1 ## RATIO AND PROPORTIONThe natural numbers: Cardinal and ordinal The theorem of the alternate proportion The theorem of the same multiple TRIGONOMETRY historically is the study of triangles. The name literally means measurement of triangles. It begins with the study of right triangles, which are triangles that have a right angle, and with the ratios-- the relationships -- of their sides. Now the meaning of a ratio depends on what we mean by the parts of a number. And we name parts with ordinal numbers: the third part, the fourth, the fifth, and so on. So with those we shall begin. Natural numbers: cardinal and ordinal The natural numbers are the counting numbers. They have two forms, cardinal and ordinal. The cardinal forms are One, two, three, four, and so on. They answer the question How much? or How many?. The ordinal forms are First, second, third, fourth, and so on. They answer the question Which one?. Parts of natural numbers We say that a smaller number is a part of a larger number if the larger number is its multiple. Here are the first few multiples of 5: 5, 10, 15, 20, 25, 30, 35. 5 is the 5 is a part of each one of those, except itself. Now, since 15, for example, is the third multiple of 5, we say that 5 is the third part of 15. We use that same ordinal number to name the part. 5 is the fourth part of 20. It is the fifth part of 25; the sixth part of 30. And so on. 5 is which part of 10? We do not say the second part. We say So with the exception of Again, 5 is not a part of itself. There is no such thing as the first part. (See Skill in Arithmetic, Lesson 15.) It is important to understand that we are not speaking here of proper fractions -- Problem 1. 7 is which part of 28? The fourth. To see the answer, pass your mouse over the colored area. Problem 2. Which part of 45 is 9? The fifth. Problem 3. 6 is which part of 12? Half. Problem 4. What number is the eighth part of 24? 3 Problem 5. 3 is the tenth part of what number? 30
The figure shows that we have divided 15 into thirds, that is, into three equal parts, and that 5, 10, 15. Therefore, 10 -- which is Now, 10 is not a Similarly, if we divide 15 into fifths, that is, into five equal parts -- -- then 3 is the fifth part of 15. 6 is 9 is 12 is And 15 is all five of its fifth parts. We can now say what we mean by The ratio of natural numbers
DEFINITION. The ratio of two natural numbers is their relationship with respect to relative size, which we can always name. Specifically, we can say that one number is a
(Skill in Arithmetic, Lesson 17.) Example 1. Multiple What ratio has 15 to 5?
That is the ratio -- the relationship -- of 15 to 5. We do not answer "3 to 1," because we want to name the ratio of 15 to 5 explicitly. It is true that 15 has the The two numbers in a ratio are called the terms; the first term and the second. Notice that we answer with a complete sentence beginning with the first term and ending with the second: "15 is three times 5." For, a ratio is a relationship. Example 2. Part What ratio has 5 to 15?
That is the inverse ratio of 15 to 5. The terms are exchanged. Example 3. Parts What ratio has 10 to 15?
Those are the three kinds of ratio: One number is a multiple of the other -- so many times it -- a part of it, or parts of it. Problem 6. What ratio have the following? Answer with a complete sentence beginning with the first term. a) 2 to 10? 2 is the fifth part of 10.
c) 7 to 1? 7 is seven times 1. d) 1 to 7? 1 is the seventh part of 7. e) 25 to 100? 25 is the fourth part of 100. f) 75 to 100? 75 is three fourths of 100. g) 12 to 6? 12 is two times 6, or twice as much as 6, or double 6. h) 6 to 12? 6 is half of 12. The names of the fractions In English, the proper fractions have the same names as the
is called "one-third" because of the ratio of 1 to 3. 1 is one third of 3.
For that reason, fractions are called rational numbers. Fractions have the same ratio to 1 as two natural numbers. Fractional symbols may therefore be regarded as ratio symbols, in that they signify the ratio of the numerator to the denominator.
1 is to 2 as 2 is to 4 as 5 is to 10, etc. Each numerator is Example 4. What ratio has 3 to 4?
On the practical side, since the student has very likely learned the names of the fractions first, then the ratio of 3 to 4 is expressed by the
A proportion is a statement that two ratios are the same. 5 is to 15 as 8 is to 24. 5 is the third part of 15, just as 8 is the third part of 24. We will now introduce this symbol 5 : 15 to signify the ratio of 5 to 15. A proportion will then appear as follows: 5 : 15 = 8 : 24. "5 is to 15 as 8 is to 24." Or, we can represent a proportion with fractional symbols --
-- and read it the same way: "5 is to 15 as 8 is to 24." Example 5. 12 : 2 = 42 : 7. ("12 is to 2 as 42 is to 7.") Why is this a proportion?
