14. Show that two of these are the same, only differently divided. Also, how many grains Troy are there in a pound Avoirdupoise? 15. What is the standard of capacity in England, and how may it be ascertained? Write out the table of capacity. 16. How is time divided, and what is the length of a year? Also, what is a lunar, and what is a calendar month? Explain the terms, 17. Who established the present calendar, and when was it revised ? Also, when was the revised calendar adopted in England? 18. What is exactly the difference between the old and new styles? Why was the alteration necessary? And why will no further change be required, though we still reckon 3651 days to a year? 19. We find mention of ancient coins and sums. State the value of each of the following:-groat, angel, noble, mark, guinea; also of moidore, pistole. 20. There are also various measures. State the contents of the following:-runlet, tierce, anker, pipe, butt, kilderkin, coom, wey, chaldron; also hogshead of wine, and hogshead of beer, 6. EXAMPLES OF CONCRETE NUMBERS. 1. Add together 5l.; 27. 14s. 61d.; 7d.; 10s. 6d.; 1d.; 2001.; 151. 14s. 63d.; 21. 12s. 71⁄2d.; 19s. 113d. 2. From 1007. take ‡d.; and from 1007, take 997. 19s. 11 d. 3. What is the entire weight of 4 barrels of sugar, the first weighing 1 ton 13 cŵt. 3 qrs. 27 lbs.; the second, 2 tons 3 cwt. 1 qr. 7 lbs.; the third 12 cwt. 1 qr.; and the fourth 4 tons 27 lbs.? Also, what is the net weight of the whole, deducting for tare 13 cwt. 2 qrs. 17 lbs. ? 4. What is the weight of three nuggets of gold, weighing respectively 13 ozs. 14 dwts. 21 grs., 3 lbs. 2 ozs. 17 dwts. 18 grs., and 541 lbs.? Also, into how many sovereigns may the whole be coined, the weight of gold in a sovereign being 5 dwts. 3 grs.? 5. How many times will a walking-stick touch the ground in walking 3 miles, if it does so at each third step, and a step is 2 feet? 6. How many minutes are there in the nine years preceding 1857 ? 7. Amongst how many men may 201. be divided, giving 5s. 4d. to each? Also, divide 157. among an equal number of men, women, and children, giving a man thrice as much, and a woman twice as much as a child, who receives 2s. 6d. 8. How long will it take to count a million of sovereigns at the rate of 100 a minute? Also, how far will they reach in a straight line if 25, laid side by side, extend one foot ? 9. What is the cost of flooring a room 22 ft. 6 in. long, by 15 ft. 3 in. broad? And what will be the cost of painting the walls of the same room, if it is 12 ft. high, at 11⁄2d. per square foot, deducting for a door 7 ft. by 3 ft., 2 fire-places, each 3 ft. 6 in. by 7 ft., and 3 windows, each 6 ft. by 2 ft.? 10. How much larger must the English sovereign be made that it may equal 1000 farthings? Also, if the sovereign is the standard, how much smaller must the penny be made that it may be equal to four mils? And what common money is nearly equal to the cent? 11. What is the value in present money of 3-2417. i. e. 3 sovereigns +2 florins+4 cents, (reckoned at 24d.) +1 mil; and of 72-1587.? 12. Put, according to this plan, 21. 6s. 94d. in a decimal form; and also 87. 2s. 6d. 13. What is the value of a beam of cedar 60 feet long, and on the average 13 feet broad and 11⁄2 feet thick, at 2s. 74d. per cubic foot? 14. Add together 29 crowns, 37 guineas, 55 florins, 14 pence, 27 farthings, 3 marks, and a moidore. 15. How many pounds of tea, at 4s. 6d. per lb., can be procured for 25 boxes of oranges, each containing 1,200, at d. each? 16. Sold out of the North Western Railway 16 shares at 94, and bought into the Great Western at 631, how many shares had I; and how much yearly dividend at the rate of 37. 10s. for 1007. ? 17. In a shoal of herrings 7 miles in length, 21⁄2 miles in breadth, and 50 yards in depth, how many herrings, allowing 75 to a solid foot? and what would be their value at 18 pence per thousand? 18. How long after firing a gun will the report be heard at a distance of 7 miles, if sound travels at the rate of 1142 feet per second? Also, if 1 minute elapse after seeing a flash of lightning before the thunder is heard, how far off is the thunder-cloud? 7. PROPORTION. 1. What is meant by ratio, or what is sometimes called geometrical ratio? Illustrate the definition from the examples of 9 to 12, and 5 to 8. Also, show that the ratio between two numbers may be expressed by a fraction. 2. Show that several numbers may have the same ratio to one another; thus, that the ratios of 9 to 12, 3 to 4, 27 to 36, 153 to 204, are the same. 3. Of how many ratios does a proportion consist? Show how these ratios are arranged so as to form a proportion; and arrange 4, 5, 15, 12, as a proportion. 