Geometry: Background You’ll Need 2

  • Solve a proportion for an unknown

Solving Proportions

Proportions are the heart of understanding how different quantities relate to one another. They allow us to compare two rates and determine how one quantity changes in relation to another. Mastering the art of solving proportions is not just a mathematical skill; it’s a critical thinking skill that applies to real-world situations such as comparing prices, calculating speed, or adjusting recipes.

A proportion equation sets two rates or ratios equal to each other, providing a clear pathway to finding unknown values. By cross-multiplying or using equivalent ratios, we can unlock these unknowns and gain insights into the relationship between the quantities at hand. Let’s explore how this powerful tool works.

rates

A rate is the ratio (fraction) of two quantities.

 

unit rate is a rate with a denominator of one.

proportion equation

A proportion equation is an equation showing the equivalence of two rates or ratios.

How to: Solve Proportions for an Unknown Value

When faced with a proportion, such as [latex]\frac{a}{b} = \frac{c}{d}[/latex], where you are solving for an unknown variable, follow these steps:

  • Identify the Unknown: Determine which part of the proportion is unknown. This will be the value you are solving for.
  • Cross-Multiply: Create an equation by cross-multiplying the terms of the proportion. Multiply the numerator of one ratio by the denominator of the other ratio, and set the two products equal to each other.
  • Set Up the Equation: After cross-multiplying, you will have an equation with one unknown. It will look like [latex]a×d=b×c[/latex] for the proportion above.
  • Solve for the Unknown: Isolate the unknown variable on one side of the equation. If necessary, use algebraic operations such as division or multiplication to solve for the variable.
Solve the proportion [latex]\displaystyle\frac{5}{3}=\frac{x}{6}[/latex] for the unknown value [latex]x[/latex].

Solve the proportion [latex]\displaystyle\frac{2}{7}=\frac{4}{x}[/latex] for the unknown value [latex]x[/latex].