We use ratios to make comparisons between two things. When we express ratios in words, we use the word "to"--we say "the ratio of something to something else." Ratios can be written in several different ways: as a fraction, using the word "to", or with a colon.

Let's use this illustration of shapes to learn more about ratios. How can we write the ratio of squares to circles, or 3 to 6? The most common way to write a ratio is as a fraction, 3/6. We could also write it using the word "to," as "3 to 6." Finally, we could write this ratio using a colon between the two numbers, 3:6. Be sure you understand that these are all ways to write the same number.

Which way you choose will depend on the problem or the situation.

• ratio of squares to circles is 3/6
• ratio of squares to circles is 3 to 6
• ratio of squares to circles is 3:6

There are still other ways to make the same comparison, by using equal ratios. To find an equal ratio, you can either multiply or divide each term in the ratio by the same number (but not zero). For example, if we divide both terms in the ratio 3:6 by the number three, then we get the equal ratio, 1:2. Do you see that these ratios both represent the same comparison? Some other equal ratios are listed below. To find out if two ratios are equal, you can divide the first number by the second for each ratio. If the quotients are equal, then the ratios are equal. Is the ratio 3:12 equal to the ratio 36:72? Divide both, and you discover that the quotients are not equal. Therefore, these two ratios are not equal.

Some other equal ratios:
3:6 = 12:24 = 6:12 = 15:30

Are 3:12 and 36:72 equal ratios?

Find 3÷12 = 0.25 and 36÷72 = 0.5

The quotients are not equal —> the ratios are not equal.

You can also use decimals and percents to compare two quantities. In our example of squares to circles, we could say that the number of squares is "five-tenths" of the number of circles, or 50%.

Here is a chart showing the number of goals made by five basketball players from the free-throw line, out of 100 shots taken. Each comparison of goals made to shots taken is expressed as a ratio, a decimal, and a percent. They are all equivalent, which means they are all different ways of saying the same thing. Which do you prefer to use?