4. Arrange these numbers variously, so as still to be proportionals; and combine them together in various ways, so that they may still be proportionals. 5. Make as many arrangements and combinations as possible out of the proportion, ab::c:d. Also, do the same with the proportion, 2:3::4:6. 6. What relation exists between the terms of a proportion upon which the "Rule of Three " is founded? 7. Three terms then of a proportion being given, the fourth may easily be found. In the following proportions supply the vacant places, and show the method: 8. Find first, second, third, and fourth proportionals to the numbers 2, 3, 4. 9. When numbers are concrete (i.e. representing denominations), show that there can be no ratio unless the denominations are the same; but that it is not necessary that all four terms of a proportion should be identical. Thus, that 7 days: 21 days: : 13 miles: 39 miles, is a correct proportion. 10. In "stating the question" in Rule of Three, the term wanted being considered as the fourth, how may the third be easily known? And the third term being placed, what is the important matter as to the arrangement of the other two terms? 11. Show that the following two questions will be sufficient to state any simple question in Rule of Three :-" Of what kind must the answer be?" and, "Must the answer be less or more?" 12. Apply this method to the statement of the following questions, and state them, but not work them : A. Find the value of 15 sheep, when 8 sheep cost 117. 178. 4d. B. How many yards of cloth may be bought for 251., when 27 yards cost 127. 17s. 3d.? C. How many men can build a wall in 15 days, if 75 men can build it in 8 days? D. How many days will it take 75 men to do a piece of work, which takes 25 men 8 days? E. How long will provisions for 6 months last a garrison of 225 men, if they are reduced from 18 ounces to 12 ounces of bread per man daily? F. How long will provisions for 6 months last a garrison of 225 men, if they are reinforced by 150 men? 13. Show how double, triple, or quadruple proportion may be stated in a similar manner, there being only one third term, and two, three, four, or more first and second terms. 14. State the following three sums, arranging the several first and second terms in a column : A. If 14 men can do a piece of work in 18 days of 12 hours B. If 100%. will last a family of 6 persons for 12 months, 15. Show how these several proportions may be combined into one; and also show how the working may be shortened by cancelling or striking out equal factors from the first and second terms. 16. Work out the above three sums in Compound Proportion, and cancel as far as possible. 8. EXAMPLES IN RULE OF THREE. 1. A bankrupt pays 12s. 8d. in the £, and his assets amount to 950.; what is the amount of his debts? 2. A bankrupt's debts amount to 204/. 16s., and his assets to 1797. 4s. ; how much can he pay in the £? 3. How many yards of cloth at 3s. 74d. per yard, must be given in exchange for 935 yards at 18s. 1d. per yard? 4. Find a fourth proportional to the numbers 1:2::3: also to 3: 3.75::40: ; and 5. What is the price of 7 cwt. 3 qrs. 18 lbs. of sugar at 4s. 41d. for 11 lbs. ? 6. What is the price of 17 gallons of oil at 377. 16s. for 84 gallons? 7. If an ounce of silver cost 88. 02d., what must be paid for 7 ingots, each 33 lbs. ? 8. What is the price of silver per lb. when 4 lbs. 10 ozs. cost 247. 17s. 10d.? 9. What is the interest upon 650l. at 4 per cent.? 10. What is the price of 7% ells English at 107. 12s. 6d. for 25 yards? 11. If 42 men perform a piece of work in 108 days, in what time will 72 men do it? 12. How many figs may be bought for 17. 178. Od. at the rate of 471. 1s. 3d. for 11⁄2 tons? 13. If 84 men mow 72 acres of ground in 15 days, how many acres will 96 men mow in 12 days? 14. What is the interest of 7501. for 3 years and 6 months at 5 cent. per annum? per 15. A solid foot of stone is 16 inches broad and 3 inches thick; required its length. 16. If 18 men eat 16s. worth of bread in 3 days, when wheat is at 9s. per bushel, what value of bread will 45 men eat in 27 days, with wheat at 7s. 6d. per bushel ? 17. If 12 men can build a wall 60 yards long, 4 feet thick, and 6 feet high, in 24 days, working 12 hours per day, what length of wall 3 feet thick and 8 feet high will 15 men build in 18 days, working 10 hours per day? 18. If 125 men can make an embankment 100 yards long, 20 feet high, and 4 feet wide, in 12 days of 7 hours each, in how many days of 10 hours each will 2,400 men make an embankment 1,000 yards long, 16 feet high, and 6 feet wide? 1 |