INTRODUCTION TO RATIO AND PROPORTION.

MENTAL EXERCISES.

1. Eight is how many times 4?

2. What is the relation of 8 to 4?

Ans. 8 is two times 4.

3. What is the relation of 12 to 3? Of 16 to 4? Of 18 to 6? Of 20 to 5? Of 24 to 6? Of 30 to 5?

4. What is the relation of 3 to 6? Of 4 to 12? Of 6 to 24? Of 1 to 35? Of 8 to 57? Of 9 to 62?

5. The measure of the relation of two numbers is called their ratio 6. What is the ratio of 12 to 4?

Ans. The ratio of 12 to 4 is three. 7. What is the ratio of 18 to 9? Of 25 to 5? Of 48 to 8? £3 to 7? Of 64 to 4? Of 70 to 10? Of 80 to 8? 8. What is the ratio of 3 to 6?

e-half.

Of 9 to

Ans. The ratio of 3 to 6 is 9. What is the ratio of 4 to 12? Of 3 to 18? Of 5 to 3 108? Of 11 to 132? Of 12 to 144? 10. What is the ratio of to? Of to? Of to Of.5 to .25? Of.2 to .04? Of.03 to .12?

11. The ratio of two numbers may be expressed by writing the colon between them; thus 8: 4 denotes the ratio of 8 to 4.

12. Required the value of 12:6; of 28:7; of 42:6; of 24: 12; of 12:24.

13. How does the ratio of 8 to 4 compare with the ratio of 12 to 6? Ans. They are equal.

14. What number has the same ratio to 12 that 18 has to 6?

15. What number has the same ratio to 20 that 40 has to 10?

16. The ratio of 9 to 36 is the same as the ratio of 15 to what number?

17. 25 is to 5 as 40 to what number? 24 is to 12 as 15 is to what number?

18. When we express the ratio of two numbers equal to the ratio of two other numbers, as, 24 is to 4 as 36 is to 6, we have a proportion. 19. What proportion can we derive from the two ratios 40 to 8 and 60 to 12?

20. How many numbers do we have in a proportion? How many ratios? Are the ratios equal or unequal?

21. The equality of two ratios may be expressed by writing the symbol between them; thus 8:4 12: 6.

=

22. Write the proportion 16 is to 8 as 24 is to 12; also 15 is to 45 as 18 is to 54.

SECTION IX.

RATIO AND PROPORTION.

RATIO.

547. Ratio is the measure of the relation of two similar quantities; thus, the ratio of 8 to 4 is 2.

548. The Symbol of ratio is the colon (:); thus, 8:4 signifies the ratio of 8 to 4. Ratio is also expressed by writing the numbers in the form of a fraction; thus, §.

549. The Terms of a ratio are the two numbers compared, called respectively the antecedent and the consequent. 550. The Antecedent is the number compared with the consequent; thus, in the ratio 8:4, 8 is the antecedent.

551. The Consequent is the number with which the antecedent is compared; thus, in 8: 4, 4 is the consequent. 552. A Ratio is found by dividing the antecedent by the consequent; thus, in 8: 4, the ratio is, or 2.

553. A Simple Ratio is the ratio of two numbers, as 6:3. A Compound Ratio is the product of two or more simple ratios; as (3:4)× (5:6), or x.

554. A Compound Ratio is usually expressed by writ (3:4) ing the simple ratios one under another; thus, 5:6)

555. Ratio exists only between similar quantities, and is always an abstract number.

NOTES.-1. The symbol of ratio (:) is supposed to be a modification of the symbol of division.

2 Ratio is usually defined as the relation of two numbers. This is indefinite, for the ratio is the measure of the relation.

3. A few authors divide the second term by the first, calling it the French Method. The method and name are both founded in error; nearly all the French mathematicians, like the German, English, etc., divide the first term by the second.

PRINCIPLES.

1. The ratio equals the quotient of the antecedent divided by the consequent.

Thus, if the antecedent is represented by a, and the consequent by c and the ratio by r, we have a÷c=r, or == r.

2. The antecedent is equal to the product of the consequent and ratio.

For, since ==r, multiplying by c, we have a=cXr.

