16. If 6 men can dig a ditch in 8 days, in how many days will 3 men do it?

The fewer men, the slower the work will go on; that is, the fewer men, the more time required. Then if but half the number of men be employed, the number of days will be doubled; or if the number of men be diminished in any ratio, the number of days will be increased in the same ratio; or as the ratio of men diminishes, the ratio of days increases. The ratio then of 6 to 4 is exactly the reverse of the ratio of 8 to the answer. Then, if I invert the ratio 6 3, making it 3: 6, it will be the same which 16 has to the answer. Thus, 3:6::8:6X8÷÷-3=16_ Ans. In this case, the fewer men, the more days. It is therefore called the case in which less requires more, and belongs, like the last, to the RULE OF THREE INVERSE.

:

Two rules are unnecessary for the two cases, but we have explained them minutely in order to render the pupil's ideas clear on the subject of two classes of questions, which he would otherwise perceive to differ, without, perhaps, un derstanding why.

The only difficulty which the pupil will now have, in the solution of problems by proportion, will consist in arranging his three given numbers in their proper order. He will see that there are two ratios, in one of which both terms are given, and in the other of which, only one term is given; the wanting term being the answer. Of this incomplete ratio, then, let him set down the given term in the third place, to stand as one of the means of the proportion. He will always know this term because it will be, of course, of the same nature with the answer. The remaining two numbers form the complete ratio; which is to be set down before the last mentioned number, making the smaller term the consequent, when the answer is to be less than that number, and the antecedent, when the answer is to be greater than that number.

[We have omitted the common rule of three as unnecessary, and as calculated to draw off the mind of the pupil from the principles of proportion. If there be any advantage in exhibiting this mode, when we have the simpler and more natural one of analysis, it must surely consist in keeping these principles in view.].

17. If 73 men consume 235 barrels of provisions in a certain time, how many barrels would 365 men consume in the same time? Proportion, 73: 365 :: 235 : .... Ans. 1,175.

Since the answer is found by multiplying 235 by 365, and dividing the product by 73, the pupil will perceive that the process is the same as it would be to multiply 235 by the fractional ratio 0.5 (§ XLVIII.) Now this fractional ratio may be reduced to the whole number 5. So that 5×235 will produce the answer.

18. If 18 barrels of flour cost 162 dollars what cost 12 barrels ? Proportion, 18:12:: 162: .... Ans. 108.

The fractional ratio here would be 12=2, and 3×162= 108 Ans.

Hence, it will often abbreviate the process, to

EXPRESS THE GIVEN COMPLETE RATIO FRACTIONALLY, REDUCE IT TO ITS LOWEST TERMS, AND MULTIPLY THE REMAINING TERM BY IT.

19. If 12 loads of wood cost 51 dollars, what cost 14? Proportion 12: 14: 51: ... Ans. 591 dolls.

:....*

Ratio 1, and X51751=717=59.
Hence the pupil will perceive, that he may

REJECT COMMON FACTORS FROM THE FIRST AND THIRD TERMS, BEFORE PROCEEDING TO FIND THE ANSWER.

20. If an army of 66,153 men require 22,051 barrels of provisions for two months, how much would a company of 100 men require ? Thus, ratio, and 100773511001331.

21. If a man's wages are 75£. 10s. pr. year, what is that for two (calendar) months? Ans. 12£. 11s. 8d.

NOTE. When there are compound numbers, the lower must be made fractions (vulgar or decimal) of the higher, or the higher must be reduced to the denomination of the lower. Also, both the terms of the first ratio must be of the same denomination. The answer will be of the denomination of the third term.

22. A goldsmith sold a tankard for £8; 12, at 5s. 5d. pr. oz. What did it weigh? A. 2 lb. 8 oz. 5 dwt.

23. Bought 8 chests of sugar, each weighing 9 cwt. 2 qrs., at 2£ 5s. pr. cwt. What did they come to? A. £171.

24. How long can I board for 32£ 32; 10, at 12s. 6d. per week? A. 1 year.

25. If 2 cwt. 3 qrs. 21: lb. of sugar cost 6£ 1s. 8d., what cost 35 cwt. 1 qr.? A. £73.

26. A man failing was only able to pay 13s. 4d, on a pound. How much could he pay on a debt of £3,475 ? A £2,316; 13; 4.

27. If a man's yearly income be 208£. 1s., what is that per day? A. 11s. 4d. 3 qrs.

28. A man failing in trade finds that he can pay only $2.00 on on every $3.00 of his debts. How much can he pay on a debt of $25.00. A. $15.75.

