What is a Ratio?

Links and useful resources

Lesson-specific resource links

Concept summary and lesson

ratios and proportions
proportional assignment of ratio parts
Converting ratios to fractions
ratios with fractions and mixed numbers

Worked examples

Working with ratios is tricky until your realize two things:

They're just fractions of a common denominator
You have to add all of them up in order to get the shared denominator.

Ratios are a kind of short hand for writing a bunch of related fractions all in a group. Example: You and three friends shared $240 in the ratio 1:2:4:5. How much money did each person get?

What that $1 : 2 : 4 : 5$ means is that you divided the money into enough equal-sized pieces so that you could give that number of pieces to each person. $1 + 2 + 4 + 5 = 12$ , so in order to share the money like this, you need to split it into $\frac{1}{12}$ sized shares.

Then, you just give each person the number of shares equal to their part of the ratio: $\frac{1}{12} : \frac{2}{12} : \frac{4}{12} : \frac{5}{12}$ . Since we have $240, one share is $\frac{1}{12} * 240 = 20$ , so the proportional shares of the money work out to $20, $40, $80, $100.

Other useful ways to look at ratios

Ratios are indeed fractions, but we can also think of them as proportions and skip the denominator for some kinds of problems. Let's say we know that Gus gets twice as many dog treats as Maggie. We can write the ratio of $G u s : M a g g i e$ dog treats as $2 : 1$ .

Now we can answer questions like If Maggie got 17 treats, how many did Gus get? We know that he got two treats for every one of hers, so we can multiply her treat count by $\frac{2}{1}$ and get 34. This still works if the numbers aren't so nice: Say we have a recipe that requires a ratio of $o i l : v i n e g a r$ that's $3 : 4$ . We measured 7 oz of vinegar, how much oil do we need? Well, we know that for every four ounces of vinegar, we need three ounces of oil. We know that we have $\frac{7}{4}$ portions of "four ounces of vinegar" (divide the amount by four, that gives how many four-ounce portions we have). Now we know that for each of those, we need three ounces of oil. That phrase suggest that we need to multiply the number of vinegar portions by three to get the amount of oil: $3 \cdot \frac{7}{4} = \frac{21}{4} = 5 \frac{1}{4}$

What did we do there? Since we know the amount for one of the ratio parts, we can see what we have to multiply the ratio number by to get the result. It's like a recipe - If you're making a double batch, you mutiply every measure by two. Well, if you have to make a $\frac{7}{4}$ batch, you multiply every measure by $\frac{7}{4}$ ! If you know one ingredient, find its multiple and then use it to figure out the rest.

The oil and vinegar could also have been solved with fractions:

Let the total amount of the mixture we make be $x$
We know that there are 7 total parts ( $3 oil + 4 vinegar$ ), so the total mix is $\frac{4}{7}$ vinegar and $\frac{3}{7}$ oil.
We know that there are 7 oz vinegar, so $7 = \frac{4}{7} x$
In order to find the oil, we want to know what $\frac{3}{7} x$ will be

\begin{aligned} \frac{4}{7} x & = 7 \\ 4 x & = 49 & multiplicative Property of Equality \\ x & = \frac{49}{4} & division property of equality \\ x & = 12 \frac{1}{4} \end{aligned}

So we know there are $12 \frac{1}{4}$ oz all together. That means that there must be $12 \frac{1}{4} - 7 = 5 \frac{1}{4}$ ounces of oil. Good thing we got the same answer!

Media resources

Guided practice

What common denominator would you need for the ratio 5:6:3
How would you convert the ratio 7:12 into equations that use plain fractions?
How many parts would you need to break a candy bar into so that you could have a 5:2:3 ratio of pieces with your friends?
If paint colors for a particular shade of purple uses $R : G : B$ in the ratio $13 : 2 : 77$ , and you know that R got 4 gallons, how many gallons did G and B get? Hint: What multiple of the basic ratio are we using here?

Homework

#hw (math) (gr7) 7.1 What is a Ratio? (p. 281) 2025-01-29