operations on the series thus obtained until we get a series of numbers all prime to one another. 4th. Find the continued product of the last series, and the factors on the left of the vertical line, which will be L. C. M. required.

N.B. We may bring out a composite instead of a prime factor when every member of the series is either divisible by or prime to this composite factor.

(5) 12, 20, 28, 18, 30, 42, 45, 63, 105, 70

(6) 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (7) 21, 15, 33, 35, 77, 105, 165, 385, 231

(8) 16, 6, 8, 2, 12, 3, 48, 24

(9) 34, 26, 65, 85, 51, 39

(10) 10, 20, 30, 40, 50, 60

(11) 105, 120, 616, 88, 24, 12, 6, 308 (12) 12, 18, 27, 63, 28

(13) 7, 11, 4, 14, 10, 5, 15

(14) 323, 247, 209, 133

(15) 6, 5, 3, 11, 35, 44, 68, 17, 14 (16) 1003, 1357

(17) 899, 961

(18) 407, 703, 444
(19) 411, 959, 2055

(20) 120, 400, 500, 375, 1500, 1000,

960

(21) What is the smallest sum of money that can be expressed either as (a) guineas or pounds? (b) crowns or half-guineas? (c) as a multiple of 15s. 9d. or of 17s. 6d. ?

(22) Find the smallest weight which can be expressed by an exact number either of lbs. troy or lbs. av.

(23) If 1 lb. of sugar is worth 5d. and 1 lb. of coffee 1s. 2d.; find the smallest number of lbs. of coffee which is worth an exact number of lbs. of sugar.

(24) Find the value of the smallest number of lbs. of coffee at 1s. 3d. that can be exchanged for an exact number of lbs. of tea at 28. 9d. each.

(25) Three travellers journey 15, 18 and 24 miles a-day respectively. How far off is the first station at which all three put up?

(26) If the year of the planet Mercury were exactly 87, that of Venus 225, and of the earth 365 of our days; how many of our years would intervene between any two occasions on which the three planets would appear to a spectator from the sun to be in a straight line?

(27) Of two cog-wheels, with 75 and 120 teeth respectively, a particular tooth of the smaller wheel comes in contact with a tooth of the larger. In how many turns of each wheel will these two teeth meet again, and how many contacts will there have been?

(28) If there be a house-door every 21 yds., and a lamp-post every 44 yds., 1 ft.; supposing a lamp-post exactly opposite one house-door, at what distance will the same occur again, and how many houses and lamp-posts will intervene ?

CHAPTER XII.

MISCELLANEOUS EXAMPLES.

1. (1) If 4 articles cost £5. 78. 8d., what will a dozen cost? (2) If 6 articles cost 17s. 34d., what will 30 cost?

(3) If 20 men can do a piece of work in 17 days, how long will 4 men take?

(4) If 7 silk handkerchiefs cost £1. 88. 101d., what will 91 cost? (5) If a dozen of wine cost £2. 2s., what will 3 bottles cost? (6) If 28 lbs. cost £7. 17s. 6d., what will 4 lbs. cost?

(7) Find the cost of 5 articles at £6. 6s. 8d. per score.

(8) If 17 articles cost £3. 18s. 71⁄2d., what will 1 cost?

(9) Find the cost of 15 articles if 8 cost £1. 7s. 4d. [Here first find the cost of 1 article, and thence of 15.]

(10) Find the cost of 35 articles, if 15 cost £7. 4s. 51d. [15 and 35 not being prime to one another (G. C. M. 5), it is shorter and easier to find the value of 5, and thence of 35.]

(11) If the provisions of a fortress will last 60 days, allowing each man 24 oz. per day, how long will the provisions last, if the allowance is reduced to 15 oz?

(12) And what may be the daily allowance, for the fortress to hold out 144 days?

(13) Find the cost of 9 articles, if 8 cost £17. 4s. 10d.

(14) Find the cost of 25 articles, if 45 cost £3. 10s. 3ąd.

(15) If 1 cwt. cost £30, what will 77 lbs. cost?

(16) If 1 ton cost £8. 8s. 4d., what will 12 cwt. cost?

(17) If for 20 shillings I can travel 150 miles, how far shall I be able to travel for 12s. ?

(18) If 2 tons, 5 cwt. cost £23. 12s. 6d., what will 1 ton, 5 cwt.

cost?

(19) If in 1 year, 8 months, I put by £45, how much shall I put by in 2 years?

(20) If 1 cwt., 2 qrs., 12 lbs. cost £3. 15s. 9d., what will 1 cwt., 3 qrs., 4 lbs. cost?

(15) 48520 × (1976+1090)

(16) (58512+7426)+(58512-7426)

(17) (58512+7426) — (58512 — 7426)

[Note that to 16 the answer is twice the greater number, and to 17 twice the

less. Why?]