29. A man failing can only pay 13£ in 15£ of his debts. How much can he pay on a debt of £17; 15. A. £15; 7; 8.

30. A man failing, can only pay 17s. on a pound. How much can he pay on 147£. 14s. A. £125; 10; 10; 3}.

31. A man failing in trade, gives up his property, amounting to $1,020.20 to his creditors, to whom he owes the following sums: viz., to A, $1,600.60; to B, $500.00; to C, $750.20; to D, $1,000; to E, $230.00. How much does he pay on a dollar, what does he pay each creditor, and what was the loss to the whole together?

Ans. $0.25 on a dollar. To A, $400.15; to B, $125.00; to C, $187.55; to D, $250.00; to E, $57.50. Total loss, $3,060.60. 32. If $7,200.00 will buy 3,600 bu. of wheat, what will 600 bu. cost? A $1,200.00.

33. If a staff 6 ft. long cast a shadow 9 ft., what is the height of a tower, which, at the same time casts a shadow 198 ft.

A. 132 ft. high.

34. If 12 horses eat up 30 bu. of oats in a week, how many bu. will 45 horses eat in the same time. A. 112 bu.

35. Bought a piece of cloth for $48.27, at the rate of $2.38 for every 2 yds. What was the number of yds.? A. 40 yds. 2 qrs. 1} na. 35. If the wages of 15 weeks amount to $64.19, what is a year's wages at that rate? A. $222.525.

37. If a man is 5 days on a journey, travelling 12 hours a day : how long would he be going the same distance, travelling 10 hours a day? A. 6 days.

38. If 12 men build a wall in 20 days, how many will do it in 3 days? A. 30.

33. A building was finished in 8 months by 120 workmen; but being destroyed by fire, it was necessary to rebuild it in 2 months. How many men did it take? A. 480.

40. If a field will pasture 6 cows 91 days, how long will it pasture 21 cows? A. 26 days.

41. If a man performs a piece of work in 15 days, when the day's are 12 hours long, how many days will it employ him, when they are 10 hours long? A. 18.

42. 2,000 seldiers, having provision to last them, at an allowance of 14 oz. a day, 12 weeks, find on examination that 105 barrels, containing 200 lb each are wholly spoiled. What must be the daily allowance to each man, that the remainder may last the same time? A. 12 oz.

43. 65 men are in garrison, with provision enough for 2 months. How many must leave the garrison that the same provision may last those who remain, 5 mo.? A. 390.

44. If $100.00 gain $6.00 interest in a year, what principal will gain the same in 8 mo. ? A. $150.00.

45. If $5.00 will pay the freight of 2 cwt. 150 miles, how far may 15 cwt. be carried for the same money? A. 20 miles.

46. How many men will finish in 3 days, what would take 15 men 20 days. A. 100 men..

47. If of a yard of cloth cost of a dollar, what cost of a yard? A. 188 of a dollar 80.4721.

NOTE. When Fractions occur they are to be treated according to the rules for multiplying and dividing that class of numbers. Perhaps, however, the mode of solving questions of this kind by analysis is best. By analysis, the above question is thus solved. If & cost, will cost half as much, that is; and, or a whole yard, 3. Then, if a yard cost 93, of a yard will cost an eighth as much, that is, and, three times as much, that is 188.

89

400

63

63

509

51. If 1 cwt. cost 35£., what cost 93 lbs. ? A. 6s. 3 d. 52. A man owning of a vessel, sold of his share for $956.00 what was the vessel worth? A. $1,794.375. 53. If 34 cwt. of iron cost $145 what is that pr. cwt.? A. $47-$4.075. 54. If of a yd. cost £., what cost of an Ell English. A. £ 178.2,2,d..

55. If 7 lb. of butter cost $15, what cost 274 lbs. ? A. 581917=$5.30241.

56. If by travelling 12 hours in a day, a man performs a journey in 7 days, how long will it take him, at the rate of 9 hours a day? A. 9173 days.

57. If of a yd. cost $43, what cost 173 yds. at that

rate? A. $178.381.

