text{Means: } 5 \text{ and } 12
Apply the property: Set the product of the means equal to the product of the extremes.
5×12=3×x5 \times 12 = 3 \times x
Solve: Simplify and divide both sides by 3.
60=3x  ⟹  x=2060 = 3x \implies x = 20
Answer: x=20x = 20. The proportion is 3:5=12:203 : 5 = 12 : 20.

Why It Matters

Knowing the names "means" and "extremes" helps you follow cross-multiplication steps in proportion problems and understand related ideas like the mean proportional (geometric mean between two numbers). These terms appear in state math glossaries and standardized test explanations.

Common Mistakes

Mistake: Mixing up which terms are the means and which are the extremes.
Correction: Think of the proportion written with colons: a:b=c:da : b = c : d. The extremes are on the extreme outside (aa and dd); the means are in the middle (bb and cc).

Related Terms