STATE the question, by setting down in a straight line the three given numbers, in the following manner, viz. so that the 2d term be that number of supposition which is of the same kind that the answer or fourth term is to be; making the other number of supposition the 1st term, and the demanding number the 3d term, when the question is in direct proportion; but contrariwise, the other number of supposition the 3d term, and the demanding number the 1st term, when the question has inverse proportion.

Then, in both cases, multiply the 2d and 3d terms together, and divide the product by the 1st, which will give the answer, or 4th term sought, viz. of the same denomination as the second term.

Note, If the first and third terms consist of different denominations, reduce them both to the same: and if the second term be a compound number, it is mostly convenient to reduce it to the lowest denomination mentioned.-If, after division, there be any remainder, reduce it to the next lower denomination, and divide by the same divisor as before, and the quotient will be of this last denomination. Proceed in

the same manner with all the remainders, till they be reduced to the lowest denomination which the second admits of, and the several quotients taken together will be the answer required.

Note also, The reason for the foregoing Rules will appear, when we come to treat of the nature of proportions.-Sometimes two or more statings are necessary, which may always be known from the nature of the question.

EXAMPLES.

1. If 8 yards of Cloth cost 1/ 48, what will 96 yards cost?

Ex. 2. An engineer having raised 100 yards of a certain work in 24 days with 5 men; how many men must he employ to finish a like quantity of work in 15 days?

ds men ds men

As 155: 24: 8 Ans.

5

15) 120 (8 Answer.

120

3. What will 72 yards of cloth cost, at the rate of 9 yards for 5/ 128? Ans. 44/ 168. 4. A person's annual income being 1467; how much is that per day?

Ans. 88. 5. If 3 paces or common steps of a certain person be equal to 2 yards, how many yards will 160 of his paces make?

Ans. 106 yds 2 ft. 6. What length must be cut off a board, that is 9 inches broad, to make a square foot, or as much as 12 inches in length and 12 in breadth contains ? Ans. 16 inches.

7. If 750 men require 22500 rations of bread for a month; how many rations will a garrison of 1200 men require ?

8. If 7 cwt 1 qr of sugar cost 26/ 108 4d; price of 43 cwt 2 qrs ?

9 The clothing of a regiment of foot of ing to 2831/58; what will the clothing of men amount to?

Ans. 36000. what will be the

Ans. 159/ 28. 750 men amounta body of 3500 Ans. 13212/ 10s. 3 ft broad, will

10. How many yards of matting, that is cover a floor that is 27 feet long and 20 feet broad?

Ans. 60 yards.

11. What is the value of 6 bushels of coals, at the rate of 1/ 14s 6d the chaldron ?

Ans. 58 9d.

12. If 6352 stones of 3 feet long complete a certain quantity of walling; how many stones of 2 feet long will raise a like quantity?

Ans. 9528.

13. What must be given for a piece of silver weighing 73 lb 5 oz 15 dwts, at the rate of 58 9d per ounce ?

T124

Ans. 253/ 108 03d. 14. A garrison of 536 men having provision for 12 months; how long will those provisions last, if the garrison be increased to 1124 men? Ans. 174 days and 64 15. What will be the tax upon 763/ 158, at the rate of 38 6d per pound sterling? Ans. 133/ 138 1d.

16. A certain work being raised in 12 days, by working 4 hours each day; how long would it have been in raising by working 6 hours per day? Ans. 8 days. 17. What quantity of corn can I buy for 90 guineas, at the rate of 68 the bushel? Ans. 39 qrs 3 bu.

18. A person, failing in trade, owes in all 9771; at which time he has, in money, goods, and recoverable debts, 4201 68 31d; now supposing these things delivered to his creditors, how much will they get per pound? Ans. 88 74d.

19. A plain of a certain extent having supplied a body of 3000 horse with forage for 18 days; then how many days would the same plain have supplied a body of 2000 horse? Ans. 27 days. 20. Suppose a gentleman's income is 600 guineas a year, and that he spends 258 6d per day, one day with another; how much will he have saved at the year's end?

Ans. 164/ 128 6d. 21. What cost 30 pieces of lead, each weighing 1 cwt 12lb, at the rate of 168 4d the cwt? Ans. 271 28 6d.

22. The governor of a beseiged place having provision for 54 days, at the rate of 141b of bread; but being desirous to prolong the siege to 80 days, in expectation of succour, in that case what must the ration of bread be? Ans. 1lb. 23. At half a guinea per week, how long can I be boarded for 20 pounds? Ans. 386 wks. 24. How much will 75 chaldrons 7 bushels of coals come to, at the rate of 1/ 138 6d per chaldron ?

Ans. 125/ 19s old.

25. If the penny loaf weigh 8 ounces when the bushel of wheat costs 78 3d, what ought the penny the wheat is at 88 4d?

loaf to weigh when Ans. 6 oz 15, dr.

