Proportions

If you want to compare the prices of goods, you must know the cost of 1 kg or 1 m or any other unit of them. You then can compare stamp collections, prizes, profits, losses, images of objects in a microscope, in a telescope, etc.

When you say that the price of rice has risen by 10 Baht, you mean that the difference between the former and present price is 10 Baht, referred to a given unit. This is a difference of two numbers.

A film is shown in the movie theatre with an enlargement of 300, that is, the ratio of the film's image to the screen image is 300 : 1(say 300 to 1). This is the ratio of two quantities - a quotient of two numbers.

Two quantities are compared by formation of their difference or quotient.

A comparison by differences is frequently more suitable, but you also use often a quotient, for example, for mixing ratios, in sport (victory 6 : 1), etc. This mode of comparison yields the Definition:

The ratio of two quantities of the same kind
is the quotient of their measures.

You can apply to such quotients the rules of the calculus of fractions.

Task: What is the ratio of 21 mm and 18 cm? 

Task: What is the ratio of 40 m² and 20 m² with numerator 1? 

Task: What is the rate of ascent of a railway track, which climbs 50 m over the horizontal distance of 2500 m? The rate of ascent is the ratio of the distances CB and AB. In the figure

50 m/2500 m = 1/50.

This is often displayed beside the rail road track as shown; it indicates the constant rate of ascent of 1 m in the horizontal distance of 50 m for 2500 m. At small rates of ascent, one indicates often the rise for the distance AC. As rule, it is stated in such a way that the numerator is 1.

You can compare different rates. The special case, when two ratios are the same, leads to the concept of proportion (Latin: proportio).

An equation between two ratios is an equation of ratios or a proportion.

Since a ratio is a quotient of two numbers, every proportion can be replaced by a quotient equation and, inversely, every quotient equation can be interpreted as a proportion: 

If you have a proportion, you read the equation in the same way, whether it has one or the other form, that is:

a is related to b as c to d

a and d are the outer, b and c the inner terms of the proportion. 

Both ratios are equal to 5, i.e.,you have an equality of ratios - a proportion. In this example, the units on the two sides differ. In what follows, in the case that a proportion of quoted quantities is involved, you should always think of them as measures, so that a, b, c, d are numbers.

An equation a/b = c/d can be transformed in many ways; you can give it the form

Product equation: ad = cd. The product of the outer terms
of a proportion equals the product of the inner terms.

This equation lets you check a given proportion. It yields the proportions 

and four more through interchange of the sides.Thus, each proportion yields seven others, whence follows the Rule:

The inner and outer terms of a proportion can be interchanged

However, a/c = c/d does not only yield the seven proportions above, but infinitely many proportions. Multiplication of both sides of an equation by an arbitrary number m, followed by addition and subtraction of an arbitrary number n on both sides, yields 

Division of the equations on the right hand side yields

(ma + nb)/(ma - nb)=(mc + nd)/(mc - nd)

Since the additions and subtractions took place in a corresponding manner on both sides of the equation, one talks of the law of corresponding addition and subtractions, which, for m = 1 and n =1, becomes the frequently used Rule of corresponding additions and subtractions:

(a + b)/(a - b) = (c+d)/(c - d).

If you know three terms in a proportion, the fourth term can be computed. The proportion appears as a determining equation 

Task: 2/3 = 4/x. Find the fourth proportional x.

In a given proportion, x can occur in another but the fourth position; it is then still called the fourth proportional.

a : b = b : c. A proportion is said to be steady, if the internal terms are equal. It is then also called the central proportional of a and c.

Task: Find x from the steady proportional x : 14 = 14 : 2! 

If more than 2 proportions are equal, one talks of a progressive proportion, which has the special mode of writing: 

Conversely, you can compute from a progressive proportion the individual proportions.

From a : b : c = a₁ : b₁ : c₁ follows a/a₁ = b/b₁, b/b₁ = c/c₁, a/a₁ = c/c₁, or, as you can interchange the inner terms, a/b= a₁/b₁, b/c= b₁/c₁, a/c= a₁/c₁.

Task: Construct out of a/b = 3/4 and b/c = 2/3 a progressive proportion.

You must first obtain equal quotients. 

whence

a/3 = b/4 = c/6 and the progressive proportion a:b:c = 3:4:6.

Exercises: 

Answers

last next