A ratio is a relationship between two quantities. Let’s say that we have 3 red squares and 2 blue circles. In this scenario, we can say that the ratio of red squares to blue circles is 3 to 2.
We can express that ratio in several different ways:
Each of these ratios means the same thing - that for every 3 red squares, there are 2 blue circles.
However, we have to pay close attention to the way we state the ratio we are looking for. Order matters! For example, the ratio of blue circles to red squares would be 2 to 3...
...rather than 3 to 2. We must make sure that the quantity we state first - in this case, the number of blue circles - comes first in our ratio.
The ratios we have been looking at so far are known as Part to Part Ratios. They define the relationship between two parts of a greater whole.
The other major category of ratios is Part to Whole Ratios, which define - you guessed it - the ratio between a part and a whole. Let’s take another look at our collection of shapes.
If we want to know the ratio of blue circles to the total number of shapes, we must first count up the total number of shapes Since there are 5 shapes in total, we can say that the ratio of blue circles to the total number of shapes is 2 : 5. Similarly, we can say that the ratio of red squares to the total number of shapes is 3 : 5. These ratios are Part to Whole Ratios, because they describe the relationship between the parts (red squares or blue circles) of the greater whole (the total number of shapes).
Let’s add some more shapes into the mix.
With the addition of 8 yellow triangles, we can now define some new ratios, such as:
Notice that our part to whole ratios for the other shapes will change when we add the yellow triangles.
The ratio of blue circles to the total number of shapes is now 2 to 13, because we have a total of 13 shapes, rather than 5. Similarly, the ratio of red squares to the total number of shapes is now 3 to 13.
Of course, the part to part ratio of blue circles to red squares hasn’t changed at all: it’s still 2 to 3, because the quantity of each of these parts has not changed.
It’s generally best to express ratios in their simplest form - that is, with the smallest integers possible in both parts of the ratio. Let’s take another look at the ratio of blue circles to yellow triangles, as an example.
We can simplify this ratio in the same way that we would simplify any other fraction. By dividing both sides of the ratio - or both the numerator and denominator of the fraction - by 2, we can see that a 2 to 8 ratio is equivalent to a 1 to 4 ratio.
So, we can also say that the ratio between blue circles and yellow triangles is 1 to 4.
Let’s move on to Proportions, which is where ratios become really useful for problem-solving.
A proportion is a statement that two ratios are equal to one another. For example, the ratios “2 to 1” and “10 to 5” are proportional, because 10 to 5 can be reduced to 2 to 1. We could write this as a proportion that looks like this.
When you see a proportion in a math problem or on a standardized test, you will typically be missing one of the values in the proportion. Solving a proportion is the process of determining what that value is, using algebra. Let’s look at an example:
The most common method of solving proportions is cross-multiplying
We multiply the numerator of one side by the denominator of the other side, and set that equal to the product of the other two values. Here, we get 4x = 165
Now we just need to solve...
16 times 5 gives us 80…
Divide both sides by 4…
And we find that x equals 20. We know that we found the right answer because the fraction ⅘ is equivalent to the fraction 16/20; we found the value of x that makes these two ratios equal.
In many cases, you’ll have to write your own proportion in order to solve a word problem. Let’s say that you’re baking a cake, and the recipe calls for 3 eggs and 4 cups of flour. Upon checking the fridge, you realize that you only have 2 eggs! How many cups of flour should you use?
We can find out by setting up a proportion expressing the idea that the ratio between eggs and flour in our adjusted recipe, should be the same as it was in our original recipe.
In our original recipe, we had 3 eggs and 4 cups of flour. The ratio of eggs to flour is 3 to 4.
In our adjusted recipe, we’ll only be using 2 eggs, and we need to determine the appropriate number of cups of flour.
To solve, we just have to cross-multiply, which gives us 3 times x equals 2 times 4
And then solve for x: 2 times 4 gives us 8. Then we divide both sides by 3 to get x by itself
So, our answer is 83, or 2 and 2/3 cups of flour.
When you set up a proportion to solve a word problem, you have to make sure that the ratios you write to represent the situation are logical.
If we wrote the proportion like this:
Our proportion would not make sense, because our ratio of eggs to flour was 3 to 4 -- not 2 to 4.
We’ve mixed and matched the numbers from our ratios in an illogical way.
We could, however, write our proportion like this:
Here, we made sure that the numbers in each ratio correspond with one another logically.
These proportions would also work:
See for yourself! Solving any of these proportions for x will give you the same value we got the first time.
As long as you write a proportion that represents the scenario accurately, you can use these skills to solve all kinds of word problems.