26. How much a year will 173 acres 2 roods 14 poles of land give, at the rate of 17 78 8d per acre?

Ans. 2401 28 7d. 27. To how much amounts 73 pieces of lead, each weighing 1 cwt 3 qrs 7 lb, at 107 48 per fother of 191 cwt?

Ans 691 48 2d 118 q. 23. How many yards of stuff, of 3 qrs wide, will line a cloak that is 1 yards in length and 34 yards wide? Ans. 8 yds 0 qrs 23 nl. 29. If 5 yards of cloth cost 148 2d, what must be given for 9 pieces, containing each 21 yards 1 quarter?

Ans. 271 is 10d. 30. If a gentleman's estate be worth 2107 128 a year; what may he spend per day, to save 500/ in the year?

Ans. 41 88 11d.

31. Wanting

31. Wanting just an acre of land cut off from a piece which is 13 poles in breadth, what length must the piece be? Ans. 11 po 4 yds 2 ft 01 in.

32. At 7s 9d per yard, what is the value of a piece of cloth containing 53 ells English 1 qu. Ans. 25/ 188 13d. 33. If the carriage of 5 cwt 14 lb for 96 miles be 17 12s 6d; how far may I have 3 cwt I qr carried for the same money ? Ans. 151 m 3 fur 3 pol.

34. Bought a silver tankard, weighing 1 lb 7 oz 14 dwts; what did it cost me at 68 4d the ounce ? Ans. 61 48 94d. 35. What is the half year's rent of 547 acres of land, at 158 6d the acre?

Ans. 211/ 198 3d.

36. A wall that is to be built to the height of 36 feet, was raised 9 feet high by 16 men in 6 days; then how many men must be employed to finish the wall in 4 days, at the same rate of working? Ans. 72 men. 37. What will be the charge of keeping 20 horses for a year, at the rate of 141⁄2d per day for each horse?

Ans. 441/ Os 10d. 38. If 18 ells of stuff that is yard wide, cost 39s 6d; what will 50 ells, of the same goodness, cost, being yard wide? Ans. 71 68 330d

39. How many yards of paper that is 30 inches wide, will hang a room that is 20 yards in circuit and 9 feet high.

Ans. 72 yards. 40. If a gentleman's estate be worth 384/ 168 a year, and the land-tax be assessed at 2s 91⁄2d per pound, what is his net annual income? Ans. 331 18 9}d.

41. The circumference of the earth is about 25000 miles; at what rate per hour is a person at the middle of its surface carried round, one whole rotation being made in 23 hours 56 minutes? Ans. 1044-816 miles.

42. If a person drink 20 bottles of wine per month, when it costs 88 a gallon; how many bottles per month may he drink, without increasing the expense, when wine costs 108 the gallon? Ans. 16 bottles.

43. What cost 43 qrs 5 bushels of corn, at 1l 88 6d the quarter. Ans. 62/ 38 33d. wide will line

44. How many yards of canvas that is 50 yards of say that is 3 quarters, wide?

ell

Ans. 30 yds.

45. If an ounce of gold cost 4 guineas, what is the value

of a grain?

Ans 2d.

46. If 3 cwt of tea cost 40 128; at how much a pound must it be retailed, to gain 10/ by the whole?

Ans. 38.

COMPOUND

COMPOUND PROPORTION.

COMPOUND PROPORTION shows how to resolve such questions as require two or more statings by Simple Proportion; and these may be either Direct or Inverse.

In these questions, there is always given an odd number of terms, either five or seven, or nine, &c. These are distinguished into terms of supposition, and terms of demand, there being always one term more of the former than of the latter, which is of the same kind with the answer sought. The method is thus:

SET down in the middle place that term of supposition which is of the same kind with the answer sought.-Take one of the other terms of supposition, and one of the demanding terms which is of the same kind with it; then place one of them for a first term, and the other for a third, according to the directions given in the Rule of Three.-Do the same with another term of supposition, and its corresponding demanding term; and so on if there be more terms of each kind; setting the numbers under each other which fall all on the left-hand side of the middle term, and the same for the others on the right-hand side.-Then, to work

By several Operations.-Take the two upper terms and the middle term, in the same order as they stand, for the first Rule-of-Three question to be worked, whence will be found a fourth term. Then take this fourth number, so found, for the middle term of a second Rule-of-Three question, and the next two under terms in the general stating, in the same order as they stand, finding a fourth term for them. And so on, as far as there are any numbers in the general stating, making always the fourth number, resulting from each simple stating, to be the second term in the next following one. So shall the last resulting number be the answer to the question.

By one Operation-Multiply together all the terms standing under each other, on the left-hand side of the middle term; and, in like manner, multiply together all those on the right-hand side of it. Then multiply the middle term by the latter product, and divide the result by the former product; so shall the quotient be the answer sought.

VOL. I.

H

EXAMPLES.