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?

3. What will 72 yards of cloth cost, at the rate of 9 yards for 5l. 12s.

Ans. 447. 168. 4. A person's annual income being 1467.; how much is that per day? Ans. Ss. 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 f

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 22,500 rations of bread for a month; how many rations will a garrison of 1200 men require? Ans. 36,000. 8. If 7 cwt. 1 qr. of sugar cost 267. 10s. 4d., what will be the price of 43 cwt. 2 qrs.?

Ans. 1591. 2s.

9. The clothing of a regiment of foot of 750 men amounting to 28317. 5s.; what will the clothing of a body of 3500 amount to? Ans. 13,212. 10s. 10. How many yards of matting, that is 2 ft. 6 in. broad, will cover a floor that is 27 feet long and 20 feet broad? Ans. 72 yds. 11. What is the value of 6 bushels of coals, at the rate of 17. 14s. 6d. the chaldron? Ans. 5s. 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 5s. 9d. per ounce? Ans. 2531. 10s. 03d.. 14. A garrison of 536 men has provisions for 12 months, how long will those provisions last if the garrison be increased to 1124 men ?

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Ans. 174 days and 1947. 15. What will the tax upon 7637. 15s. be, at the rate of 3s. 6d. per pound sterling? Ans. 133. 13s. 1 d. 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 6s. the bushel? Ans. 39 qrs. 3 bu. 18. A person failing in trade, owes in all 9777; at which time he has in money, goods, and recoverable debts, 4202 6s. 34d; 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. 2). Suppose a gentleman's income is 500 guineas a year, and that he spends

19s. 7d. per day, one day with another; how much will he have saved at the year's end? Ans. 1674 12s. ld 21. What cost 30 pieces of lead, each weighing 1 cwt. 12 lb., at the rate of 16s. 4d. per cwt. ? Ans. 271. 2s. 6d. 22. The governor of a besieged place having provisions for 54 days, at the rate of 11⁄2lb. 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. 1 lb.

23. At half a guinea per week, how long can I be boarded for 20 pounds? Ans. 38 weeks.

24. How much will 75 chaldrons 7 bushels of coals come to, at the rate of 17. 13s. 6d. per chaldron ? Ans. 125/ 19s. 01d. 25. If the penny loaf weigh 9 ounces when the bushel of wheat cost 6s. 3d.; what ought the penny loaf to weigh when the wheat is at 8s. 24d.?

Ans. 6 oz. 13137 dr. 26. How much a year will 173 acres, 2 roods, 14 poles of land give, at the rate of 11. 78. 8d. per acre ? Ans. 2401. 28. 7d. 27. To how much amounts 172 pieces of lead, each weighing 3 cwt. 2 qrs. 171⁄2 lb., at 87. 17s. 6d. per fother of 191⁄2 cwt. ? Ans. 2861. 4s. 4d,

28. How many yards of stuff, of 3 qrs. wide, will line a cloak that is 5 yards in length and 14 yard wide ? Ans. 9 yds. 0 qrs. 23 nl

29. If 5 yards of cloth cost 14s. 2d., what must be given for 9 pieces, containing each 21 yards 1 quarter? Ans. 27. ls. 101d. 30. If a gentleman's estate be worth 2107% 12s. a year; what may he spend per day, to save 500% in the year? Ans. 44. 8s. 1d. 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. 014 in. 32. At 138. 2 d. per yard, what is the value of a piece of cloth containing 523 ells English? Ans 434. 10s. 1134

33. If the carriage of 5 cwt. 14 lb. for 96 miles be 17. 12s. 6d. ; how far may I have 3 cwt. 1 qr. carried for the same money? Ans. 151 m. 3 fur. 33. pol. 34. Bought a silver tankard, weighing 1 lb. 7 oz. 14 dwts.; what did it cost me at 6s. 4d. per ounce ? Ans. 61. 4s. 94d.

35. What is the half year's rent of 547 acres of land, at 15s. 6d. the acre? Ans. 2117. 19s. 3d

36. A wall that is to be built to the height of 27 feet, was raised 9 feet high by 12 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. 36 men.

37. What will be the charge of keeping 11 horses for a year, at the rate of 114d. per day for each horse? Ans. 1924. 7s. 81d.

38. If 15 ells of stuff that is yard wide cost 37s. 6d.; what will 40 ells of the same goodness cost, being yard wide? Ans. 64. 13s. 4d.

