8. A, B, and C hire a pasture for $300. A puts in 10 oxen for 20 weeks, 15 cows for 14 weeks, and 99 sheep for 26 weeks; B puts in 7 oxen for 24 weeks, 12 cows for 20 weeks, and 66 sheep for 25 weeks; C puts in 25 oxen for 8 weeks, 12 cows for 12 weeks, and 33 sheep for 15 weeks. Now, if 11 sheep are reckoned as 1 cow, and 3 cows as 2 oxen, what is the cost per week for a sheep? a cow? an ox? How many dollars does each man pay for sheep? cows? oxen? What part of the rent does each man pay? How many dollars?

Ans. Cost per week for a sheep, 1 c.; a cow, 16 c.; an ox, 24 c. A pays for sheep, $37.44; for cows, $33.60; for oxen,

$48. B pays for sheep, $24; for cows, $38.40; for oxen, $40.32. C pays for sheep, $7.20; for cows, $23.04; for oxen, 48. A pays 18 = $119.04; B, 21 $102.72; C, 163 $78.24.

=

19. 9. J. Fox and S. Low enter into partnership. January 1, Fox puts in $5000, but Low puts in nothing until May 1; what shall he then put in that the partners may be entitled to equal shares of the profits at the close of the year?

10. Jan. 1, 1853, A, B, and C form a partnership for 1 year, and each furnishes $3000; March 1, A furnishes $1000 more; June 1, B withdraws $500, and C adds $500; Sept. 1, A withdraws $2000 and C $500, and B adds $1500. Having gained $4000, at the close of the year the partnership is dissolved. What is each partner's share of the gain?,

11. A, B, and C traded in company. A at first put in $1000, B $1200, and C $1800; in three months A put in $500 more and B $300, and C took out $400; in 7 months from the commencement of business, A withdrew all his stock but $700, B put in as much as he at first put in, and C withdrew as much as A at any time had in the firm. At the end of a year they found they had gained 10 per cent. on the largest total stock at any one time in trade. What is the total gain? What fractional part shall each have? How many dollars?

Ans. Total gain, $440.

A's part, $107.63107.

B's part,

C's part,
Proof,

(313 a.

cost?

EXAMPLES IN ANALYSIS.

1. If 6 barrels of flour cost $42, what will 11 barrels

2. If g of a cask of wine cost $35, what will 7 casks cost? 3. Twenty is of what number?

6. If of a ton of hay cost 95 shillings, what will a ton cost?

7. If 33 of a cask of oil is worth $74, what is the value of 5 casks?

8. Sixty-four is of how many times 12?

9. Seventy-two is % of how many times 4?

10. A man sold a watch for $63, which was of its cost; what was its cost?

11. A pole is in the mud, in the water, and 6 feet above water; what is the length of the pole?

12. A ship's crew have provisions sufficient to last 12 men 7 months; how long would they last 24 men?

13. A can build 35 rods of wall in 33 days, but B can build 9 rods while A builds 7; how many rods can B build in 44 days? of how many fifths of 55?

14. of 28 is 15. of 44 is

of how many thirds of 15?

16. 3 of 27 is 2

17. A fox has

of how many twelfths of 60?

39 rods the start of a hound, but the hound runs 27 rods while the fox runs 24; how many rods must the hound run to overtake the fox?

Ans. 351.

18. A bare has 32 rods the start of a hound, but the hound runs 12 rods while the hare runs 8; how many rods will the hare run before the hound overtakes him?

19. A man being asked how many sheep he had, replied that if he had as many more, as many more, and 2 sheep he should have 100; how many had he?

20. A detachment of 2000 soldiers was supplied with bread sufficient for 12 weeks, allowing each man 14 ounces a day, but

finding 105 barrels, containing 200 lb. each, wholly spoiled, how many ounces may each man eat daily, that the remainder may last them 12 weeks?

Ans. 12 oz. of their bread

21. A detachment of 2000 soldiers, having spoiled, were put upon an allowance of 12 oz. each per day for 12 weeks; what was the whole weight of their bread, good and bad, how much was spoiled?

22. A detachment of 2000 soldiers having lost 105 barrels of bread, weighing 200 lb. each, were allowed but 12 oz. each per day for 12 weeks; but if none had been lost, they might have had 14oz. daily; what was the weight, including that which was lost, and how much was left to subsist on? 1st Ans. 147000 lb.

