gallons) for £60, and retails it at 358. a dozen: what profit does he make? (6 bottles = a gallon).

(79) A fire engine throwing 3 gallons a second begins to play upon a fire, and after an hour a second one begins to work, throwing 6 gallons a second. In another half-hour a third arrives which throws 4 gallons, and in another hour the fire is extinguished. How many gallons of water have been thrown upon it?

(80) Divide £13 8s. 9d. among three persons so that the first may have 11s. more, and the second 9s. less, than the third.

(81) If the fore-wheel of a carriage be 8 ft. in circumference, and the hind-wheel 11 feet, how many revolutions will the former make more than the other in 13 miles?

(82) 21 first-class passengers charged 2d. per mile, and 33 second-class charged 1 d., pay altogether for a journey £31 58. 3. How long is it?

(83) How many seconds are there from 8 hrs. 15′ 23′′ P.M. on Sept. 11th, 1853, to 10 hrs. 27′ 29′′ A.M. on Jan. 23rd, 1866?

(84) The postage to Canada for a half-ounce letter is 8d., at what rate is that per ton?

(85) Determine the difference in the price of a field of 23 acres, at £53 178. 41⁄2d. per acre, and at 6s. 8 d. per pole.

(86) If a man row in still water three miles an hour, how long will he take to row 8 miles up a river, and back, when the stream is flowing at the rate of a mile an hour?

(87) The traffic receipts of the railways in the United Kingdom for a week in 1866 amounting on 12306 miles to £594790; and for the corresponding week in 1865 to £571104 on 11898 miles, find (1) how many miles of railway had been opened in the year? and (2) whether the average receipts per mile for the week had increased or decreased, and by how much?

(88) Reduce 82180 inches, and 77900 inches, into poles. (89) 1106 tons of copper ore from a mine A are sold for £5930 18s. 6d.; 355 tons from a mine B for £1757 5s.; and 180 tons from a mine C for £908 12s. 6d. Which is the most valuable ore, and which the least, and how much

does the value of a ton of the one exceed that of a ton of the other?

(90) Divide £9 between A, B, and C, so that A may receive 155. more than B, and B five times as much as C.

(91) An exhibition was open for a week at 5s. for the first two days, half a crown for the next two, and a shilling for the last two; and the numbers admitted on the several days were 573, 897, 5821, 4239, 10379, and 9768. Determine the amount of money taken.

(92) Two trains start at nine o'clock from the ends of a railway, A, B, 150 miles long, that from A travelling at the rate of 35, and that from B at 25, miles per hour; at what time will they meet and at what distance from A?

(93) The weekly amount of wages at a factory where an equal number of men, women, and boys were employed was £41 8s. The men received 4s. 6d. per day, the women the boys 18. 2d.: how many were there of each?

2S.,

(94) If a sum of five guineas were made up of equal numbers of bronze pence, halfpence, and farthings, how far would the coins reach if placed side by side in a line; 5 pence measuring 6 inches, 5 halfpence 5 inches, and 5 farthings inches?

(95) A boy sets off to run 300 yards, taking three steps in a second, each step 3 feet long; and is followed after an interval by a dog which makes in every second two bounds of 6 feet each: how long a start, both in time and distance, must the boy have so as not to be overtaken by the dog?

(96) Express 7095924-square feet in acres, roods, and poles.

(97) It is new moon on March 16th at 9 hrs. 37′ P.M., and again on April 15th at 7 hrs. 3' A.M. If the lunar months were exactly equal, when would it be new moon again?

(98) If two millions of rivets were used in the construction of the Britannia bridge, each weighing 15 oz., what was their whole weight? and supposing that two men could drive a rivet every 3', how many days of 10 hrs. each would it take 1000 men to fasten them all?

(99) The returns of a gold mine are 241 tons of ore, yielding 2 oz. 1 dwt. 15 grs. of fine gold per ton, and 193 tons yielding 1 oz. 12 dwts. 9 grs. per ton: find the value of the whole yield at £3 178. 10d. per oz.

(100) The number of miles run by trains on a certain railway in the half-year ending June 30th was 1858574, and in the half-year ending Dec. 31st it was 385892 miles more than this. The total working expenses and renewals for the two half-years were £243179 and £299619. Find the average current expense (on the whole year) of a train per mile.

CHAPTER III.

MEASURES AND MULTIPLES.

14. A number is said to be a factor or measure of another number, when it is contained in that number an exact number of times; thus 3 is a measure of 12, 15, &c.; 12 is a measure of 24, 36, &c.

A number which is divisible by no factor except itself and unity is called a prime number; as 5, 7, 23,

&c.

A number which contains other factors besides itself and unity is called a composite number; as 10, which is composed of the factors 2 and 5; 21, of 3 and 7; &c.

The prime factors of a composite number are the prime numbers contained in it as factors, and by the product of which, repeated if necessary, it is formed. Thus the prime factors of 30 are 2, 3, 5; of 12, they are 2, 3.

RULE. To separate a composite number into its prime factors.—Divide the given number by any prime number that is contained in it exactly; and then the quotient by any prime number contained in it; and so on, until the quotient is itself a prime number. The divisors and the last quotient are the prime factors required.

Ex. Separate 60 into its prime factors.

.. The prime factors are 2, 3, 5; and 60=2 X2 X3 X5.

15.-A common measure of two or more numbers is a number which is a measure of each of them. Thus 5 is a common measure of 20 and 35; 8 is a common measure

of 64, 48, and 32. The greatest common measure of two or more numbers is often written for shortness G. C. M.

Two numbers which have no common measure except unity, are said to be prime to one another. Thus 12 and 35 are prime to one another. The greatest common measure of two numbers may sometimes be known on inspection: thus 12 might be seen to be the G. C. M. of 36 and 84. But if not, the following rule must be adopted.

RULE.-To find the greatest common measure of two numbers.-Divide the greater by the less, and then the divisor by the remainder left, and so on until there is no remainder the last divisor will be the G. C. M. required. Ex. Find the G. C. M. of 42 and 154.

42) 154 (3

126

28) 42 (1

28

14) 28 (2

28

14 is the G. C. M. required.

This method depends on the facts:

(a) that any measure of a number will be also a measure of every multiple of that number; thus 3, which is a measure of 12, is a measure also of 24, 36, &c.;

(B) that any common measure of two numbers will be also a measure of their sum or their difference; thus 5, which is a common measure of 30 and 45 is a measure of 75 or of 15.

Applying these in the above example: we see that since 14 is a measure of itself and of 28, it is, by (8), a measure of 42. Therefore it is a measure of 126, by (a); and therefore by (8) a measure of 126 + 28, i. e. 154. Hence it is a common measure of 42 and 154.

To show that it is the greatest common measure, we notice that every common measure of 42 and 154, being by (a) a measure of 126, is by (B) a measure of 154-126, i. e. 28. Similarly every such measure, being now a common measure of 42 and 28, is a measure of 42 – 28, i. e. 14.

Therefore the greatest common measure of 42 and 154 is contained in, and therefore cannot be greater than, 14: so