Problem 7. Complete this proportion, 3 : 12 = 7 : ?
The theorem of the alternate proportion The numbers in a proportion are called the terms: the 1st, the 2nd, the 3rd, and the 4th. 1st : 2nd = 3rd : 4th We say that the 1st and the 3rd are corresponding terms, as are the 2nd and the 4th. The following is the theorem of the alternate proportion:
Since 1 : 3 = 5 : 15, then alternately, 1 : 5 = 3 : 15. (Skill in Arithmetic: Lesson 17, Question 2.) Example 6. Complete this proportion: 4 : 5 = 12 : ?
12 is four fifths of. Alternately, however, 4 is a third of 12 -- or we could say that 4 has been 4 : 5 = 12 : 15 That is, 4 : 5 = 3 × 4 : 3 × 5 As one 4 is to one 5, so any number of 4's will be to an equal number of 5's. Three 4's are four fifths of three 5's. This is the theorem of the same multiple.
Example 7. Complete this proportion, 5 : 8 = 35 : ?
Problem 8. Complete this proportion, 4 : 9 = 24 : ?
54. The term corresponding to 4 is We shall often make use of this basic property of the square root radical: A radical multiplied by itself produces the radicand. (See Lesson 26 of Algebra, Radicals.) Problem 9. Complete this proportion: : 2 = 3 : ? 2. For, has been multiplied by ; therefore 2 also must be multiplied by . Example 9. Solve this proportion -- 8 : 12 = 2 : ?
8 : 12 = 2 : 3 The theorem of the same multiple may be inverted to mean that we may divide both terms by the same number. An all too common method these days is to make this an algebra problem.
The student is taught to cross-multiply and solve for Example 10. Solve this proportion -- 2 : 11 = 9 : ?
Similar figures Trigonometry depends on the meaning of similar figures.
To say then that figures ABCDE, PQRST are similar, is to say that the angle at A is equal to the angle at P, the angle at B is equal to the angle at Q, etc.; and, proportionally, as AB is to BC, so PQ is to QR. And so on, for each pair of equal angles and the sides that make them. For triangles to be similar, however, it is sufficient that they be equiangular. (Theorem 15 of "Some Theorems of Plane Geometry.") From that it follows: Right triangles will be similar if an acute angle of one In the right triangles ABC, DEF, if the acute angle at B is equal to the acute angle at E, then those triangles will be similar. Therefore the sides that make the equal angles will be proportional. For example, BC : CA = EF : FD. If BC were twice as long as CA, then EF would also be twice as long as FD. And alternately, the corresponding sides are proportional: BC : EF = CA : FD. If BC were half of EF, then CA would also be half of FD. Problem 10. The right triangles HJK, LMN are similar, with HJ = 3 cm, JK = 4 cm, and KH = 5 cm. If LM = 12 cm, then how long are MN and NL? Alternately, LM : HJ = MN : JK. Since LM is four times HJ, then MN will be four times JK -- it will be 16 cm. Similarly, NL will be four times KH, it will be 20 cm. To put the whole thing more simply: The side corresponding to HJ has been multiplied by 4. Therefore, since the triangles are similar, every side will be multiplied by 4. Problem 11. Triangles ABC, DEF are similar.
AB = 6 cm, BC = 11 cm, CA = 7 cm, and DE = 18 cm. How long are EF and DF? The side corresponding to AB has been multiplied by 3. Therefore every side will be multiplied by 3. EF = 33 cm, and DF = 21 cm. * Our definition of a proportion has been in terms of the ratio of natural numbers. The sides of a triangle, however, are not natural numbers. They are lengths, which are magnitudes -- things that have size -- they are not numbers. And magnitudes do not necessarily have the same ratio as natural numbers. They can be "incommensurable." Hence a new theory of ratio and proportion is required which will be applicable to magnitudes. The interested student is referred to Topic 11 of The Evolution of the Real Numbers, in particular the section "The new theory of proportions." Next Topic: Definitions of the Trigonometric Functions Copyright © 2018 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |