Multi-way ratios

Links and useful resources

Concept summary and lesson

Ratios with more than two parts
Operations on ratio groups (multiply, divide) and why they are allowed
Converting all terms of a ratio to integers
comparing parts of a ratio (How many times bigger is one than the other?)

Worked examples

Multi-part ratios are just ratios

The only difference between a multi-part ratio and a two-part ratio is that you're breaking the total up in to more categories. It still works exactly the same way: They are all shorthand for fractions of the total, and the exact fraction of the total can be found by adding up all of the ratio terms and using it as the denominator:

if $a : b : c = 2 : 3 : 7$ , then we can use $2 + 3 + 7 = 12$ for the denominator. We need to make some kind of variable so we can represent the total, so let's call it $total= T$ :

\begin{aligned} a & = \frac{2}{12} T \\ b & = \frac{3}{12} T \\ c & = \frac{7}{12} T \end{aligned}

Now we can see that they are all just proportionally-sized parts of whatever the total amount is. You can also see that you can compare them to each other:

\begin{aligned} \frac{a}{b} & = \frac{\frac{2}{12} T}{\frac{3}{12} T} \\ = \frac{2 T}{3 T} \\ = \frac{2}{3} \end{aligned}

So that means you can instantly see how much bigger or smaller one part of the ratio is compared to another by just dividing the ratio terms!

Comparing parts of a ratio to each other

When you have a ratio, it tells you what the numerators are if you put all of the pieces over a single common denominator. We learned to add up all of the ratio terms to find a common denominator that makes the parts all add up to one, but in truth the ratio is true no matter what denominator you use!

So, let's say you have the ratio a:b:c = 7:3:8 for some problem. If the question gives you a value for one of the terms, you can easily figure out the others by noticing that the ratio tells you how much bigger one is compared to the other.

In the example, $8$ is $\frac{8}{3} \cdot 3$ , so c is $\frac{8}{3}$ times bigger than b, no matter what you're given for c or b. Let's say that you're told c is 24... Then to find b, all you do is divide by $\frac{8}{3}$ , or multiply by $\frac{3}{8}$ : $24 \times \frac{3}{8} = 9$ .

What about comparing one part to several other parts? It's as easy as adding the parts together and dividing:

If $a : b : c : d : e = 2 : 3 : 6 : 4 : 7$ , what is $b : c + a$ ? Since these can all be converted to fractions of a whole, we can use the same trick from above to do the math (the sum of all of the ratio parts is 22...):

\begin{aligned} b & = \frac{3}{22} T \\ c & = \frac{6}{22} T \\ a & = \frac{2}{22} T \\ \frac{b}{c + a} & = \frac{\frac{3}{22} T}{\frac{6}{22} T + \frac{2}{22} T} \\ = \frac{3}{6 + 2} \\ = \frac{3}{8} \end{aligned}

It works out so that you just add the ratio terms! No need for any fancy fractions.

When to use what approach

We have two basic approaches here: One of them lets of solve problems by comparing parts of the ratio to each other, and the other one lets us solve problems by comparing parts of the ratio to the total amount.

So if your problem gives you the total, you'll usually be better off working with fractions of the whole (find the denominator and make a total variable).

If your problem doesn't give you the total, but you have some information about the parts, then you can still use the fractions approach, but sometimes it's easier to use the one where you compare each piece to the others directly.

Media resources

Guided practice

Reduce the ratio 4:8:32:16 to simplest terms
If you have A:B:C = 4:2:8, how many times bigger is C than B? How many times bigger is A than B? C than A?
If A:B:C:D = 3:7:5:11, how much bigger is C than A? How about comparing C to D?

Homework

#hw (math) (gr7) 7.2 Multi-way Ratios (p. 287) 2025-01-31