24 yards is to be divided into allotments, each containing 1 rood 11 perches 18 yards. How many will there be?

12. If a bale of wool containing 4 cwt. 2 qr. 20 lb. sell for £68 15s. 6d., how much is that per lb.?

13. What is the area of a room broad, and what length of carpet 27 in. 14. Reduce 3 tons 7 cwt. 11 lb. to

to weeks.

16 ft. 6 in. long, and 10 ft. 3 in. wide would be required for it? ounces, and 7843291 seconds

15. Multiply £13 15s. 2od. by 35, and divide £816 15s. 71⁄2d. by 59. 16. A number is multiplied by 4, 8 is added to the product, and the sum being divided by 12 gives as a quotient 3. Find the number.

17. A watch is set right at noon on the 9th of August. If it is 15/ minutes too fast at 6 P.M. on the 1st of September, how many seconds has it gained per day?

18. Two men at 5s. and a boy at 2s. 6d. a day were employed in putting up the framework of a wooden house, in which were used 1200 ft. of timber at 23s. per hundred feet, and 120 lb. of nails and bolts at an average price of 21d. per lb. The total cost being £21 6s., find how long they took to do the work.

19. If a man walk at the rate of 110 yards a minute, and for 4 hours a day, how many miles will he go in 27 days?

20. Find the difference between the value of a million farthings and 209 five pound notes.

21. Reduce 8974 grains troy to lbs., and 67893 pints to loads.

22. What will it cost to paper a room 18 ft. 6 in. long, 13ft. 9 in. broad, and 10 ft. 6 in. high, with paper 27 in. wide, at 41d. per yard? 23. Reduce 763254 drams to tons, and 213762975 seconds to years. 24. Multiply £25 2s. 1d. by 36, and divide the product by 75. 25. A lady buys velvet at 9s. 4d., and half as much silk at 4s. 10d., and altogether spends £8 4s. 91d. How much of each does she buy? 26. Find the number of acres in a square space each side of which is 700 feet.

27. A wine merchant mixes 100 gallons of sherry at 14s. with 30 gallons at 20s. At what price must he sell the mixture so as to gain 68. on every £1 of his outlay?

28. What is the value of 7 boxes of tea, each containing 14 lb. 12 oz. at 3s. 2d. per lb.?

29. To each of several poor men is given at Christmas a purse containing a sovereign, a crown, a shilling, a sixpence, a penny, a halfpenny, and a farthing. The whole amount paid being £30 12s. 10d., what was the number of poor men?

30. The water in a mill-lead has an uniform width of 15 and depth

of 3 inches. If it flows at the rate of 1 miles an hour, how many gallons will be discharged in a day, 10 gallons being taken to contain 2772 cubic inches?

31. Divide £36 4s. 3d. among 4 persons, so that the first may have £2 5s. 3d. more than the second, the second £3 1s. 4d. more than the third, and the third £2 12s. 6d. more than the fourth.

32. A book contains 232 pages of 36 lines, consisting on the average of 14 words each. How long would it take a man to make a copy of it, supposing he could write in an hour 112 lines of 12 words each, and worked 6 hours a day?

33. Multiply £18 13s. 43d. by 13, and divide the product by 29.

34. If a gentleman can save £27 out of every £100 he receives, and in 3 years he has thus saved £425 5s., how much has he spent per day during this period?

35.

Reduce 2 acres 1 rood 17 perches 3 yards to square feet.

36. Divide £25 16s. O‡d. among 10 men and 11 women, giving each man twice as much as each woman.

37. Find the number of ells in 645 poles, and the number of ounces avoirdupois in 875 dwts. troy.

38. What will be the expense of papering a room 19 ft. 4 in. long, 16 ft. 8 in. broad, and 12 ft. high, with paper a yard wide, and costing 16d. per piece of 12 yards?

39. Sound travels at the rate of 1140 feet per second. What interval of time will there be between the flash and report of a gun fired 2 miles off?

40. Multiply 17 cwt. 1 qr. 12 lb. 9 oz. 13 dr. by 28.

41. The wards of a hospital are 12 feet high, allowing 600 cubic feet of air to each patient, and 500 yards of matting 3 feet 6 inches wide are laid down between the beds, altogether covering a quarter of the floor. Find how many patients the hospital can accommodate.

42. A gentleman living in London, where there is no Sunday post, spends on the average 1s. 3d. per day on postage. What does he spend in this way in 28 years, regard being had to Leap years?

