What are Equivalent Ratios?
A ratio is a way to describe the relationship between two different quantities without having to know the exact values of said quantities. Equivalent ratios are two ratios that express the same relationship between numbers.
You may recall from my previous video on ratio’s that a ratio is a way to describe the relationship between two different quantities where we may not know the exact value of either quantity. A ratio notation looks like this. Let’s say we have two numbers, A and B, and the ratio of A to B, written as A:B, is, for instance, let’s say 3 to 2.
This tells us that for every 3 in A, there are 2 in B. It can be applied to a broad range of things. Let’s say that we have a bag of marbles. This bag contains red marbles, blue marbles, and white marbles. The example that I’m going to use here, I’m going to demonstrate two different possible types of ratio problems that you might encounter.
Let’s say that we’re told that we have 18 red marbles. R equals 18. We’re told that the ratio of red marbles to blue marbles is the same as the ratio of blue marbles to white marbles. For the first type of problem here, we’ll say that we’re given that this ratio of red to blue marbles is 3 to 5. This is one piece of information we could be given.
Let’s solve the problem using that piece. We’re told that the ratio of red marbles to blue marbles is 3 to 5 and that the number of red marbles is 18. We can set up a set of equivalent fractions and cross multiply to find the missing quantity. This ratio notation can be rewritten as R over B is equal to 3 over 5.
We know that R equals 18, so we can just rewrite this as 18 over B equals 3 over 5. Now we can cross multiply. We’ll multiply 18 times 5 and that’ll give us 90. We’ll cross multiply the other way. That gives us 3B. To solve for B, we can just divide both sides by 3. That gives us B equals 30. From this ratio, we knew that for every 3 red marbles there were 5 blue marbles.
That led us to find the number of blue marbles. Now, because we know that this ratio is the same as this ratio, we can apply the same ratio one more time. We can say that B over W is equal to 3 over 5. We have the value for B here and it’s 30. 30 over W is equal to 3 over 5.
Once again, we can cross multiply and that will give us 150 equals 3W or W equals 50. We went through this process twice and found the values of of each marble count. The other way that you can be given some information in a ratio problem like this, we’re already given that there are 18 red marbles and that the ratio of red marbles to blue marbles is the same as that of blue marbles to white marbles.
What if instead of being given this ratio, the value of this ratio, what if we’re given the value of one of the other marble counts? Let’s say that instead of everything below here, we’re told that R equals 18 and B equals 30. Now we have to set up an equivalent fraction equation using this ratio here. We’re given the values for R and B.
All we have to do to find the number of white marbles is plug in R and B into this equation here. We’ll have R over B equals B over W, which is the same as 18 over 30 is equal to 30 over W. We aren’t given the ratio, but because we’re given to the quantities here, we have to find the ratio before we can solve for W.
18 over 30 is essentially the ratio. To get the true ratio, we need to reduce it. 18 and 30 can both be divided by 6. If we divide 18 by 6 we get 3, and if we divide 30 by 6 we get 5. In doing this, we found the ratio and now we can just cross multiply with these two fractions. That will give us the value for W.
We’ll have 5 times 30 is 150 and that is equal to 3W or W equals 50. Essentially, there were four quantities involved in both of these problems. The numbers of red, blue, and white marbles and the ratio of each set of marbles. We had to use different methods depending on which pieces of information were missing.
We found the ratio in the white over here and the blue and the white over here. There’s going to be some variability in what particular variables you’re given when you’re asked to solve a ratio problem like this. There’s actually one further twist you might see on occasion.
That would be that instead of being given red and blue, you might be given red and white. If you’re given red and white, you no longer have three out of the four values in this equivalent ratio here. What you’ll end up having is, if you’re given red and white instead of red and blue, you’ll have an equation that looks like this: 18 over B is equal to B over 50.
In this case, if you cross multiply, what you end up with is a B squared that’s equal to 900. B squared equals 900. You then have to take the square root of both sides. That involves kind of the next level of calculation here. The square to B squared is B and the square root of 900 is, of course, 30. That’s one of the twists you might see on these equivalent ratio problems.