How to Find an Unknown in a Proportion

Instructor: Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

Artists, engineers, athletes, and accountants all use proportions in their daily lives, and setting them up is really quite easy, once you know the rules. In this lesson, you'll learn how to solve proportions when there is an unknown value and work on some practice problems.

Finding the Unknown in a Proportion

Imagine you've been commissioned to create a mural on the side of a building. You decide on a replica of Piet Mondrian's Composition with Red, Yellow, and Blue, as shown below.

Mondrian: Composition with Red, Yellow, and Blue

While you feel confident replicating the colors, you're a little unsure about the size of the rectangles. You know that the painting's dimensions are 28.5 x 27.1 inches, and the wall is 9 feet, or 108 inches, tall. So, how do you proportion your mural reproduction to fill the width of the wall while maintaining the integrity of Mondrian's original composition and size?

What is a Proportion?

A proportion is simply two ratios that are equal to each other. We define a ratio as a comparison of any two numbers. In Composition with Red, Yellow, and Blue, we can compare the height of the painting to the width of the painting. We can also compare the height of the painting to the height of the mural, or the width of one of the painting's rectangles to the width of the entire painting.

To make these comparisons, we'll use ratios, which can be written in several ways. Take a look at the examples below that show the ratio of innings to outs in a baseball game.

  • 9 innings : 54 outs
  • 9 innings / 54 outs
  • 9 innings to 54 outs

In the innings: outs ratio, we know how many outs there are in a 9-inning game. But we might want to know how many outs there would be in a rain-shortened, 7-inning game. To do so, we would:

  1. Identify the known ratio, where both values are known.
  2. Identify the ratio with one known value and one unknown value.
  3. Use the two ratios to create a proportion.
  4. Cross-multiply to solve the problem.

In our innings example:

  1. 9 innings / 54 outs represents the two known values.
  2. 7 innings / x outs represents the one known value and one unknown value.
  3. 9x = 54 x 7.
  4. 9x = 378.
  5. x = 42.

As a result of our calculations, we know that there'd be 42 outs in a 7-inning game.

Calculating Proportions

Let's go back to the hypothetical mural we talked about at the beginning of the lesson and the ratio of the original painting height to wall height. This ratio is key, as it will allow us to take any dimension of the original and set up a proportion to figure out the mural dimension.

In terms of height, the ratio of the original painting to the mural is 28.5 inches / 108 inches. Now that we know the height of our mural, let's figure out the width.

  1. 28. 5 inches / 108 inches represents the painting height / mural height, or our known ratio.
  2. 27.1 inches / x inches represents the painting width / mural width, or the known and unknown values.
  3. 28.5 inches / 108 inches = 27.1 inches / x inches.
  4. 28.5x = 108 x 27.
  5. 28.5x = 2926.8.
  6. x = 102.7 inches.

Therefore, by our calculation, the width of the mural will be 102.7 inches or 8 feet, 6.7 inches.

Now, for the sake of accuracy, let's say we want to mark off the horizontal and vertical lines in the painting. For example, if the first vertical line on the left is 3.8 inches from the edge, where would it be on the mural?

  1. 28.5 inches / 108 inches
  2. 3.8 inches / x inches
  3. 28.5 inches / 108 inches = 3.8 inches / x inches
  4. 28.5x = 108x 3.8
  5. 28.5x = 410.4
  6. x = 14.4 inches

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