PROPORTION

PROPERTIES OF A PROPORTION

44. Definition. A proportion is a statement of the equality of two ratios, as A: B = X: Y.

These four magnitudes are said to form a proportion, of which A and Y are the extremes, and B and X the means; and Y is called the fourth proportional to the three terms A, B, and X. The proportion is sometimes read thus: 4 is to B as X is to Y.

The next three theorems are concerned with the establishment of certain general "rules of inference," by which, from a given proportion, certain other proportions can be at once derived. They are the Rules of Equi-multiplication, Alternation, and Composition.

Equi-multiples of homologous terms.

45. THEOREM 12. If two ratios are equal, and if any like multiples of the antecedents are taken, and also any like multiples of the consequents, then the multiple of the first antecedent is to the multiple of the first consequent as the multiple of the second antecedent is to the multiple of the second consequent.

Given

to prove

A: B = X: Y;

MA:nB = mx: nY.

To compare the latter two ratios, take the pth multiple of each antecedent, and the qth multiple of each consequent, and compare the order of size of the two pairs of resulting multiples

According to 9 (9), these may be written in the form

Now these two pairs of multiples are in the same order of size, because the ratios 4: B and X: Y are equal.

Therefore the former pairs of multiples are in the same

order of size, whatever whole numbers P and q may be.

NOTE. This corollary may be stated in words as follows:

If four magnitudes form a proportion, and if the first is any multiple, or part, or multiple of a part, of the second, then the third is the like multiple, or part, or multiple of a part, of the fourth.

Rule of alternation.

47. THEOREM 13. If four magnitudes of the same kind form a proportion, then the first is to the third as the second is to the fourth.

Let A, B, C, D be four magnitudes of the same kind such that

To prove

A: BC: D.

A: CB:D.

Since the ratio of two magnitudes equals the ratio of their like multiples,

hence

MA: MB = nC : nD.

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Therefore, by comparison of homologous terms in equal ratios, the two pairs

and

ms, no

mB, nD

are in the same order of size.

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Now m and n are any whole numbers; hence, by definition

Rule of composition.

48. THEOREM 14. If four magnitudes form a proportion, then the sum of the first and second is to the second as the sum of the third and fourth is to the fourth.

Given

to prove

A:BX: Y;

A+ BB = X + Y: Y.

In order to compare the latter two ratios take any like multiples of the antecedents, and any like multiples of the consequents; and then compare the order of size in the two resulting pairs

The order of the first pair of multiples is not altered by subtracting mB from each; and the order of the second pair is not altered by subtracting my from each.

Therefore the above pairs of multiples are in the same order, respectively, as the pairs

Now, from the hypothesis, these are in the same order of size; hence the above pairs are in the same order of size. Next, let n be not greater than m.

Then the pairs of multiples in question are evidently both in descending order.

are always in the same order of size whatever m and n are.

49. Cor. In the same case AB: B = X - Y: Y.

The complete statement and proof are left to the student.

50. It is convenient to insert here the following restatement of theorem 3, to be called the "rule of reciprocation."

51. If four magnitudes form a proportion, then the second is to the first as the fourth is to the third.

TWO OR MORE PROPORTIONS

52. The next three theorems are concerned with rules of inference from two or more proportions. They are the Rules of Combination, of Succession, and of Addition.

Rule of combination.

53. THEOREM 15. If there are any number of equal ratios, all the magnitudes being of the same kind, then as any of the antecedents is to its consequent so is the sum of all the antecedents to the sum of all the consequents.

Given

to prove

A: BA': B' = A": B";

A: BA + A' + A" : B + B' + B''.

From the hypothesis, and the definition of equal ratios, the three pairs of multiples

are in the same order of size; hence the pair

M (A + 4' + 4''), n(B + B' + B'')

is also in the same order as any of the preceding pairs, whatever m and n are (axioms I. 25, 32; II. 9, 10; III. 38). Therefore A: B = A + A ' + A'' : B + B' + B'.

54. Definitions. A set of ratios will be called successive when the consequent of each is the antecedent of the next. The first antecedent and the last consequent are called the extremes of the set. E.g., the ratios A : B, B: C, C : D, D : E are a set of successive ratios, whose extremes are A and E.

Rule of succession.

55. THEOREM 16. If there are any number of like magnitudes and an equal number of any other like magnitudes, such that the successive ratios in the first set are equivalent respectively to the corresponding successive ratios in the second set, then the ratios of the extremes in the two sets are equal.

1. Let there be three magnitudes in each set; and let them be

Let the successive ratios A: B and B : C be equal to the successive ratios X: Y and Y: Z, respectively.

i.e.

To prove that the ratios of the extremes are equal,

A: CX : Z.

Take any like multiples of the antecedents, and any like multiples of the consequents; and compare the order of size in the two pairs

and

i.e.

mA, no mx, nz.

First, suppose the first pair to be in descending order,

mA > nc;

then, comparing each of these magnitudes with the same consequent mB, MA: MB > nC: mB.

Now, by hypothesis and rule of equi-multiples,

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Thus the second pair of the above multiples are also in

descending order of size.