23. A detachment of 2000 soldiers, having lost of their bread, had each 12 oz. per day for 12 weeks; what was the weight of their bread, including the part lost, and how much per day might each man have had, had none been lost?

24. A gentleman left his son an estate, of which he spent in 7 months, and of the remainder in 3 months more, when he had only $5000 remaining; what was the value of the estate?

25. The quick-step in marching being 2 paces of 28 inches each per second, what is the rate per hour? and in what time will a detachment of soldiers reach a place 60 miles distant, allowing a halt of 13 hours? Ans. 32 m. 2014 h.

26. Two men and a boy engage to reap a field of rye; one of the men can reap it in 10 days, the other in 12, and the boy in 15 days? In how many days can the three together reap it? 27. A merchant bought a number of bales of hops, each bale containing 246 lb., at the rate of $3 for 11 lb., and sold them at the rate of $5 for 12 lb., and gained $248; how many bales did he buy?

18

133

Ans. 7.

28. Suppose I pay 331 cents per bushel for carting my wheat to mill, the miller takes for grinding, it takes 4 bushels of wheat to make a barrel of flour, I pay 25 cents each for barrels and $14 per barrel for carrying the flour to market, where my agent sells 60 barrels for $3671⁄2 out of which he takes 25 cents per barrel for his services; what do I receive per bushel for my wheat? Ans. 87 cents.

RATIO.

314. RATIO is the relation of one quantity to another of the same kind; or, it is the quotient which arises from dividing one quantity by another of the same kind.

315. Ratio is usually indicated by two dots; thus, 8: 4 expresses the ratio of 8 to 4.

The two quantities compared are the terms of the ratio; the first term being the antecedent, the second the consequent, and the two terms, collectively, a couplet.

316.

Most mathematicians consider the antecedent a divi

dend, and the consequent a divisor:

but others take the antecedent for the divisor, and the consequent for the dividend;

NOTE 1. The first method is often called the English method, and the other the French; but there appears to be no good reason for such a distinction.

NOTE 2. The first is a direct ratio; the second is an inverse or reciprocal ratio. The first being considered the more simple and natural, is adopted in this work.

317. The antecedent and consequent being a dividend and divisor, it follows that any change in the ANTECEDENT causes a LIKE change in the value of the ratio, and any change in the CONSEQUENT causes an OPPOSITE change in the value of the ratio (Art. 84, 85, and 131). Hence,

1st. Multiplying the antecedent multiplies the ratio; and dividing the antecedent divides the ratio (Art. 83, a and b).

314. What is ratio? 315. How indicated? What are the terms? The 1st? The 2d? The two collectively? 316. Which term is divisor? Is the custom uniform? Which method is here taken? Why? What is a direct ratio? An inverse ratio? 317. Explain and illustrate Art. 317 fully.

2d. Multiplying the consequent divides the ratio; and dividing the consequent multiplies the ratio (Art. 83, c and d).

3d. Multiplying both antecedent and consequent by the sume number, or dividing both by the same number, does not affect the ratio (Art. 84, a and b).

318. The antecedent, consequent, and ratio are so related to each other, that, if either two of them be given, the other may be found; thus; in 12: 3 = 4, we have

antecedent ÷ consequent ratio,

319. When there is but one antecedent and one consequent the ratio is said to be simple; thus, 15: 5=3, is a simple ratio. 320. When the corresponding terms of two or more simple ratios are multiplied together the resulting ratio is said to be compound; thus, by multiplying together the corresponding terms of the simple ratios,

pound ratio, 48:4=12 or 480:12= 4 0.

we have the com

A compound ratio is always equal to the product of the simple ratios of which it is compounded.

NOTE. A compound ratio is not different in its nature from a simple ratio, but it is called compound merely to denote, its origin.

4. What is the inverse ratio of 2 to 9? 5. What is the ratio compounded of 8 to 6 and 9 to 2? 6. Which is the greater, the ratio of 9 to 7 or of 19 to 14? 7. Which is the greater, the ratio of 5 to 4 or of 15 to 13?

318. What of antecedent, consequent, and ratio? 319. What is simple ratio? 320. Compound ratio? Its value? Its nature? Why called compound?