Ratios Part 2: What Can I Do with Ratios?

Ratios allow us to compare some pretty unexpected things — did you know that we can use ratios to tell if an asteroid came from outside our solar system?

A picture of the solar system.

Image by Dakota Monk

From measuring thousands of meteorites, we have found that there is a constant ratio of the abundances of isotopes for various elements in our solar system (O-17 to O-16, for example). This is because our solar system was all formed from a single cloud of dust. Therefore, if we measure the ratio of a certain isotope in a meteorite, and the ratio is different enough from the observed common ratio of our solar system, then we know it originated outside of the solar system.

Here is a bit more about how ratios work. Ratios compare a part to another part or a part to a whole. 

A kitten watches the camera intently. A dog stares wide-eyed at the camera.

Images by Brodie Vissers and Scott Murdoch

Example: part to part

apples to oranges
cats to dogs
pies to cookies
oxygen to hydrogen

Example: part to whole

red apples to all apples
tabby cats to all cats
apple pies to all pies

You can also use proportions to find values that you are missing either in the number of items that you have or in the relationship between them.

Example

You have a deck of cards that come in only green, blue, and red. For every 10 cards, 3 are green, 6 are blue, and the rest are red. How many out of every 10 are red?

Because we are given the ratio of two out of the three possible colors for every 10 cards, we can easily calculate the ratio of red cards per 10 total cards by subtracting the other two ratios from the whole:

\( \frac{10}{10}  –  \frac{6}{10}  –  \frac{3}{10}  =  \frac{1}{10}\)

Therefore, 1 out of every 10 cards is red.

A deck of cards fanned out on a table.

Image by Sarah Pflug

Example

Let’s try another one. You have a deck of cards that come in green, blue, and red. \( \frac{1}{2} \) are green, \( \frac{1}{4} \) are blue, and the rest are red. If you have 52 cards, how many are red?

If \( \frac{1}{2} \) of our 52 cards are red, there must be 26 red cards (52 x \( \frac{1}{2} \) = 26). If \( \frac{1}{4} \) of our 52 cards are blue, there must be 13 blue cards (52 x \( \frac{1}{4} \)). That means 26 + 13 = 39 cards are either green or blue. If the rest are red, 52 – 39 = 13 cards are red.

The slightly faster way to get the same answer is to recognize that a whole deck of cards is 1 deck. If \( \frac{1}{2} \) of the cards are green and \( \frac{1}{4} \) are blue, that means \( \frac{1}{2} \) + \( \frac{1}{4} \) = \( \frac{3}{4} \) of the cards are either green or blue (out of 1 deck). 1 – \( \frac{3}{4} \) = \( \frac{1}{4} \) of the cards must be red. 52 cards times \( \frac{1}{4} \) is 13, which means there are 13 red cards. 

You can even find missing values in ratios that include more than two numbers.

Example

If 1:2:5 is proportional to x:6:y, what are x and y?

The only value that we know in both the first and second ratios is the middle value — it starts off as a 2 and, through multiplication, becomes a 6. 2 x 3 = 6, so the constant of proportionality that will scale our first ratio up to our second ratio is 3. If we multiply every value in our first ratio by 3, we will have our answer. 1 x 3 : 2 x 3 : 5 x 3 = 3 : 6 : 15

Therefore, x must be 3, and y must be 15.