3 ## PROPORTIONSThis Lesson continues from Lesson 2. The theorem of the alternate proportion The theorem of the same multiple The theorem of the common divisor The theorem of extremes and means 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." Problem 1. Read the following. Why is each one a proportion? a) 2 : 6 = 10 : 30 "2 is to 6 as 10 is to 30." Because 2 is the third part of 6, just as 10 is the third part of 30. b) 12 : 3 = 24 : 6 "12 is to 3 as 24 is to 6." Because 12 is four times 3, just as 24 is four times 6. c) 2 : 3 = 10 : 15 "2 is to 3 as 10 is to 15." Because 2 is two thirds of 3, just as 10 is two thirds of 15. Problem 2. Complete each proportion.
AB, CD are straight lines, and AB is three fifths of CD. Express that ratio as a proportion. AB : CD = 3 : 5
Example 1. If, proportionally,
Proportions
Problem 4. Explicitly, what ratio has a) b) c) 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:
(Euclid, VII. 13.) For example, since 1 : 3 = 5 : 15, then alternately, 1 : 5 = 3 : 15. (Skill in Arithmetic: Lesson 17, Question 2.) Problem 5. State the alternate proportion.
This leads to: The theorem of the same multiple Let us complete this proportion, 4 : 5 = 12 : ? 4 is four fifths of 5 (Lesson 2), but it is not obvious of what number 12 is four fifths. Alternately, however, 4 is the third part of 12 -- or we could say that 4 has been 4 : 5 = 12 : That is, 4 : 5 = 3 × 4 : 3 × 5. This is called the theorem of the same multiple. 4 is four fifths of 5. But each 4 has that same ratio to each 5. Two 4's, then, upon adding them, will have that same ratio to two 5's. Three 4's will have that same ratio to three 5's. And so on. Any number of 4's will have that same ratio, four fifths, to an equal number of 5's. Here is how we state the theorem:
(Euclid, VII. 17.) Problem 6. Write five pairs of numbers that have the same ratio as 3 : 4. Create them by taking the same multiple of both 3 and 4. For example, 6 : 8, 9 : 12, 12 : 16, 15 : 20, 18 : 24 Problem 7. Complete each proportion.
Problem 9. Complete this proportion, 2 Since 2.45 has been multiplied by 100, then 7 also must be multiplied by 100. PQ is two fifths of RS. If PQ is 12 miles, then how long is RS?
PQ : RS = 2 : 5. If PQ is 12 miles, then PQ : RS = 2 : 5 = 12 miles : ? miles. That is, 12 miles corresponds to PQ and 2. And since 12 is PQ : RS = 2 : 5 = 12 miles : 30 miles. RS is 30 miles. Or, since PQ : RS = 2 : 5, then inversely, RS : PQ = 5 : 2. Now, what ratio has 5 to 2? 5 is
AB is three fourths of CD. Specifically, AB is 24 cm. How long is CD?
AB : CD = 3 : 4 = 24 cm : ? Since 24 is The theorem of the common divisor Since we may multiply both terms by the same number, then, symmetrically, we may 25 : 40 = 5 : 8 upon dividing both 25 and 40 by 5. Explicitly, then, we see that 25 is Problem 11. Explicitly, what ratio has 16 to 40? Express that ratio so that the terms have no common divisors (except 1).
Upon dividing both terms by 8, 16 : 40 = 2 : 5. Lowest terms When the terms of a ratio have no common divisors except 1, then we have expressed their ratio with the lowest terms. They are the smallest terms -- the smallest pair of numbers -- that have that ratio. Problem 12. Explicitly, what ratio have the following? Express each ratio with the lowest terms. a) 6 is three fourths of 8, upon dividing each term by 2.
The theorem of extremes and means
For, if
Which is what we wanted to prove. By working the proof backwards, we could show that, conversely, if
* This theorem, or at any rate its algebraic version, seems to be the only one taught in the schools, and it has become the mechanical method for solving all ratio problems. The student should resist that tempatation and should understand the facts of ratio and proportion. We include it here only for the purpose of explaining the following:
Example 3. If then what ratio has
Problem 13. If eight
Five eighths. The language of ratio Example 4. Joan earns $1600 a month, and pays $400 for rent. Express that fact in the language of ratio.
That sentence, or one like it, expresses the ratio of $400 to $1600, of the Example 5. In Erik's class there are 30 pupils, while in Ana's there are only 10. Express that fact in the language of ratio.
This expresses the ratio of 30 pupils to 10. Example 6. In a class of 24 students there were 16 B's. Express that fact in the language of ratio.
This expresses the ratio of the Problem 14. Express each of the following in the language of ratio. Use a complete sentence. a) In a class of 30 pupils, there were 10 A's. A third of the class got A. b) Out of 120 people surveyed, 20 responded No. A sixth of the people surveyed responded No. c) The population of Eastville is 60,000, while the population of The population of Eastville is three times the population of Westville. d) Over the summer, John saved $1000, while Bob has saved only $100. Over the summer, John saved ten times more than Bob. e) At a party, there were 12 girls and 4 boys. At that party, there were three times as many girls as boys. f) In a class of 28 students, there were 21 A's. Three fourths of the students got A. g) In a survey of 60 people, 40 answered Yes. Two thirds of the people surveyed answered Yes. h) In a class of 40 pupils, 25 got a B. Five eighths of the pupils got B. i) Of the 2100 students who voted, 1400 voted for Harrison. Two thirds of the students voted for Harrison. j) This month's bill is $50, while last month's was only $20. This month's bill is two and a half times last month's. k) Sabina makes $24,000 a year, while Clara makes only $16,000. Sabina makes one and a half times what Clara makes. l) In the past thirty years, the population grew from 20,000 to 70,000. In the past thirty years, the population grew three and a half times. Next Topic: Continuous versus discrete Please make a donation to keep TheMathPage online. Copyright © 2017 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |