Math Help: Proportions and Ratios

Help with Ratios and Proportions

Ratio Relationships

Ratios express a relationship between two quantities. They can be written in fraction form (1/4) or in standard ratio notation (1:4). Ratios are used in lots of everyday activities, like cooking. For instance, a recipe might call for one cup of sugar for every three cups of flour, which could be represented with the ratio 1:3.

Ratios are often used to help complete household projects too. For example, if it takes two cans of paint to cover four walls, the ratio of cans of paint to walls covered is 2:4. You're also likely to encounter ratios in sports. For instance, you might set a goal of running five kilometers in 30 minutes (5:30).

Notice that both of these ratios can be written as fractions and reduced. For instance, 2:4 can be written as 2/4, which reduces to 1/2. This ratio tells you that one can of paint covers two walls. Likewise, 5:30 = 5/30 = 1/6, which means that running five kilometers in 30 minutes is equivalent to running one kilometer in six minutes.

Proportional Relationships

Proportions state that two ratios are equal. Using the previous example, we could write a proportion stating that 5/30 = 1/6. This shows that someone who runs five kilometers in 30 minutes is running at the same speed as someone who runs one kilometer in six minutes.

Now, let's imagine you want to know how far you would go if you ran at a speed of one kilometer every six minutes for two hours (120 minutes). To find out, we need to use a technique called cross-multiplication. Follow these steps to solve the problem:

1. Set up the proportion, like this: 1/6 = x/120. Notice that like units must be used, and they must appear in the same part of each fraction. In this example, the times in minutes are on the bottom (in the denominator) and the distances in kilometers are on the top (in the numerator).
2. Cross-multiply each numerator by the opposite denominator and write the results on either side of an equals sign. For this problem, the result will be 6x = 120.
3. Solve for x. Here, we need to divide both sides of the equation by six to find that x = 20. This means that you would cover 20 kilometers if you ran at the same speed for 120 minutes.