Considering numbers in terms of parts and multiples, it is possible to express a relationship between these whole numbers called natural numbers. As we can see, 5 is one third part of 15; 10 is two third parts of 15; 10 is half of 20, etc. More specifically, the relationship between any two numbers is called their ratio.
For example; what is the relationship of 1 to 2?
2 is two times 1; or 2 is twice as large as 1.
Or, what relationship does 5 have to 15?
15 is three times 5; or 15 is three times as large as 5.
Or, what relationship does 15 have to 5?
5 is one third part of 15.
The numbers involved in expressing a ratio are called the terms of the ratio. The ratios above have two terms, such as 15 to 5 or 1 to 2. If expressing the ratio of a greater number to a smaller number, like 5 to 15, it can be seen in terms of how much greater the larger is than the smaller. If a smaller number is being expressed in relation to a larger number, such as 2 to 3, then the smaller can be expressed in terms of being parts of the larger. For example, 2 is two third parts of 3, and 6 is six seventh parts of 7.
A ratio can be expressed using the colon symbol (:). Thus, the ratio of 1 to 2 can be written as follows:
1:2
Note | Ratio is often used for scale models. Imagine a company wishes to design a new airplane. Rather than waste time and money experimenting on full-size airplanes in the design stage, it's better to create a miniature version and then later, when manufacturing the final airplane, the design will be scaled to a larger size. The relationship between the size of the model and the size of the final airplane will be in a ratio. The model will be so many times smaller than the actual product. |
A proportion is two ratios that are the same. So, two quantities are said to be proportional when their ratios are identical. Consider the following two ratios:
3:9 | 4:12 |
These ratios are the same, and it is said that 3 is to 9 as 4 is to 12. From these numbers, it should be clear that 9 is three times 3, and 12 is four times 3. In other words, 3 is the third part of 9 just as 4 is the third part of 12. Thus, these two ratios are the same and are therefore proportional.
Just as the two numbers featured in a ratio are called terms, so are the two ratios in a proportion. Every number is called a term. Consider the following ratios:
5:20 | 10:40 |
In these ratios 5 is to 20 as 10 is to 40. 5 is the fourth part of 20 just as 10 is the fourth part of 40. 5 is considered to be the first term of the proportion, 20 is the second term, 10 is the third, and 40 is the forth. The first and third terms (5 and 10) are said to be corresponding terms, and so are the second and fourth terms (20 and 40). These corresponding terms are also in a ratio. From this we can derive an important fact: Where four numbers are proportional, the first term is to the third as the second term is to the fourth. From our example we can see that 5 (the first term) is half of 10 (the third term), just as 20 (the second term) is half of 40 (the fourth term).
If each term in a single ratio is multiplied by the same number, the two terms will still be in the same ratio after multiplication. In the ratio 5:10, both terms are multiplied by 2 to make 10:20. The ratio of 10:20 and 5:10 are the same, and both ratios are therefore proportional. This rule of multiplication applies to any ratio.
Just as any two terms in a single ratio can be multiplied by a common number to make an equivalent ratio, so they can also be divided by a common number. This is called reducing, or simplifying, a ratio. For example, the ratio 3:15 is the same as 1:5. 15 is five times greater than 3, or 3 is the fifth part of 15. In the same way, 3 is three times greater than 1 and 1 is one third part of 3. In most cases, when given any ratio it is standard practice to reduce it to its lowest terms where possible. This means to find a common divisor (a single number to divide both terms by) and rewrite the ratio using the lowest numbers possible. In our example, what number could 3:15 be divided by to rewrite the ratio as1:5? This process is called reducing to its lowest terms. In our example, both 3 and 15 can be divided by 3 to produce 1:5.
Note | Once a ratio is in its lowest terms it cannot be simplified any further. You can determine whether any ratio is in its lowest terms because, in such a ratio, each term can be divided by no common number except 1. |
Two quantities are said to be inversely proportional if, when one quantity changes, the other changes in the opposite ratio. Thus, when one quantity halves (2: 1), the other doubles (1: 2). In these two ratios, the positions of the terms have been exchanged; one ratio is the inverse of the other.
The weight of a vehicle is related to the speed it can travel. The heavier it is, the slower it will move. A vehicle twice as heavy as another will move twice as slowly. The relationship between weight and speed is therefore said to be inversely proportional.
Ratios are typically employed to solve two types of problems. The first is where, given a ratio and a specific amount of one quantity, you must calculate the value of the remaining quantity. The other is where you must share a total quantity in a given ratio. Let's begin with the first kind.
Determining quantities
If a scale model of an airplane is constructed with a ratio of1:2 meters, then 1 meter of the model represents 2 meters of the real, full-size airplane. The scale model therefore could be said to be half the size of the full-size airplane. Thus, if given a full-size airplane whose length is 30 meters, what is the length of the scale model?
We know the model is half as large as the full-size airplane, so the length of the model in meters must be half of 30, which is 15. Likewise, if we knew the size of the model and needed to determine the size of the full-size airplane, in this case we examine the ratio to establish that the model is twice as large. So if the model were 30 meters in length, the 30 would need to be doubled (multiplied by 2) and the full-size airplane would therefore be 60 meters in length.
Sharing quantities
The ratio of women to men is2:8.Given 400 people, how many women are there, and how many men?
From this we can establish that the ratio of women to men is two-eighths, or there are four times as many men as there are women. Together, the terms of the ratio add up to 10 since 2 + 8 = 10. These make the complete ratio of men and women. Of this total, two-tenths are women, and eight-tenths are men. If 400 (the total number of people) were divided by 10 (so 10 people per part), there would be 40 parts. We also know that for each part there are 2 women and 8 men. Using our knowledge of multiplication, multiples, and parts, it can be determined that 40 × 2 (women) = 80 women and 40 × 8 (men) = 320 men.