Ratios and proportions are closely linked to each other as concepts. A ratio tells you how much of one quantity there is compared to another quantity, whereas a proportion tells you that two ratios are equal. If you’re making a drink from a concentrate with one part concentrate to five parts water, the ratio is 1:5. If you make the same drink in a ratio of 2:10, the two finished drinks will have the same strength of flavor. The two ratios are proportionate. In other words, you can multiply both parts of one ratio by the same number to arrive at the second ratio. Learning to calculate ratios and proportions can help you solve many problems in real life and in math class.
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Calculate problems involving ratios by multiplying both parts by the same number to scale the ratio up or down. To turn ratios into real-world values, find one “part” in the ratio by adding its two sides together and dividing the total real-world amount by this number. Multiply your value for one part by both sides of the ratio to find the ratio as a real world amount.
Solve problems involving proportions by equating two ratios and using an algebraic symbol in place of the unknown quantity. Rearrange the equation to find an expression for the unknown quantity, then calculate the result to find the answer.
How to Calculate Ratios
Calculating ratios involves either scaling the ratio up (or reducing it) or translating the ratio into real-world quantities. Ratios can be expressed in three ways, either separated by a colon (e.g. 2:1), separated by the word “to” (e.g. 2 to 1) or as a fraction (e.g. 2/1), and all of these tell you the same information.
Scale a ratio either up or down by multiplying or dividing both parts of the ratio by the same number. For example, if a pancake recipe uses three cups of flour to two cups of milk, the ingredients are in a ratio of 3:2. To make twice as many pancakes without ruining the consistency of the mix, you need twice as much of both ingredients. Multiply both sides of the ratio by 2 to find the ratio you need:
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3 × 2 : 2 × 2 = 6:4
Make the pancakes with six parts flour to two parts water to scale up the recipe. Similarly, if you’re using a recipe that serves six, with a ratio of 9 to 6, but you only have two people, divide both parts of the ratio by three to find the ratio you need:
9 ÷ 3 : 6 ÷ 3 = 3:2
Turn a ratio into a real-world quantity involves working out what “one part” corresponds to in real-life and then working from there. For example, imagine two friends agree to share $150 in prize money in the ratio 3:2. Calculate this by looking at the total number of parts in the ratio. In this case, 2 + 3 = 5, so one part is equal to one fifth of the money. Calculate $150 ÷ 5 = $30 to find the real-world value of one part. From here, multiply this quantity by the number of parts on each side of the ratio to find how the money is distributed:
$30 × 3:$30 × 2 = $90:$60
So one friend receives $90, and the other receives $60.
How to Calculate Proportions
You can also solve problems involving scaling by using the proportionality between the ratios. For example, if two eggs are needed to make 20 pancakes, then how many eggs do you need to make 100 pancakes?
Note that the ratios have to be equivalent (i.e. in proportion) in order for the recipe to work. Because of this, you can write the given ratio as proportionate to the second ratio (including the unknown quantity of eggs, which you call x). The ratio is:
Eggs / pancakes
This has to equal the ratio for the larger serving, so you can insert the numbers you know and set them to equal:
2 / 20 = x / 100
Turn this around so that the unknown quantity is on the left (only for clarity; this doesn’t affect the math):
x / 100 = 2 / 20
Solve this equation for x to calculate the number of eggs you need. To do this, you multiply the known quantity on the same side as x (in this case the 100 in the denominator) by the opposite quantity on the other side (in this case the 2 in the numerator), otherwise called taking a cross product.
In the stricter terms of the rules of algebra, you’re actually multiplying both sides of the equation by the same number. Here, multiply both sides by 100:
(x / 100) × 100 = (2 / 20) × 100
Since the 100s on the left hand side cancel, this leaves:
x = 200 / 20
So this means you need 10 eggs to make 200 pancakes using this recipe.
The Link Between Ratios and Proportions
It’s worth stressing that ratios and proportions tell you very similar information. The ratio of one quantity to another can easily be turned into a proportion by multiplying both parts of the ratio by the same number, and then setting the two expressions to be equal. For a ratio of 4:6, multiplying both parts by 2 gives 8:12. These two ratios are equivalent, so they are proportional, and you can write:
4 / 6 = 8 / 12
And the fraction format makes this proportionality clear. If you put these two fractions under the same common denominator, they are clearly equivalent, because:
4 / 6 = 2 / 3 × 2 / 2 = 2 / 3
8 / 12 = 2 / 3 × 4 / 4 = 2 / 3