Ratio

A ratio is the relation between two like numbers or two like values. A ratio may be written as a fraction (3/4), as a division expression (3÷4), or with a colon (3:4), or it can be stated as '3 to 4'. A ratio is simply a fraction. The ratio of the number a to the number b (b ≠ 0) is a/b.

The order of the terms of a ratio is important. For example, the ratio of the number of months with exactly 30 days to the number with exactly 31 days is  4/7, not 7/4.

Because a ratio can be regarded as a fraction, multiplying or dividing both terms of a ratio by the same number doesn’t change the value of the ratio. So, 1:4 = 2:8 = 4:16. To reduce a ratio to its lowest terms, simplify the ratio as you would a fraction.

Ratios often come up up in word problems. Suppose an auto manufacturer ships a total of 160 cars to two dealerships at a ratio of 3 to 5. This means that for every three cars that go to Dealer 1, five cars ship to Dealer 2. To determine how many cars each dealership receives, add the terms of the ratio, or 3 + 5, to get the total number of fractional parts each dealership will get: 3 + 5 = 8. The first dealership will receive 3/8 of 160 cars which equals 60. The second dealership receives of 5/8 160 cars which equals 100.

Proportion

A proportion is a relationship between two equal ratios. It may be written as the proportion sign :: or with an equal sign. So you can read 1:4 :: 2:8 as '1 is to 4 as 2 is to 8'.

So, a proportion is a statement that two ratios are equal; for example, 2/3 = 8/12 is a proportion. The first and last terms in a proportion are called the extremes, and the second and third terms are called the means. If you multiply the means together and multiply the extremes together and then compare the products, you find that the products are the same.

Anytime you know three terms of a proportion, you can find the missing term first by multiplying either the two means or the two extremes (depending on which are known) and then dividing the product by the remaining term. This is also known as cross-multiplying.

For example, to solve for n in the proportion 2/3 = n/12, cross multiply, obtaining 24 = 3n; then divide both sides by 3, to get n = 8.