с

3. The consequent is equal to the quotient of the antecedent divided by the ratio.

For, since == r, a=c × r, from which we see that c=

с

MENTAL AND WRITTEN EXERCISES.

What is the ratio of

3:95

?

9. What is the value of the compound ratio 2:4 SOLUTION.―This compound ratio equals (2: 4) × (3:9), which equals

(5:2) 18:65

?

Ans. 33.

what is the

Ans. 3.

12. The antecedent is 24, the consequent 8; ratio? 13. The consequent is 8 and ratio 9; what is the antecelent?

Ans. 72. 14. The antecedent is 36 and ratio 4; what is the consequent? Ans. 9. 15. The consequent is 15 and ratio ; what is the antecedent? Ans. . 16. The antecedent is 15 and ratio ; what is the consequent? Ans. 1.

17. Can you express the ratio between $24 and 6 lb.? Why not?

18. The antecedents of a ratio are 5 and 6, and the conse quents 10 and 14; what is the ratio? Ans.

SIMPLE PROPORTION.

556. A Proportion is the expression of equality between equal ratios, the terms of the ratios being indicated.

557. The Symbol for proportion is the double colon, (::), which expresses an equality of ratios; thus, 8: 4 :: 6:3, means the same as 8:4 6:3.

558. A Proportion is read in two ways; thus, 8:4:: 6:3 is read "the ratio of 8 to 4 equals the ratio of 6 to 3;" or "8 is to 4 as 6 is to 3."

559. The Terms of a proportion are the four numbers used in the comparison. The first and fourth terms are

the Extremes; the second and third are the Means.

560. The Couplets are the two ratios compared. The first couplet consists of the first and second terms. The second couplet consists of the third and fourth terms.

561. Proportion may be Simple or Compound. In Simple Proportion both the ratios compared are simple; in Compound Proportion one or both of the ratios are compound.

562. A Simple Proportion is the expression of the equality of two simple ratios.

563. The Principles of proportion are the truths relating to a proportion. They enable us to find any one term when the other three are given.

PRINCIPLES.

1. In every proportion the product of the means equals the product of the extremes.

In any proportion, as 6:3::8:4, we have } = {, and multiplying these equals by 4 and 3 we have 6 × 48 × 3; that is, the product of the two means 8 and 3, equals the product of the two extremes 6 and 4. 2. Either extreme equals the product of the means divided by the other extreme.

For, from the proportion 6:3::8:4, we have 6 × 43 × 8; hence, 63 x 84, or 43 × 8÷6. Therefore, etc.

3. Either mean equals the product of the extremes divided by the other mean.

For, from the proportion 6:3:: 8:4, we have 6 x 43 x 8; hence, 86 x 48, or 86x 43. Therefore, etc.

4. The first term of a proportion equals the second term multiplied by the ratio of the third to the fourth.

For, from the proportion 8:6::12:9, we have §=1a; hence, 8= Y × 6, or 12:9 multiplied by 6. Therefore, etc.

5. The fourth term of a proportion equals the third term divided by the ratio of the first to the second.

For, from the proportion 8:6:12:9, we have 8 × 9 = 6 × 12, 01 9 6 x 128, which equals 12 × §, which equals 12÷, or 124 (8:6). Therefore, etc.

NOTES.-1. Let the pupils be required to demonstrate these principles by using symbols of any numbers; that is, by letters. French authors usually represent the unknown term by z; the same is done in this work. 2. Principle 1 may be demonstrated by showing that in a proportion we have 2d term X ratio : 2d term: 4th term X ratio: 4th term; in which we see the factors in the means are the same as the factors in the extremes.

MENTAL EXERCISES.

1. Write a proportion and point out the different terms and couplets. Write a proportion and show that the ratios are equal.

2 If we multiply the antecedent of one couplet, what must we do to the other couplet to make the ratios equal?

3. If we divide the antecedent of one couplet, what must we do to the other couplet to make the ratios equal?

4.. Write a proportion and illustrate Prin. 1; Prin. 2; Prin.3; Prin. 4; Prin. 5.

5. Show that if we change the two means one for the other, or the two extremes, the four numbers will still form a proportion.

6. Take some proportion and show that we may invert the terms of the couplets, and the four terms will still be in proportion.

WRITTEN EXERCISES.

Find the terms denoted by x in each of the following proportions: