Last Updated on March 8, 2024
Ratios are a common topic for the word problems that you can expect to see on the GRE. This article will discuss some of the basic properties of ratios and help you recognize and solve both part-to-part and part-to-whole ratio questions.
Here are the topics we’ll cover:
- What is a Ratio?
- What Information Can We Gather from a Ratio?
- Solving Ratio Questions
- Key Takeaways: Part to Part and Part to Whole Ratios
- What’s Next?
To start, let’s discuss what a ratio is.
What Is a Ratio?
In its simplest form, a ratio compares two quantities. For example, if there are 5 cats and 6 dogs in a room, we can say that the ratio of the number of cats to the number of dogs is 5 to 6. Equivalently, we can say that for every 5 cats in the room, there are 6 dogs.
Now suppose instead that the room contains 10 cats and 12 dogs. By dividing the cats and dogs each into two equal groups, we can still say that for every 5 cats in the room, there are 6 dogs. So, the ratio of the number of cats to the number of dogs is still 5 to 6.
KEY FACT:
A ratio compares two quantities.
We can express ratios in three equivalent ways. Using the previous example, we can express the ratio of cats to dogs as any of the following:
- cats to dogs = 5 to 6
- cats : dogs = 5 : 6
- cats/dogs = 5/6
While the GRE might use any of these three notations, we will use the fractional ratio when solving ratio questions in the problems that follow.
What Information Can We Gather from a Ratio?
To successfully solve ratio questions on the GRE, we need to understand the information a ratio provides. A ratio allows us to determine both of the following relationships:
- How one part of a ratio relates to the other part
- How one part of a ratio relates to the whole or total
Going back to the cat and dog example, we express the part-to-part ratio of cats to dogs as 5 to 6, or 5/6, because there are 5 cats for every 6 dogs in the room.
We can also use this information to determine a part-to-whole ratio. Because we have 5 cats and 6 dogs, we know that the total number of animals in the room is 6 + 5 = 11. So, the part-to-whole ratio of cats to total is 5 to 11, or 5/11, and the part-to-whole ratio of dogs to total is 6 to 11, or 6/11.
KEY FACT:
A part-to-part ratio shows how one part relates to another part, and a part-to-whole ratio shows how one part relates to the whole.
The major takeaway is that if we have a ratio that compares a part to a part, we can use that information to determine the ratio that compares a part to a whole.
Let’s practice with another example:
If the ratio of girls to boys in a class is 4 to 3, and all students in the class are either boys or girls, then what is the part-to-part ratio of girls to boys, and what is the part-to-whole ratio of girls to the total number of students in the class?
The part-to-part ratio of girls to boys is Girls/Boys = 4/3.
The total number of students in the class is 4 + 3 = 7. Thus, the part-to-whole ratio of girls to the total number of students is Girls/Total = 4/7.
So, in general, if we have a part-to-part ratio of A to B, we can say that the corresponding part-to-whole ratios are as follows:
- A/(A + B)
- B/(A + B)
Now, let’s discuss solving ratio questions.
Solving Ratio Questions
In our cat-and-dog example, we calculated that the ratio of cats to dogs was 5 to 6 when there were 11 animals in the room. However, what if we were in a new room with 18 dogs, and we were told that the ratio of cats to dogs was still 5 to 6? Could we determine the number of cats in the new room?
The answer is yes. Since the problem gives us a new number of dogs, we need only to extend the cats/dogs = 5/6 ratio to the new situation by letting the new number of cats be C and saying the following:
cats/dogs = 5/6 = C/18
We can cross-multiply in this equation, and solve for C:
(5)(18) = (6)(C)
90 = 6C
15 = C
In this part-to-part ratio question, we were given the part-to-part ratio of cats to dogs and a new number of dogs, and we were able to use that information to determine the new number of cats.
However, what if instead of being given a new number of dogs, we were given a new total number of animals? Suppose in a new room, the ratio of cats to dogs is still 5/6, while the total number of animals is 33. Can we still calculate the number of cats?
To do this calculation, we must re-express our part-to-part ratio of 5/6 as the part-to-whole ratio of cats/total = 5/11, which we will use to solve the problem by letting C again equal the new number of cats.
cats/total = 5/11 = C/33
(5)(33) = (11)(C)
165 = 11C
15 = C
Thus, there are 15 cats in this new room.
Now, let’s look at some more examples of ratio problems.
Example 1
In a particular refrigerator, the ratio of bottles of soda to bottles of water is 2 to 3. If there are 24 bottles of water in the refrigerator, how many bottles of soda are there?
Solution:
We are given a part-to-part ratio of bottles of soda to bottles of water, and we are given the number of bottles of water. Thus, we can solve this problem with only part-to-part information, letting S = the number of bottles of soda in the refrigerator and setting up the following ratio:
soda/water = 2/3
Since there are 24 bottles of water, we have:
S/24 = 2/3
We can cross-multiply to obtain:
(S)(3) = (2)(24)
3S = 48
S = 16
Thus, there are 16 bottles of soda.
Let’s try one more.
Example 2
If the ratio of boys to girls in a certain class is 4 to 3, and if there are 35 total students in the class, all of whom are either girls or boys, how many boys are in the class?
Solution:
In the problem stem, we should notice that we are given the part-to-part ratio of boys to girls, but we are provided with the total number of students in the class. Thus, we need to convert the part-to-part ratio to a part-to-whole ratio, and then we will be able to determine the number of boys in the class.
Boys/Total = 4/(4 + 3) = 4/7
Since there are 35 students in the class, we can let B = the number of boys and set up the following equation:
B/35 = 4/7
Cross–multiplying gives us the following:
(B)(7) = (4)(35)
7B = 140
B = 20
Thus, there are 20 boys in the class.
Key Takeaways: Part to Part and Part to Whole Ratios
Here are the key points to know to understand and solve ratio questions:
- Ratios tell us proportional relationships between quantities.
- From a ratio, we can determine how individual parts of the ratio relate to each other and to the total.
- To solve ratio questions, we can set up a proportion using the given ratio and either a value of one of the parts of the ratio or the value of the total.
What’s Next?
In this article, we have reviewed only a small portion of the topic of ratios and how they can be tested on the GRE. If you would like to learn more about GRE Ratio questions, you can check out the top-rated Target Test Prep GRE Prep Course.