(19) (125796+18043 +237509 +2759286)÷602

(20) (106033 +112568)÷(45+62)

(21) (106033 +112568)÷(437 +598 +612+396)

(22) (436 × 436 — 157 × 157)÷(436 — 157)

(23) (8424 × 7056 × 102)÷9072

(24) (3492 × 2049 × 867)÷15606

(25) (34936 × 816 x 2046)÷209616

(26) (£1. 12s. 84d. × 63 × 112)÷£1. 1s. 94d.
(27) (£1. 12s. 81d. × 63 × 112)÷392

(28) (£597. 138. 11d. × 845) — (£597. 138. 11d. × 843) (29) 28519 × 63 - - 28519 x 53

(30) 62743 × 1509-62740 × 1509

(31) (14688 × 1045 × 10110)÷108851

(32) (812345+109876 +234567 +321098 +456789 +159576)÷(3257 × 643) (33) £14. 7s. 3 d. × 79+£14. 7s. 3 d. × 21

3. (1) London contains 400778 houses, each inhabited by 8 persons on an average. Find the total number of inhabitants.

(2) The water-works of London supply 26 gallons a-day for each person. How much is supplied in a year?

(3) How much will this water weigh, at 10 lbs. per gallon?

(4) In 1868 the births were in London 115744, and the deaths 74908. What was the increase of the population from this source? (5) The area of the metropolis is 77997 acres. How many square miles is this? (1 square mile = 640 acres.)

(6) The total revenue of the United Kingdom in 1868 was £71,860,677. 12s. 8d., and the expenditure was £74,082,280. 5s. 5d. Find the deficit.

(7) The different sources of inland revenue yielded in 1868: excise, £20,173,288; stamps, £9,461,010; taxes, £3,450,318; income-tax, £6,184,166. The income from the same sources in 1867 was £39,159,781. Find the increase or decrease.

(8) The income-tax was levied at 6d. in the pound. Find the amount of income taxed.

(9) Divide £15,000 among A, B, C and D, giving to A £1961. Os. 8d. more than to either of the others, who have equal shares. (10) At an election, the Liberal candidate obtained 1375 votes more than the Conservative. The total number of votes polled was 7209. How many voted for the Conservative?

(11) A leaves an estate of £100,000, of which he disposes as follows: building lodging-houses, £1750; endowment of parish school, 1000 guineas; hospitals, £500; to the church, £500; to each of 4 chapels, £125; to each of 7 old servants, 19 guineas; the residue to be divided amongst his family, giving one-third to his widow, one-fourth to his daughter, and the balance in equal shares to his 2 sons. Find their respective shares.

(12) Find the weekly wages of 325 men, 108 women and 75 children, who receive 4s. 3d., 2s. 10d., 1s. 1d. per day, for each man, woman and child respectively.

(13) A exchanges with B 1 cwt., 11 lbs. of coffee, at 103d. per lb., for 5 cwt., 1 qr., 12 lbs. of sugar, at 34d. per lb. The difference is to be paid in money. How much is to be paid, and by whom?

(14) Find the average age of 7 persons, aged respectively 47, 38, 52, 45, 41, 49, 43 years. [An average is the sum of a number of quantities divided by the number of these addenda.]

(15) Find the average height of the following Peaks: Monte Viso, 12580 ft.; Genèvre, 11780 ft.; Cenis, 11457 ft.; Izéran, 13266 ft.; M. Blanc, 15732 ft.; Matterhorn, 14835 ft.; Rosa, 15150 ft.; Gallenstock, 12475 ft.; Vogelberg, 10866 ft.; Ortlerspitz, 12852 ft.; Groszglockner, 12776 ft.; Finster-Aarhorn, 14109 ft.; Jungfrau, 13176 ft.

(16) A mixes 4 gallons of spirit at 10s. 9ąd., with 6 gallons at 16s. 11⁄2d., and 5 gallons at 7s. 6d. each. Find the average cost per gallon.

(17) Find the average age of a school in which there are 20 boys at 9 years old, 4 at 10, 10 at 11, 12 at 12, 11 at 13, 2 at 14, and 1 at 15?

(18) A piece of translation consists of 32 pages, averaging 21 lines of 15 words each. What will be the charge at the rate of

3s. 6d. for every 72 words? words?

Also at the rate of 3s. 6d. for every 96

(19) I mixed 2 cwt., 2 qrs., 20 lbs., at 6d. per lb., with 4 cwt., 2 lbs., at 4d. per lb. Find the price per lb. of the mixture.

(20) What is my income, if at 7d. in the £, I pay £11. 1s. 8d. income-tax?

(21) £11. 18. 8d. 7.

(22) A's income is 500 guineas. What will be left him after paying the income-tax of 5d. in the £?

(23) How many times will a coach wheel, of 13 ft., 9 in. in circumference, turn round in going from London to Brighton, 50 miles?

(24) In exchange for 833 articles at 1s. 4d. each, I gave 39 guineas and 100 articles. What was the cost of each of the latter?