58. A man sold 51 would 173 cwt. cost? 59. If bu. of corn rate. A. $0.955.

cwt. of sugar for $587.00 What A. $198.1993.

cost $5, what cost 1 bu. at that

60. If of a barrel of flour cost 14 £., what cost 43ğ

barrels? A. 93£. 9s. 7d. 29 qrs.

NOTE. When examples containing Fractions are solved by proportion, and

the first term, or that by which it is necessary to divide, is a Fraction this term may be inverted and used as a multiplier (§ LI. Rule 11.)

61. If of a bushel cost of a dollar, what cost 1 bushel. A. $0.561.

62. If cwt. cost $75, what cost 1 cwt.? A. $933. 63. If 3 yd. of silk cost $15 what cost 50 yds.?

A. $31.25. 64. If cwt. of sugar cost $, what cost 40 cwt.?

A. $82.274.

§ XCII. FELLOWSHIP, by analysis, has been already explained. (§ XLIX.) Examples in Simple Fellowship may also be solved by Simple Proportion. No particular rule is necessary.

1. A, and B purchase goods in company, A laying out $350.00, and B, $470.00. They gain $250. How much should each have? Ans. A, $106.707+ B, $143.293

2. Three merchants trade, A furnishing $240.00, B $360.00, and C, 600.00. They gain $325.00. What is each one's share?

Ans. A's $65.00. B's $97.50. C's $162.50. 3. Four men trade, A furnishing $140.00, B, 260.00, C, $300, and D, $100. They gain $307. What is each one's share?

Ans. A's, $53.725, B's $99.775, C's $115.125, D's $38.375. 4. A and B trade and gain $100.00. A turnished $640.00 stock, and B, so much that he must have $60.00 for his share of gain, What amount of stock did B furnish? Ans. $960.00.

5. A, B, and C traded. A furnishes $140.00, B, $250.00, and C, 120 yds. of cloth at cost. They gained $230.00, of which $100 fell to C as his share. What was the cost of C's cloth per yd.? Ans. $2.50 per yd..

6. In a trading company, A puts in $175.00 capital, B, $200.00, and C, $500.00. They lose $250.00. What is each one's loss? Ans. A's $50.00, B's $57.1429, C's $142.8574.

In the same manner, money or property, which is to be shared in a given ra tio by several individuals, may be divided. This comprehends the division of an insolvent debtor's estate, the distribution of prize money, dividing an inheritance among heirs in a given ratio, the assessment of taxes, &c. &c. It will be obseryed that all these calculations may be made analytically, as well as by proportion.

7. A captain, mate and 20 seamen took a prize worth $3,501.00, of which the captain takes 11 shares, the mate 5 shares and each sailor 1 share. What must each have?

Ans. Capt. $1,069.75. Mate $486.25. Each sailor $97.25. 8. Divide the number 360 into parts which shall be to each other as 2, 3 and 4. Ans. 80; 120; 160.

9. Three persons share $600.00, so that the second has twice as much as the first; and the third, twice as much as both first and second. How much has each ?

Ans. First $66.6663. Second $133.333. Third $400.00. 10. A bankrupt gives up his whole property $600.00, to his creditors, to whom he stands indebted as follows: to A, $780.00 to B, $460.00, and to C, $760.00. What is each creditor's share?

Ans. A's, $234.00, B's $138.00, C's $228.00.

NOTE. In practice, it is customary to find first how much the bankrupt can pay on a dollar, and in speaking of the amount of the deficiency, it is customary to say he pays so much on a dollar, or sometimes so much per cent. Finding what he pays on 1 dollar, and afterwards on a larger debt, is evidently not proportion, but analysis.

11. An insolvent gives up property to the amount of $1,557.50, to be divided among the following creditors; A, to whom he owes $300.00; B, to whom $1,100.00; C, to whom $1,150.00; D, to whom $505.00; E, to whom $25.00; and F, to whom $35.00. At what per ct. does he pay, how much on a dollar, and how much on each debt? Ans. 50 per ct. 50 cts. on a dollar. To A, $150.00; to B, $550.00; to C, $575.00; to D, $252.50; to E, 12.50, and to F, $17.50.

12. A certain rich man left $243,000 to be distributed among children, of the following ages, 3, 5, 7, 9, 11, 13 and 15 years re