Ex. 19. What are 256 reams of paper worth, at 33s. 6d. per ream? Ex. 20. Fifty thousand larks have been sold in a single season in London: what did they fetch, supposing they were bought at 14d. each?

Ex. 21. The circumference of the Earth, in the latitude of London, is 15,120 miles, which is the space we pass over in 24 hours, by the diurnal motion of the earth: how much space do we pass over in a minute?

Ex. 22. Three thousand ounces of gold are imported into England annually how many pounds and grains are imported in 50 years, at this rate, and what is the value of it at 31. 185. per ounce?

Ex. 23. To work the silver mines in South America, 40,000 negroes are imported annually: how many of these poor creatures have perished in this work during the last century?

Ex. 24. The duty on hops amounted, at 11⁄2d. per lb., in a certain year, to 26,3571. 9s. 9d.: how many hops were grown that season? Ex. 25. The battering ram employed by Titus to demolish the walls of Jerusalem, weighed 100,000 lbs.: how many tons did it contain ?

Ex. 26. The copper mines in the island of Anglesey produce 1500 tons annually, and those in Comwall 4000 tons: what is the value of the whole at 9d. per ib.?

Ex. 27. Mr. Bolton coined 40,000,000 penny-pieces, each weighing an ounce: how many pounds of copper were used for them: how much was the value of these in pounds sterling; and what was gained by this coinage, supposing the copper and expense of coining to be estimated at 12d. per pound?

Ex. 28. In the year 1794, 43,259,746 yards of Irish linen were exported from Ireland: how many packages did they make, each package containing 20 pieces, and each piece 261 yards? How many shirts would this linen make, at the rate of 33 yards per shirt?

Ex. 29. The circumference of the earth is estimated at 24,912 miles: how many barley-corns, (three of which make an inch), would fill up this space?

Ex. 30. The territory of the United States of America contains a million of square miles, or 640 millions of square acres; of these, about 56 millions are water: what number of acres, roods, and perches of land, do the United States contain, and how many inhabitants will they support, allowing to each 41 acres?

Ex. 31. There are now in England, Scotland, and Wales, 23 millions of acres of waste land: how many farms might these be divided into, allowing to each 75 acres :-and allowing 5 persons to each farm, how many souls would these waste acres support?*

NOTE.

"England and Wales contain 73,334,400 acres, and 8,873,000 inhabitants, Scotland has 1,600,000, and Ireland about 4,250,000 inhabitants. England and Wales have 152 inhabitants for each square mile: Scotland 55, and Ireland 146.

"England contains 78 millions of acres: its rents are estimated

Ex. 32. Between the 5th of July, 1810, and the same day, 1811, there were brewed, by 12 brewers only, 939,900 barrels of porter: how much would this quantity sell for when retailed out at 5d. per quart?

Ex. 33. How many hours, minutes, and seconds have elapsed since the birth of Christ, which is 1808 years, supposing 365 days in a year? Ex. 34. It is said the Small-pox carries off in London, by death, 50 persons a week? how many (if the disease is not checked) will it destroy in ten years?

Ex. 35. There are about 10,540 tons of cheese imported into London annually: how much do they sell for at the average price of 7 d. per lb. Ex. 36. It is computed that there are 50,000 tons of butter annually consumed in London: what is the expense, supposing the average price 103d. per lb. ?

Ex. 37. About 120,000 persons are employed in the cotton trade: if of these one-fourth are men, who earn 3s. 6d. a day, and one-fourth women, who earn 1s. 1d. a day, and the rest children, who earn, each, 3s. per week, how much is earned by manual labour in the cotton manufacture every year?

Ex. 38. There have been 20,000,000 lbs. of tea imported in a single year from China: what was the value of it, supposing the average price 4s. 9d. per lb. ?

Ex. 39. The consumption of tobacco in this country is about 169,000 cwt.: how much is expended on this article at 14d. per oz.?

Ex. 40. Sir R. Phillips, (the publisher of this Arithmetic), caused to be printed, for various books, between the years 1798 and 1808, as many sheets of paper as would, if joined together, extend round the world. Considering each sheet as 21 inches in length, how many reams of paper did he use in that time? and what was the value of the paper, reckoning it at thirty shillings per ream? See Ex. 29.

Ex. 41. The consumption of milk is not less than 6,980,000 gallons annually, in London: how much is expended on this article at 2d. per pint?

Ex. 42. In London alone, 630,000 chaldron of coals are burnt: what is the cost at 42d. per peck?

Ex. 43. The iron rails round St. Paul's cost 11,202l. os. Ed., and they weighed 200 tons and 81 lbs.: what was the iron charged per lb. ?

Ex. 44. Westminster-bridge cost 389,500l. in building: how soon would it have been paid for by foot passengers, at a halfpenny each, supposing 2420 went over each day?

NOTE.

at about 29 millions, but are in reality 50. The stock on the land is estimated at 145 millions; the money in the country 50; the shipping 190: merchandize and manufactures 60:- of the land 13 millions of acres are inclosed, 11 arable; 64 waste in England, 1 in Wales, 14 in Scotland. For eight millions of inhabitants, the country produces 11 ounces of wheat, and 7 of meat per day."-See Middleton's Survey of Middlesex. The above estimate was taken in 1793.

PROPORTION,

OR

THE RULE OF THREE.

This Rule is called the Rule of Three, because, by threenumbers being given we find a fourth; and it is either the Rule of Three Direct or Inverse.