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 yds. 40. If a gentleman's estate be worth 3847. 16s. a year, and the land-tax be assessed at 2s. 94d. per pound, what is his nett annual income ?

Ans. 3314 1s. 93d. 41. The circumference of the earth is about 25,000 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

miles

42. If a person drink 20 bottles of wine per month, when it costs 8s. a gallon; how many bottles per month may he drink, without increasing the expense, when wine costs 10s. the gallon?

Ans. 16 bottles.

43. What cost 43 qrs. 5 bushels of corn, at 17. 8s. 6d, the quarter?

Ans. 621. 38. 33d.

44. How many yards of canvass that is ell wide, will line 20 yards of say that is 3 quarters wide?

Ans. 12 yds.

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

Ans. 2.

46. If 3 cwt. of tea cost 40%. 12s.; at how much a pound must it be retailed, to gain 10%. by the whole? Ans. 333.

COMPOUND PROPORTION.

COMPOUND PROPORTION teaches how to resolve such questions as require two or more statings by Simple Proportion: and that, whether they be 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.

RULE.-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 sup position, 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 from 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 of the next following one. So shall the last resulting num. ber 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 gether 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.

EXAMPLES.

1. How many men can complete a trench of 135 yards long in 8 days when 16 men can dig 54 yards in 6 days ?

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2. If 100%. in one year gain 57. interest, what will be the interest of 7501. for years? Ans. 262/. 10s.

3. If a family of 9 persons expend 1207. in 8 months; how much will serve a family of 24 people 16 months? Ans. 640/ 4. If 27s. be the wages of 4 men for 7 days; what will be the wages of 14 men for 10 days?

Ans. 67. 15s.

12 hours long; Ans. 9 days

5. If a footman travel 130 miles in 3 days, when the days are in how many days, of 10 hours each, may he travel 360 miles? 6. If 120 bushels of corn can serve 14 horses 56 days; how many days will 94 bushels serve 6 horses? Ans 1021 days.

7. If 3000 lb. of beef serve 340 men 15 days; how many lbs. will serve 120 men for 25 days? Ans. 1764 lb. 11 oz. persons 12 days; how

8. If a barrel of beer be sufficient to last a family of 7 many barrels will be drunk by 14 persons in the space of a year?

Ans. 60 barrels.

9. If 240 men, in 5 days, of 11 hours each, can dig a trench 230 yards long,

3 wide, and 2 deep; in how many days, of 9 hours long, will 24 men dig a trench of 420 yards long, 5 wide, and 3 deep? Ans. 278 days.

OF VULGAR FRACTIONS.

A FRACTION, or broken number, is an expression of a part, or some parts, of something considered as a whole.

It is denoted by two numbers, placed one below the other, with a line between them;

The Denominator, or number placed below the line, shows how many equal parts the whole quantity is divided into; and represents the Divisor in Division. And the Numerator, or number set above the line, shows how many of those parts are expressed by the Fraction; being the remainder after division.—Also, both these numbers are, in general, named the Terms of the Fraction.

Fractions are either Proper, Improper, Simple, Compound, or Mixed.

A Proper Fraction, is when the numerator is less than the denominator; as †, or, or †, &c.

An Improper Fraction, is when the numerator is equal to, or exceeds, the denominator; as }, or 4, or 3; &c.

A Simple Fraction, is a single expression denoting any number of parts or the integer; as, or .

A Compound Fraction, is the fraction of a fraction, or several fractions connected with the word of between them; as of }, or ‡ of § of 3, &c.

A Mixed Number, is composed of a whole number and a fraction together; as 34, or 124, &c.

A whole or integer number may be expressed like a fraction, by writing 1 below it, as a denominator; so 3 is, or 4 is †, &c.

A fraction denotes division; and its value is equal to the quotient obtained by dividing the numerator by the denominator; so is equal to 3, and 2o is equal to 4.

Hence then, if the numerator be less than the denominator, the value of the fraction is less than 1. If the numerator be the same as the denominator, the fraction is just equal to 1. And if the numerator be greater than the denominator, the fraction is greater than 1.

REDUCTION OF VULGAR FRACTIONS.

REDUCTION of Vulgar Fractions, is the bringing them out of one form on denomination into another; commonly to prepare them for the operations of Addition, Subtraction, &c. of which there are several cases.

PROBLEM.

To find the greatest common measure of two or more numbers.

THE Common Measure of two or more numbers, is that number which will divide them both without a remainder: so 3 is a common measure of 18 and 24; the quotient of the former being 6, and of the latter 8. And the greatest number