Proportions

Links and useful resources

Concept summary and lesson

proportional relationship
equal ratios

Proportions tell you how to keep a ratio between things stable as you change the amounts of some of them. If you add some to part A, you'll have to add some to part B as well to keep the ratio equal, and you can use the proportion to calculate exactly how much you need.

Remember that a ratio is written as a sequence of numbers separated by colons: $2 : 4 : 1 : 9$ for example. Each number is fundamentally just the numerator of a fraction, and the denominator for all of them is just the sum of the ratio parts. That makes the whole deal add up to 1, which is handy for figuring out how much of each thing you have if you know the total.

Proportions tell you how to change the amounts so that you keep a matching ratio. It works by noticing that each pair of ratio terms has a "time-as-much-as" kind of relationship. In the $a : b : c : d = 2 : 4 : 1 : 9$ ratio, $d$ has 9 times as much as $c$ . So if you know that $d = 18$ , you can figure out that $c = 2$ because $c = \frac{1}{9} d$ , and $\frac{1}{9} \times 18 = 2$ .

You can figure out the proportion between any two elements of the ratio by just dividing one by the other. Usually, you're going to want to know one of them when you already know the other, so which fraction do you use? Ratios work like unit conversions (in fact, they often are unit conversions!). You can think of it like this: I'm going to multiply the thing I have by a conversion factor in order to get the thing I want. That means I need to cancel out the unit from the thing I have, and add in the unit for the thing I want. Since you are multiplying, the way you cancel something out is by dividing by it. So, when you're doing proportion calculations, think in terms of multiplying by the conversion factor of $\frac{what I want}{what I have}$ .

Example: A recipe calls for 3 cups of water and 5 cups of flour to make 15 servings. I need to make 25 servings, so what do I do? Start with the ratio: $water (c) : flour (c) : servings = 3 : 5 : 15$ . Given that I'm making 25 servings, I want to know how much water and flour I need. The proportion of $flour(c) : servings = 5 : 15$ , so I need to multiply $\frac{what I want}{what I have} = \frac{5 cups of flour}{15 servings}$ by $what I have$ : $\frac{5}{15} \times 25 = \frac{125}{15} = 8 \frac{1}{3} cups of flour$

The same process works for the water, but in that case the proportion is $\frac{3 cups of water}{15 servings} \times 25 servings = 5 cups of water$ .

If it all worked out, we should have our flour and water in the correct proportion as well: $flour : water = 5 : 3$ , so $flour = \frac{5}{3} water$ . Now we check, is $8 \frac{1}{3} = \frac{5}{3} \times 5$ ? $8 \frac{1}{3} = \frac{25}{3}$ and $5 \times \frac{5}{3} = \frac{25}{3}$ , so it all works out! We can use these amounts, and the recipe will make 25 servings.

Worked examples

An architect is designing a new building. The building will be 30 feet tall, and each window will be 8 feet high. On the drawing, the building is 8 inches tall. How tall are the windows on the drawing?

Start by noting that we have proporitional items here, because we are scaling something up or down. The true size of the building is 30 feet and the true size of each window is 8 feet, so they have a proportion of $30 : 8 = 15 : 4$ . We can also say that the windows are $\frac{4}{15}$ of the height of the building, or the building is $\frac{15}{4}$ of the height of the windows.
We know that the building is 8 inches tall on the drawing, so if we use our proportion of $window = \frac{4}{15} building$ , then we can just multiply $8 \times \frac{4}{15}$ to get $\frac{32}{15} = 2 \frac{2}{15}$ inches.

The map has a scale that says 1/4 inch is 5 miles. If the route we're taking to New Orleans is 24 inches long on the map, how many miles will we be driving to get to the cruise terminal?

The ratio of map-distance (inches) to real distance (miles) is $\frac{1}{4} : 5 = 1 : 20$ . That means the proportion of real world miles per map inch is $\frac{20}{1}$ , so if we want to know the distance, we can use that proportion
$\frac{20}{1} \times 24 = 480$ , so we will be driving 480 miles.

Media resources

Guided practice

If we are in Mexico, and the exchange rate is $1 = 24 pesos, how many dollars do we need to bring if we want to have at least 2000 pesos?
A recipe calls for $2 \frac{1}{4}$ cups of flour and 4 eggs. If we only have three eggs, how much flour should we use to make the reduced recipe?
My wallet-size photo Gus is 3 cm wide and 5 cm tall. If I want a larger photo to put on my wall, and I want the area of the photo to be 135 square centimeters, then how many centimeters wide should the larger photo be?

Homework

#hw (math) (gr7) 7.3 Proportions (p. 293) 2025-02-03