43. A spirit merchant mixes brandy at 24s. per gallon, with 12 gallons of inferior spirit at 8s. If he gains £15 18s. by selling the mixture at 26s. 4d. per gallon, find what was the original quantity of brandy.

44. The first of three partners in business receives a share half as much again as that of the second, and twice that of the third. How should £1300 be divided between them?

45. If goods are bought at £3 3s. per cwt., and the cost of carriage is £1 58. per ton, and they are sold at 8d. per lb., what is the profit on each quarter?

D

46. A man buys 126 dozen of apples, partly at 3d. and partly at 4d. a dozen. He sells them all for £2 12s. 6d. and finds that he has gained at the rate of 10d. per 6 dozen. How many are there of each kind?

47. On the Metropolitan Railway, trains run each way every quarter of an hour, from 6 A M. to 12 P.M. Supposing the average number of passengers to be 150, and the average fare paid 31d., what would be the weekly receipts?

48. Supposing 10 gallons to contain 2772 cubic inches, how many gallons of beer would there be on the cooling floor of a brewery 22 yards long and 7 yards broad, if the depth of beer were 24 inches?

49. After paying income tax at the rate of 7d. in the £ a gentleman has £533 19s. 2d. remaining, what amount of tax did he pay?

50. What is the area of a gravel walk 4 ft. 8 in. wide, which encloses a grassplot 80 yards long and 35 yards broad?

51

CHAPTER III.

ABSTRACT NUMBERS CONTINUED, AND FRACTIONS.

IN Chapter I. Abstract Numbers were sufficiently treated of to render their first applications to practical purposes intelligible, and these first applications were contained in Chapter II. It is now necessary to return to the subject of Abstract Numbers, and to consider some of their properties which have not hitherto been explained, but which must be understood before the more advanced parts of Arithmetic, and especially the nature and use of fractions, can be proceeded with.

Taking as an instance the number 1425, it has been already shown how it should be expressed either in words or figures, what is signified by each of the figures composing the number, and how it may be either added to or subtracted from another number, or used to multiply or divide any numerical quantity, abstract or concrete. But there is something more that may be known about it. Let it be divided by 5, then 1425÷5=285, or 1425=5x285. Let 285 be divided by 5, then 285=5 × 57. Let 57 be divided by 3, then 57=3 x 19. Combining these results, we should have 1425=5 × 5 × 3 × 19, and thus four factors have been found whose product=1425. It is not every number that can be so separated into factors; some are not divisible by any numbers except themselves and unity, and such are called prime numbers. Thus 23, 41, 59, are prime numbers. Those numbers, on the other hand, that can be separated into factors, are called composite numbers, and any composite number may be called a multiple of any of its factors.

Thus 28 is a composite number, and a multiple of 4, 7, and 14. To separate a number into as many factors as possible, care must be taken that none of the factors are themselves composite numbers, and the number is then said to be separated into its prime factors. This process is very useful, because by its aid calculation is frequently shortened to a very considerable extent, at the same time, it is not absolutely necessary, inasmuch as all operations might be performed without reference to it, the work being, indeed, often excessively long and cumbersome, in place of short and easy, but nevertheless being quite certain and accurate. It may be observed also, that there is no general rule known, by which any number may be separated into its factors. Practically, however, though a perfect knowledge of this subject is perhaps impossible, enough is known, and can be made the subject of rules and processes, which are true as far as they go, to afford us nearly the same advantages as would result from our knowing it thoroughly.

It is always easy to determine whether any number is divisible without remainder by any of the numbers from 2 to 12 inclusive, with the exception of 7. The following are the rules for this purpose: :

1. A number is divisible by 2, if the last figure is divisible by 2. Thus 87624 is divisible by 2, because 4 is. Any number divisible by 2 is termed an even number, and a number not so divisible is called an odd number.

II. A number is divisible by 4 or 8, if the number formed by the last two or last three figures respectively is divisible by 4 or 8. Thus 87624 is divisible by 4, because 24 is divisible by 4. And 87624 is divisible by 8, because 624 is divisible by 8.

III. The figures composing a number are called its digits. A number is divisible by 3 or 9, if the sum of its digits is divisible by 3 or 9. Thus 87624 is divisible by either 3 or 9, the sum of its digits being 27. And 73842 is divisible by 3, but not by 9, the sum of the digits being 24.