II. Take as many times this quotient as the divisor is contained times in the next higher unit.

408. When the right hand figure or figures of the divisor are an aliquot part of 10, 100, 1000, etc.

OPERATION.

1875) 601387

4

4

7500) 2405548

4

310000) 962 2192

3201887, Ans.

ANALYSIS. Since 33 is

of 100, we multiply both dividend and divisor by 3, (117, III), and we obtain a divisor the component factors of which are 100 and 37. We then divide after the manner of contracted division, (112).

ANALYSIS. Multiplying both dividend and divisor by 4, we obtain a new divisor, 7500, having 2 ciphers on the right of it. Multiplying again by 4, we obtain a new divisor, 30000, having 4 ciphers on the right. Then dividing the new dividend by the new divisor, we obtain 320 for a quotient, and 22192

for a remainder. As this remainder is a part of the new dividend, it must be 4 × 4

16 times the true remainder; we therefore divide

it by 16, and write the result over the given divisor, 1875, and annes the fraction thus formed to the integers of the quotient.

From these illustrations we derive the following

RULE. I. Multiply both dividend and divisor by a number or numbers that will produce for a new divisor a number ending in a cipher or ciphers.

II. Divide the new dividend by the new divisor.

NOTE. If the divisor be a whole number, or a finite decimal, the multiplier will be 2, 4, 5, or 8, or some multiple of one of these numbers.

409. Ratio is the relation of two like numbers with respect to comparative value.

NOTE. There are two methods of comparing numbers with respect to value; 1st, by subtracting one from the other; 2d, by dividing one by the other. The relation expressed by the difference is sometimes called Arithmetical Ratio, and the relation expressed by the quotient, Geometrical Ratio.

410. When one number is compared with another, as 4 with 12, by means of division, thus, 12 ÷ 4 = 3, the quotient, 3, shows the relative value of the dividend when the divisor is considered as a unit or standard. The ratio in this case shows that 12 is 3 times 4; that is, if 4 be regarded as a unit, 12 will be 3 units, or the relation of 4 to 12 is that of 1 to 3.

411. Ratio is indicated in two ways:

1st. By placing two points between the two numbers compared, writing the divisor before and the dividend after the points. Thus, the ratio of 8 to 24 is written 8: 24; the ratio of 7 to 5 is written 7: 5.

2d. In the form of a fraction.

written 24; the ratio of 7 to 5 is

.

Thus, the ratio of 8 to 24 is

412. The Terms of a ratio are the two numbers compared. The Antecedent is the first term; and

The Consequent is the second term.

:

The two terms of a ratio taken together are called a couplet. 413. A Simple Ratio consists of a single couplet; as 5 15. 414. A Compound Ratio is the product of two or more simple ratios. Thus, from the two simple ratios, 5 : 16 and 8: 2, we 5:16 8: 2

may form the compound ratio 5x8:16x2, or 16 × 3 = 32 = 1. 415. The Reciprocal of a ratio is 1 divided by the ratio; or, which is the same thing, it is the antecedent divided by the consequent. Thus, the ratio of 7 to 9 is 7:9 or 2, and its reciprocal is 7.

NOTE. The quotient of the second term divided by the first is sometimes called a Direct Ratio, and the quotient of the first term divided by the second, an Inverse or Reciprocal Ratio.

416. One quantity is said to vary directly as another, when the two increase or decrease together in the same ratio; and one quantity is said to vary inversely as another, when one increases in the same ratio as the other decreases. Thus time varies directly as wages; that is, the greater the time the greater the wages, and the less the time the less the wages. Again, velocity varies inversely as the time, the distance being fixed; that is, in traveling a given distance, the greater the velocity the less the time, and the less the velocity the greater the time.

417. Ratio can exist only between like numbers, or between two quantities of the same kind. But of two unlike numbers or quantities, one may vary either directly or inversely as the other. Thus, cost varies directly as quantity, in the purchase of goods; time varies inversely as velocity, in the descent of falling bodies. In all cases of this kind, the quantities, though unlike in kind, have a mutual dependence, or sustain to each other the relation of cause and effect.

418. In the comparison of like numbers we observe,

I. If the numbers are simple, whether abstract or concrete, their ratio may be found directly by division.

II. If the numbers are compound, they must first be reduced to the same unit or denomination.

III. If the numbers are fractional, and have a common denominator, the fractions will be to each other as their numerators; if they have not a common denominator, their ratio may be found either directly by division, or by reducing them to a common denominator and comparing their numerators.

419. Since the antecedent is a divisor and the consequent a dividend, any change in either or both terms will be governed by the general principles of division, (117). We have only to substitute the terms antecedent, consequent, and ratio, for divisor, dividend, and quotient, and these principles become

GENERAL PRINCIPLES OF RATIO.

PRIN. I. Multiplying the consequent multiplies the ratio; dividing the consequent divides the ratio.

PRIN. II. Multiplying the antecedent divides the ratio; dividing the antecedent multiplies the ratio.

PRIN. III. Multiplying or dividing both antecedent and consequent by the same number does not alter the ratio.

420. These three principles may be embraced in one

GENERAL LAW.

A change in the consequent by multiplication or division produ ces a LIKE change in the ratio; but a change in the antecedent produces an OPPOSITE change in the ratio.

421. Since the ratio of two numbers is equal to the consequent divided by the antecedent, it follows, that

I. The antecedent is equal to the consequent divided by the ratio; and that,

II. The consequent is equal to the antecedent multiplied by the ratio.

EXAMPLES FOR PRACTICE.

1. What part of 28 is 7?

1 has to .

; or, 28: 7 as 1:4; that is, 28 has the same ratio to 7

8. What is the ratio of 1 mi. to 120 rd.?

9. What is the ratio of 1 wk. 3 da. 12 h. to 9 wk.?

Ans.

Ans.

Ans.

Ans.

Ans. 4

Ans.

Ans.

10. What is the ratio of 10 A. 60 P. to 6 A. 110 P.?

11. What is the ratio of 25 bu. 2 pk. 6 qt. to 40 bu. 4.5 pk 12. What is the ratio of 18° to 45′ 30′′?

15. Find the reciprocal of the ratio of 42 to 28.

640

Ans. 1

16. Find the reciprocal of the ratio of 3 qt. to 43 gal. 17. If the antecedent be 15 and the ratio, what is the co

quent?

Ans. 1

18. If the consequent be 34 and the ratio 7, what is the a cedent?

Ans. 1

19. If the antecedent be of and the consequent .75, is the ratio?

20. If the consequent be $6.12 and the ratio 25, what is antecedent? Ans. $.24

21. If the ratio be and the antecedent, what is the c quent?

22. If the antecedent be 13 A. 145 P. and the ratio 4, is the consequent? Ans. 6 A. 90