Article: Ratios can be reduced and simplified like fractions by removing any common factors of the terms in the ratio. To reduce a ratio, divide all the terms in the ratio by the common factors they share until no common factor exists. However, when doing this, it's important not to lose sight of the original quantities that led to the ratio in the first place. In the classroom example above, 5 girls to 10 boys (5: 10), both sides of the ratio have a factor of 5. Divide both sides by 5 (the greatest common factor) to get 1 girl to 2 boys (or 1: 2). However, we should keep the original quantities in mind, even when using this reduced ratio. There are not 3 total students in the class, but 15. The reduced ratio just compares the relationship between the number of boys and girls. There are 2 boys for every girl, not exactly 2 boys and 1 girl. Some ratios cannot be reduced. For example, 3: 56 cannot be reduced because the two numbers share no common factors - 3 is a prime number, and 56 is not divisible by 3. One common type of problem that employs ratios may involve using ratios to scale up or down the two numbers in proportion to each other. Multiplying or dividing all terms in a ratio by the same number creates a ratio with the same proportions as the original, so, to scale your ratio, multiply or divide through the ratio by the scaling factor. For example, a baker needs to triple the size of a cake recipe. If the normal ratio of flour to sugar is 2 to 1 (2: 1), then both numbers must be increased by a factor of three. The appropriate quantities for the recipe are now 6 cups of flour to 3 cups of sugar (6: 3). The same process can be reversed. If the baker needed only one-half of the normal recipe, both quantities could be multiplied by 1/2 (or divided by two). The result would be 1 cup of flour to 1/2 (0.5) cup of sugar. Another common type of problem that incorporates ratios asks you to find an unknown variable in one ratio, given the other number in that ratio and a second ratio that is equivalent to the first. The principle of cross multiplication makes solving these problems fairly simple. Write each ratio in its fractional form, then set the two ratios equal to each other and cross multiply to solve. For example, let's say we have a small group of students containing 2 boys and 5 girls. If we were to maintain this proportion of boys to girls, how many boys would be in a class that contained 20 girls? To solve, first, let's make two ratios, one with our unknown variables: 2 boys: 5 girls = x boys: 20 girls. If we convert these ratios to their fraction forms, we get 2/5 and x/20. If you cross multiply, you are left with 5x=40, and you can solve by dividing both figures by 5. The final solution is x=8.
What is a summary of what this article is about?
Reduce a ratio to its simplest form. Use multiplication or division to "scale" ratios. Find unknown variables when given two equivalent ratios.