will 500? (672) What will 1000: (840) What will 2? (168)

A. $2059,68.

9. If 60 cents buy 4 lbs. of tobacco, how much will 30 cents buy? (2) How much will 90 cents? (6) How much will 120 cents? (8) How much will $2,10? (14) How much will

$2.40? (16) 4. 46 lbs.

10. If pair of gloves cost 75 cents, what will 1 dozen pair cost? (9) What will 14 doz.? (1350) What will 2 doz. (18) What will 2 doz.? (2250) What will 3 doz. ? (27)

11. If 3 dez pair of shoes cost cost? (75) What will 2 doz.? (15) What will 14 doz.? (1350)

A. $90. 27 dollars, what will 1 pair (2250) What will 2 dòz. ? What will 1 doz. ? (9)

4. $63,75

12. If 5 tons of hay will keep 25 sheep over the winter, how many sheep can be kept on 7 tons, at the same rate? (35) On 8 tons? (40) On 15 tons? (75) On 60 tons ? (300) On 80? (400). A. 850.

13. Boarding at $2,25 a week, how long will $9 last me? (4) How long will $13,50? (6) How long will $18? (8) How long will $20,25? (9) How long will $49,50? (22) A. 49 weeks.

14. If a man receive $50 for 2 months' wages, what is that a year? (300) What will 8 months' come to? (200) 16 months come to? (400) 1 years' come to? (450) 2 years' come to? (600) 10 years' come to? (3000) A. $4950.

15. What will 6 pieces of cloth, each piece containing 20 yards, come to, at $1.50 a yard? (180) What will 1 piece come to? (30) What will 3 pieces? (90) What will 5 pieces? (150) What will 10 pieces? (300) 2. $750.

16. Bought 5 bhds of um, each containing 60 gallons, for $2 a gallon; what do they come to? (600) What will 4 hlids. come to? (480) What will 20 hds.? (2400) A. $3480.

17. William's income is $1500 a year, and his daily expenses are $2,50; how much will he have saved at the year's end? A. $587,50.

18. If William's income had been $2000, how much would he have saved? (108750) If $2500 (158750) If $3600 ? (268750) If $4000 (308750) A. $8450.

19. If a hhd. of molasses cost $20,16, how much is it a gal lon? (Divide by the number of gallons in a hhd.) (32) How much is it a quart? (Divide by the number of quarts in a hhd.) (8) How much is it a pint? (4) How much is it a gill? (1) A. 45 cents.

The foregoing questions have been solved by a method terined analysis This method is though to accord with the natural operations of the human mind. Men in business earcely recognise any other. The formality of state ments is rarely if ever made by thein; and, when it is made, they do it more the sake of testing the eoriziness of the other method, than for any practs

eal purpose. They may have adopted a statement in the commencement of their business, from the circumstance of having been taught it at school; but the longer they continue in business, the less occasion they have for it. There is another method, which consists in ascertaining the ratio or relation which one number has to another. This is used more or less by all, but more exten sively by scientific men, and those well versed in mathematical principles.. 20. If 8 pair of shoes cost 63 cents, what will 24 pair cost? of 63 = 73 cents, the price of 1 pair, which we multiply by 24 to get the price of 24 pair; thus, 24 × 77 = $1,89. But, since 7 is a fraction, it would be more convenient to multiply by 24 first, and divide by 8 afterwards, as this cannot make any difference; and, that we may make no mistake in the operation, we will make a statement, by writing the 63 cents on the right,

as a third term (see operation); on the left of which we write the multiplier, 24, as a 2d term, and, as a first term, the divisor, 8: then, 63 × 24 = 1512 ÷ 8 = $1,89, the Ans., as before.

252

126

8) 1512

Ans., $1,89

21. If 3 yards of cloth cost 24 cents, what will 6 yards cost

OPERATION.

yds. yds. cts.

3 6 24

>

6

24 X 6144348, the Ans OR, as we know that 6 yards cost 2 times as much as 3 yds., that is, §=2, by simply multiplying 24 by 2, it makes 48, the answer, the same as before. This is a much shorter process; and, could we discover the principle, it would oftentimes render operations very simple and short In searching for this, we shall naturally led to the consideration of ratio, or relation; that is, the relation which necessarily exists between two or more numbers.

3)144

Ans., $,48

A. 1.

Q. What part of 4 is 3?

Q. What is the finding what part one number is of another called? A. Finding the ratio, or relation of one number to another. What is ratio, then? A. The number of times one number, or quantity, is contained in another.

Q. What part of 10 is 97 or, what is the ratio of 10 to 9?

Q. What is the ratio of 18 to 17?

A. 17.

Q. What part of 3 oz. is 12 oz.? or, what is the ratio of 3 oz. to 12 oz.? A. 4, ratio.

Q. What part of four yards is 9 yds.? or, what is the ratio of 4 to 9? A. =24.

Q. Hence, to find the ratio of one number to another, how do you proceed? A. Make the number which is mentioned last (whether it be the larger or smaller), the numerator of a fraction, and the other number the denominator; that is, always divide the second by the first.

