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Ratios & Percentages

Ratios & Percentages

At a Glance - Solving Proportions

When solving a problem that has a ratio, proportions are often a lot easier to work with. There are a few ways to solve these problems.

Method 1: cross multiplication. This is where you’ll multiply two proportions in a criss-cross to help solve the equation. Check out the examples below.

Method 2: multiply the first proportion in order to make equivalent fractions.

The really important part for both methods: compare apples to apples and oranges to oranges. If you’re comparing girls to boys on one side of the equal sign, you must also compare girls to boys on the other side (not boys to girls). The units for both numerators must match and the units for both denominators must match.

Check out this baffling baking situation:

Your great-grandma’s chocolate bourbon ball cookie recipe calls for 3 cups of flour and 6 tablespoons of brown sugar. Unfortunately, you only have 2 cups of flour, so you will have to make a small batch. You need to match the proportion of flour to brown sugar in the recipe (great-grandma wouldn’t have it any other way). How much brown sugar do you need?
  • The recipe calls for a ratio of 3 cups flour to 6 tablespoons brown sugar. That is a ratio of 3c/6t, or 1c/2t. (It’s ok to set up ratios where the measurements are different, like cups and tablespoons.)
  • You only have 2 cups of flour, but the ratio of flour to brown sugar needs to stay constant. The unknown quantity, the amount of brown sugar we need, will be represented with the variable,x. So, 1c/ 2t = 2c/ x. As you can see the units of the numerators match and the units of the denominators match, with cups of flour on top and tablespoons of brown sugar on the bottom for both.

We can solve this two different ways:

    Cross MultiplicationMaking Equivalent Fractions

    First cross-multiply. Multiply the numerator of one fraction with the denominator of the other. We will conveniently disregard the units for now. But, we will come back to them

    cross multiply

    Now set these equal to each other.

    1x = 4

    Finally, divide by the number in front of the variable.

    x = 4 / 1
    x = 4

    Wait, we're not done. We need to remember the units. We were solving for tablespoons, so we need 4 tablespoons of brown sugar.

    In this method we are making equivalent fractions, or ratios. Since we can multiply the numerator of the first ratio by 2 to get the numerator of the second, all we have to do is multiply the denominator of the first by 2 as well.

    1c/2t = 2c/x

    Since 2t x 2 = 4t, we need 4 tablespoons of brown sugar.

Example 1 (Using Cross Multiplication)

4/5 = x/15

Cross multiply and set equal to each other.

Example 1 (Using Equivalent Fractions)

4/5 = x/15

Example 2 (Using Cross Multiplication)

2/3 = x/4.5

Cross multiply and set equal to each other.

Example 2 (Using Equivalent Fractions)

2/3 = x/4.5

Here's one more example of how proportions are used in the real world:

An Alaskan wildlife preserve Ranger wants to figure out how many moose live in the preserve. To do this, she randomly catches and tags 50 moose and releases them. A few months later, she catches 75 moose and sees that 20 of them are tagged. Approximately how many moose live on the preserve?

A tricky example. Sometimes math problems like to throw in extra info to make you think.

The ratio of girls to boys in your class is 4:5. Assuming that this ratio is consistent throughout the school, approximately how many boys would be in a school of 525 students?

Solving Proportions Exercise 1

10/3 = 25/x

Solving Proportions Exercise 2

In a large bowl of Halloween candy the ratio of Snickers to Skittles if 3:2. If there are 36 packages of Skittles in the bag, how many total candies are in the bowl?

Solving Proportions Exercise 3

36/x = 8/3

Solving Proportions Exercise 4

Six out of every ten orders at the ice cream shop are for some sort of chocolate ice cream (this includes chocolate chip, triple brownie fudge, chocolate cookie dough, and all other chocolaty deliciousness). If the store sells 1230 cones on a particularly busy weekend, about how many of these are NOT of the chocolate variety?