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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.

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We can express that ratio in several different ways:

- Using the word “to,” as in “a 3 to 2 ratio”
- Using a colon
- Or even as a fraction

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Each of these ratios means the same thing - that for every 3 red squares, there are 2 blue circles.

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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...

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...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.

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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.

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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.

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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).

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Let’s add some more shapes into the mix.

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With the addition of 8 yellow triangles, we can now define some new ratios, such as:

- The ratio of yellow triangles to red squares
- The ratio of blue circles to yellow triangles
- Or, the ratio of yellow triangles to the total number of shapes

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Notice that our part to whole ratios for the other shapes will change when we add the yellow triangles.

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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.

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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.

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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.

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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.

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So, we can also say that the ratio between blue circles and yellow triangles is 1 to 4.

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Let’s move on to Proportions, which is where ratios become really useful for problem-solving.

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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.

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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:

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The most common method of solving proportions is cross-multiplying

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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 4*x* = 165

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Now we just need to solve...

16 times 5 gives us 80…

Divide both sides by 4…

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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.

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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?

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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.

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In our original recipe, we had 3 eggs and 4 cups of flour. The ratio of eggs to flour is 3 to 4.

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In our adjusted recipe, we’ll only be using 2 eggs, and we need to determine the appropriate number of cups of flour.

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To solve, we just have to cross-multiply, which gives us 3 times *x* equals 2 times 4

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And then solve for *x*: 2 times 4 gives us 8. Then we divide both sides by 3 to get x by itself

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So, our answer is 83, or 2 and 2/3 cups of flour.

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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.

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If we wrote the proportion like this:

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Our proportion would not make sense, because our ratio of eggs to flour was 3 to 4 -- not 2 to 4.

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We’ve mixed and matched the numbers from our ratios in an illogical way.

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We could, however, write our proportion like this:

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Here, we made sure that the numbers in each ratio correspond with one another logically.

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These proportions would also work:

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See for yourself! Solving any of these proportions for *x* will give you the same value we got the first time.

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As long as you write a proportion that represents the scenario accurately, you can use these skills to solve all kinds of word problems.

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