1. What part of $1 is 50 cents? or, what is the ratio of $1 to

50 cents?

A. $1100 cents; then, 2. What part of 5 s. is 2 s.

to 2 s. 6 d.?

2 s. 6 d. ratio, Ans.

30 d., and 5 s.

%, the ratio, Ans. 6 d.? or, what is the ratio of 5 s.

:60 d.;

3. What is the ratio of £1 to 15 s.? 4. What is the ratio of 2 to 3? .

. Of 4 to 20? A. 5. Of 20 to 4? A. }. Of8 to 63? A.7. Of 200 to 900? A. 44 Of 800 to 900? 4. 14. Of 2 quarts to 1 gallon? A. 2. Let us now apply the principle of ratio, which we were in pursuit of, to practical questions.

PROPORTION. 22. If 2 melons cost 8 cents, what will 10

cost?

It is evident, that 10 melons will cost 5 times as much as 2; that is, the ratio of 2 to 10 is 125; then, 5 X 8 == 40, Ans. But, by stating the question as before, we have the following proportions:-

Q. When, then, numbers bear such relations to each other, bat are the numbers said to form? Ans. A proportion.

Q. How may proportion be defined, then? A. It. is an equal ity of ratios.

Q.

How many numbers must there be to form a ratio?

A: Two.

Q. How many to form a proportion? A. At least, three. To show that there is a proportion between three or more numbers, we write them thus:

melons. melons. cents. cents.

2 : 10: 8 : 40,

which is read, 2 is to 10 as 8 is to 40; or, 2 is the same part of 10 that 8 is of 40; or, the ratio of 2 to 10 is the same as that of to 40.

Q. What is the meaning of antecedent? A. Going before. Q. What is the meaning of consequent? A. Following. Q. What is the meaning of couplet? A. Two, or a pair. Q. What may both terms of a ratio be called? A. A couplet. Q. What may cash term of a couplet be called, as 3 to 4? A. The 3, being first, may be called the antecedent; and the 4, being after the 3, the consequent.

Q. In the following proportion, viz. 2: 10: 8: 40, which are the antecedents, and which are the consequents? A. 2 and are the antecedents, and 10 and 40 the consequents.

Q. What are the ratios in 2 : 10 8 40?

and

In the last proportion, 2 and 40, being the first and last terms, are called extremes; and 10 and 8, being in the middle, are called the means. Also, in the same proportion, we know that the extremes 2 and 40, multiplied together, are equal to the product of the means, 10 and 8, multiplied together, thus; 2 X 40 =80, and 10 X 880. Let us try to explain the reason of this. In the foregoing proposition, the first ratio, 40, (= 5), being equal to the second ratio, 4, (= 5), that is, the fractional ratios being equal, it follows, that reducing these fractions to a common denominator will make their numerators alike; thus, 40 and 4 become ; in doing which, we multiply the numerator 40 (one extreme) by the denominator 2 (the other extreme), also the numerator 10 (one mean) by the denominator, 8, (the other mean); hence the reason of this equality. Q. When, then, any four numbers are proportional, what may we learn respecting the product of the extremes and means? A. That the duct of the extremes will always be equal to the product of the means. Hence, with any three terms of a proportion being given, the fourth or absent term may easily be found. Let us take the last example:

melons. cts.
10 : : 8 :

cts.

40

pro

Multiplying together 8 and 10, the two means, makes 80; then 80 ÷ 40, the known extreme, gives 2, the other extremo required, or first term. Ans. 2.

are

Again, let us suppose the 10 absent, the remaining terms

By multiplying together 40 and 2, the extremes, we have 80; which, divided by 8, the known mean, gives 10, the 2d term, or mean, required. Let us exemplify this principle more fully by a practical example.

23. If 10 horses consume 30 bushels of oats in a week, how many bushels will serve 40 horses the same time?

In this example, knowing that the number of bushels eaten are in proportion to the number of horses, we write the propor tion thus:

finding the 4th term in this example. The ratio of 10 to 40 is 8=4, that is, 40 horses will consume 4 times as many bushels as 10; then 4 X 30 bu. 120 bushels, the 4th term, or extreme, as before.

Q. When any three terms of a proportion are given, what is the process of finding the fourth term called? . The Rule of

Three.

Q. How, then, may it be defined? A. It is the process of finding, by the help of three given terms, a fourth term, be tween which and the third term there is the same ratio or propor tion as between the second and first terms.

It will sometimes be necessary to change the order of the terms; but this may be determined very easily by the nature of the question, as will appear by the following example:24. If 8 yards of cloth cost $4, what will 2 yards cost?

In this example, since 2 yards will cost a less sum than 8 yards, we write 2 yards for one mean, which thus becomes the multiplier, and 8 yards, the known extreme. for the divisor; for the less the multiplier, and the greater the divisor, the less will be the quotient; then, 2 X488$1, Ans. But multiplying by the ratio will be much casier, thus; the ratio of 8 to 2 is; then, 4 X = $1, Ans.; as before.