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🔟elementary algebra review

Fundamental Theorem of Proportions

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The Fundamental Theorem of Proportions states that if four quantities are in proportion, then the product of the means (the middle terms) is equal to the product of the extremes (the first and last terms). This theorem is a foundational principle in the study of proportional relationships and their applications.

5 Must Know Facts For Your Next Test

  1. The Fundamental Theorem of Proportions is used to solve for an unknown term in a proportion when the other three terms are known.
  2. This theorem is a crucial tool in solving problems involving similar figures, as the corresponding sides of similar figures are in proportion.
  3. The theorem can be expressed mathematically as: if a/b = c/d, then ad = bc.
  4. Proportions and the Fundamental Theorem of Proportions are often used to solve real-world problems, such as scale drawings, map distances, and rates.
  5. Understanding the Fundamental Theorem of Proportions is essential for solving a wide range of applications, including those involving similar triangles and other similar shapes.

Review Questions

  • Explain how the Fundamental Theorem of Proportions can be used to solve for an unknown term in a proportion.
    • The Fundamental Theorem of Proportions states that if four quantities are in proportion, the product of the means (the middle terms) is equal to the product of the extremes (the first and last terms). This means that if you know three of the terms in a proportion, you can use the theorem to solve for the unknown fourth term. For example, if you know that a/b = c/x, you can use the theorem to solve for x by setting ad = bc, where a, b, and c are the known terms and x is the unknown.
  • Describe how the Fundamental Theorem of Proportions is applied in the context of solving problems involving similar figures.
    • The Fundamental Theorem of Proportions is a key tool for solving problems involving similar figures, as the corresponding sides of similar figures are in proportion. If two figures are similar, their corresponding sides are in the same ratio. This means that if you know the lengths of three corresponding sides, you can use the Fundamental Theorem of Proportions to solve for the length of the fourth side. For example, if two triangles are similar and you know the lengths of two sides of one triangle and the corresponding side of the other triangle, you can use the theorem to find the length of the fourth side.
  • Analyze how the Fundamental Theorem of Proportions can be used to solve real-world problems, such as those involving scale drawings, map distances, and rates.
    • The Fundamental Theorem of Proportions has a wide range of applications in solving real-world problems. In the context of scale drawings, the theorem can be used to determine the actual dimensions of an object based on its scaled representation. For map distances, the theorem can be applied to calculate the actual distance between two points on a map, given the scale of the map. Additionally, the theorem can be used to solve problems involving rates, such as finding an unknown quantity when the other three terms in a proportion are known (e.g., miles per gallon, cost per item, etc.). By understanding the Fundamental Theorem of Proportions and its applications, students can effectively solve a variety of practical problems.