THE RULE OF THREE DIRECT teaches, from three given numbers, to find a fourth, which shall have the same proportion to the second, as the third has to the first; that is, if the first be greater than the third, the second will be greater than the fourth; and, if the first be less than the third, the second will be less than the fourth.

RULE I. STATE THE QUESTION: that is, place the given numbers so that the first and third may be of the same kind, and the second the same as the number required.

2. Bring the first and third numbers into the same denomination, and the second into the lowest denomination` mentioned.

3. Multiply the second and third numbers together, and divide the product by the first, and the quotient will be the * in the same denomination as that in which the second number was left.

answer,

NOTE.

* If there be a remainder after division, it is always of the same denomination as that of the middle number, and must be brought into the next lower denomination, and then divide by the first number as before.

Ex. 1. What is the value of a pipe of wine, if 5 gallons cost £4. 17s.?

gal, £. s. pipe.

54 17:1

97

20

2

63

126

97

882

1134

In stating the question, I first consider what is known, viz. that 5 galls, cost 4l.17s., and the demand is, what a pipe will cost at the same rate? I therefore 2 say, if 5 galls: cost 41.17s., what cost 1 pipe? for such is the meaning of the statement: 5 gal. : 4l. 17s. :: 1 pipe. The first term is gallons; I must accordingly bring the third term, or the pipe into gallons: the second, or middle term, is a mixed number;"I bring it therefore to its lowest denomination, or shillings, and then multiply the 126 galls. by 97 shillings, and divide the product by the first term 5, and the answer is, 2444 shillings, because the middle number is shillings, and there is a remainder of 2: this I bring into pence, and divide again by 5; there is now 4 remaining, this I bring into farthings, and divide again by 5, and the answer is 2444s. 42d., or by bringing the shillings into pounds, 1227. 4s. 42d..

5)12222

2,0)244,4 2

122,4

12

5)24

44

4

5)16

-1

Ex. 2. If can buy 271b. of sugar for £1. 13s. how much can I purchase for thirty guineas?

£ S. lb. guineas.
I 13: 27: 30
20
21

33

630

In this example, it is known that 17. 13s. will purchase 27 lb., these will therefore be the first and second terms; and as the demand is, how many pounds can be purchased for 30 guineas, the second, or middle term, must be pounds. Having stated the question, I bring the first and 4410 third terms into the same denomination, shillings, 1260 lbs. and then multiply the second and third terms to33)17010(515.gether, and divide by the first; the quotient, or

answer, is 515lb.; but there being a remainder of 15, I múltiply this by 16, because 16 ounces make a lb.; dividing again, the quotient is 7 oz., with a remainder of 9, which I might bring into drams; but sugar is never bought or sold with such accuracy. The answer is, therefore, 515lb. 7 oz., or by bringing the lbs. into cwts, and qrs., the answer is, 4 cwt. 2 qrs. 11 lb. 7 oz. 3.

231

9

Ex. 3. What is the value of 28 ells of cloth, if 4 eils cost 18s.?

ells. shill. ells.

4 : 18 :: 28 18

224

28

4)504 2,0)12,6

6,6

All questions of this kind, in which the first and third terms are of the same denomination, and either of them is a unit, may be solved by Multiplication only. Thus if 1lb. cost 9d., what cost 28lb.? I multiply the 28 by 9, and the answer is found in pence.

It often happens that the first or third terms may be reduced to a unit, by dividing both by a common number, and then the question is solved by Multiplication only. In the example before us, it is instantly seen, that 4 will divide 4 and 28: then the statement is, 1187, and 18 multiplied by 7, gives us 126 shillings, or 6l. 6s. for the answer as before.

Ex. 4. If six yards of cloth cost 24 shillings, what will 81 yards cost ?*

Ex. 5. If 8 bushels of coals cost 9s. 6d., what is the value of 35 chaldrons ?

Ex. 6. If 5lb. of potatoes cost 4d., what is the worth of 1 cwt. on the same terms?

Ex. 7. If 5lb. of potatoes cost 3d., how many can I buy for 40s.?† Ex. 8. If 10 ells of cloth cost 2l. 10s., what is the value of 5 pieces, each containing 26 yards?

Ex. 9. If 16 yards of muslin cost 10 guineas, how many ells can I buy for 451.?

Ex. 10. If I can purchase 25 books for 2l. 8s., how many can I have for a 101. note?

Ex. 11. If a servant's wages be 25 guineas a-year, how much has he to receive for 87 day's service?

NOTES.

* In this example, the first and second terms may be divided by 6, and then it becomes a question in Multiplication: the original statement is, 6 yds. : 24 shil. :: 81 yds.; but, by Division, it is 1 yd. : 4s. :: 8 yds., and the answer is, 324 shillings, or 161. 4s. The general rule therefore is, "Divide the first, and either the second or third term, but not both, by some common measure, that is, by some number that will divide the two without leaving a remainder, and use the results instead of the original terms.

In the statement to this example, viz, 3d. : 5lb. :: 40 shillings, neither the second nor third terms are divisible by 3: but when the third, or 40 shillings, is reduced to pence, then it is divisible by 3, and the statement, 3d : 5lb.:: 480d., or as 1 : 5 :: 160, and the question is answered by multiplying 160 by 5, which gives 800 lbs., or 7 cwt. qrs. 16lb. for the answer.

From these hints the pupil will frequently see that the labour of the operation may be very much shortened.