Proportion at a Glance

A proportion is a statement of the equality of two ratios. Unlike the amateur portion who tells us that he kept that status just so he could compete in the portion Olympics, a proportion tells us when two ratios are equivalent.

Let's dive a little deeper, shall we? Here we have a small school of fish. There's one fish, there's two fish, there's a red fish, and hey, there's a blue fish. We're a regular Dr. Seuss.

The ratio of red to blue is , while the ratio of orange to green is . Because  is equivalent to , these ratios are proportional.

What about comparing red fish to green fish and blue fish to orange fish? Well, the ratio of red fish to green fish is  while the ratio of blue fish to orange fish is . These fractions are not equivalent, so the ratios are not proportional.

Sample Problem

Spongebob and Patrick posted new photos on their Facebook timelines. Spongebob got 7 likes and 4 comments on his photo, while Patrick got 14 likes and 9 comments on his photo. Are the ratios of likes to comments on each photo proportional?

The ratio of likes to comments on Spongebob's post is 7 to 4, whereas Patrick's friends responded in a ratio of 14 to 9. The ratios 7 to 4 and 14 to 9 are not equivalent, which means they're not proportional.

Because proportions are equivalent, we can use cross-multiplication to solve them for any missing parts. Cross-multiplication, you're probably reminding your friends right now, is when you multiply the numerator on the left side by the denominator on the right side, which equals the left denominator times the right numerator.

If we had the ratios  and , we could cross-multiply to find that 3 × 10 = 5 × 6. This handy little trick will help us solve proportion problems where one of the four numbers is missing.

Let's say that you're a zoologist working on a groundbreaking formula for something epic, like, a magic powder that will bring unicorns back from extinction. (Yeah, there's an awful lot wrong with that statement, but let's just go with it.)

We know that it will take, among other things, both seaborgium and hafnium (they're real, look 'em up) in a ratio of 5 parts Sg to 8 parts Hf. We have an unlimited supply of hafnium, but only 15 grams of seaborgium. If we want to use up all of the available seaborgium (so that we can bring back as many unicorns as possible), how much hafnium will we need to use?

To solve this problem, and save a few unicorns, we can set up a proportion.

Use cross-multiplication to solve for x.

5x = 8 × 15
5x = 120
x = 120 ÷ 5
x = 24

We'll need to use 24 grams of hafnium. Bring on the unicorns!

Sample Problem

Find the value of n that makes this statement a proportion.

We'll solve for n using cross-multiplication. That is, 3 × 28 = 7n. Or, 84 = 7n. Divide both sides by 7 to find 12 = n. The proportion would be .

Sample Problem

The ratio of sapphires to rubies and the ratio of rubies to total gemstones are proportional. We know we have 1 sapphire and 3 rubies. If the rest are diamonds, how many diamonds do we have?

Set up a proportion.

Cross-multiply to find g × 1 = 3 × 3, or g = 9. There are 9 total gemstones. Since we already know that 4 of them aren't diamonds, we know that 5 of them are. Let's hope they're obscenely big diamonds. Hint, hint.