Ex. (2) To solve the equation a−1=0.

Two of the roots are evidently +1 and −1.

Hence, dividing by (x-1) (x+1), that is by x2-1, we obtain x2+1=0, of which the roots are √-1 and −√-1.

Hence the four roots are 1, -1, √-1, and -√-1.

The equation x6-6x3-7 will in like manner have six roots, for it may be reduced, as in Art. 335, to two cubic equations, x3-7=0 and x3+1=0,

each of which has three roots, which may be found as in Ex. (1).

XXVII. ON RATIO.

338. IF A and B stand for two unequal quantities of the same kind, we may consider their inequality in two ways. We may ask

(1) By what quantity one is greater than the other?

The answer to this is made by stating the difference between the two quantities. Now since quantities are represented in Algebra by their measures (Art. 33), if a and b be the measures of A and B, the difference between A and B is represented algebraically by a-b.

(2) By how many times one is greater than the other? The answer to this question is made by stating the number of times the one contains the other.

NOTE. The quantities must be of the same kind. We cannot compare inches with hours, nor lines with surfaces.

339. The second method of comparing A and B is called finding the RATIO of A to B, and we give the following definition.

DEF. Ratio is the relation which one quantity bears to another of the same kind with respect to the number of times the one contains the other.

340. The ratio of A to B is expressed thus, A: B. A and B are called the TERMS of the ratio.

A is called the ANTECEDENT and B the Consequent.

341. Now since quantities are represented in Algebra by their measures, we must represent the ratio between two quantities by the ratio between their measures. Our next step then must be to shew how to estimate the ratio between two numbers. This ratio is determined by finding how many times one contains the other, that is, by obtaining the quotient resulting from the division of one by the other. If a and b, then, be any two numbers, the fraction will express the ratio of a to b. (Art. 136.)

a

342. Thus if a and b be the measures of A and B respectively, the ratio of A to B is represented algebraically by the fraction.

a

m

α

343. If a or b or both are surd numbers the fraction may also be a surd, and its approximate value can be found by Art. 291. Suppose this value to be where m and n are whole numbers: then we should say that the ratio A: B is approximately represented by

m

n

n

344. Ratios may be compared with each other, by comparing the fractions by which they are denoted.

Thus the ratios 3: 4 and 4: 5 may be compared by comparing the fractions and

3

2.

EXAMPLES.-CXXVIII.

1. Place in order of magnitude the ratios 2: 3, 6 : 7, 7 : 9. Compare the ratios x+3y : x+2y and x+2y : x+y. Compare the ratios x-5y: x-4y and x- -3y: x-2y. What number must be added to each of the terms of the ratio a: b, that it may become the ratio c:d?

3.

4.

5. The sum of the squares of the Antecedent and Consequent of a Ratio is 181, and the product of the Antecedent and Consequent is 90. What is the ratio?

345. A ratio of greater inequality is one whose antecedent is greater than its consequent.

A ratio of less inequality is one whose antecedent is less than its consequent.

This is the same as saying a ratio of greater inequality is represented by an Improper Fraction, and a ratio of less inequality by a Proper Fraction.

346. A Ratio of greater inequality is diminished by adding the same number to both its terms.

Thus if 1 be added to both terms of the ratio 5: 2 it be6 comes 6:3, which is less than the former ratio, since that 3'

And, in general, if x be added to both terms of the ratio ab, where a is greater than b, we may compare the two ratios thus,

347. We may observe that Art. 346 is merely a repetition of that which we proposed as an Example at the end of the Chapter on Miscellaneous Fractions. There is not indeed any necessity for us to weary the reader with examples on Ratio: for since we express a ratio by a fraction, nearly all that we might have had to say about Ratios has been anticipated in our remarks on Fractions.

348. The student may, however, work the following Theorems as Examples.

(1) If a b be a ratio of greater inequality, and x a positive quantity, the ratio a-x: b-x is greater than the ratio a: b.

:

(2) If a b be a ratio of less inequality, and x a positive quantity, the ratio a+x: b+x is greater than the ratio a : b.

:

(3) If a b be a ratio of less inequality, and x a positive quantity, the ratio a-x: b-x is less than the ratio a : b.

349. In some cases we may from a single equation involving two unknown symbols determine the ratio between the two symbols. In other words we may be able to determine the relative values of the two symbols, though we cannot determine their absolute values.

Thus from the equation 4x=3y,

Find the ratio of x to y from the following equations:

1. 9x=6y.

4. x2+2xy=5y3.

2. ax=by,

3. ax-by=cx+dy.

5. x2-12xy=13y2. 6. x2+mxy = n2y2.

7. Find two numbers in the ratio of 3: 4, of which the sum is to the sum of their squares :: 7 : 50.

8. Two numbers are in the ratio of 6: 7, and when 12 is added to each the resulting numbers are in the ratio of 12 : 13. Find the numbers.

9. The sum of two numbers is 100, and the numbers are in the ratio of 7: 13. Find them.

10. The difference of the squares of two numbers is 48, and the sum of the numbers is to the difference of the numbers in the ratio 12: 1. Find the numbers.

11. If 5 gold coins and 4 silver ones are worth as much as 3 gold coins and 12 silver ones, find the ratio of the value of a gold coin to that of a silver one,

12. If 8 gold coins and 9 silver ones are worth as much as 6 gold coins and 19 silver ones, find the ratio of the value of a silver coin to that of a gold one.

350. Ratios are compounded by multiplying together the fractions by which they are denoted.

and

and

Thus the ratio compounded of a : b and c d is ac: bd.

EXAMPLES.-CXXX.

Write the ratios compounded of the ratios

1. 2:3 and 4: 5.

2. 37, 14: 9 and 4: 3.

3. x2—y2; x3+y3 and x2−xý+y2:x+y.

4. a2-b2+2bc-c2: a2-b3-2bc-c2 and a+b+c:a+b−c.

5. m3 + n3: m3-n3 and m-n: m+n.

6. x2+5x+6: y2-7y+12, and y2 - 3y: x2+3x.

351. The ratio a2 : b2 is called the DUPLICATE RATIO of a: b.

Thus 100: 64 is the duplicate ratio of 10 : 8,

36x2: 25y2 is the duplicate ratio of 6x : 5y.

The ratio a3: b3 is called the TRIPLICATE RATIO of a: b.

Thus 64 : 27 is the triplicate ratio of 4 : 3,

343x3: 1331y3 is the triplicate ratio of 7x : 11y.