MISCELLANEOUS EXAMPLES.

1. What number is that to which, if of be added, the sum will be 1? Ans.

2. A certain gentleman, at the time of his marriage, agreed to give his wife of his estate, if, at the time of his death, he left only a daughter, and if he left only a son, she should have of his property; but as it happened, he left a son and a daughter, in consequence of which the widow received in equity $2400 less than she would have received if there had been only a daughter. What would have been his wife's dowry if he had left only a son? Ans. $2100.

3. A gentleman being asked what o'clock it was, said that it was between 5 and 6; but, to be more particular, he said that the minuteband had passed as far beyond the 6 as the hour-hand wanted of having reached the 6; that is, that the hour and minute-hands made equal acute angles with a line passing from the 12 through the 6. Required the time of day. Ans. 32m. 186s. past 5.

4. Divide 97deg. 55m. 7fur. 35rd. 4ft. 6in. by 6.

5. A, B, and C are to share $100,000 in the proportion of 3, 4, and, respectively; but C's part being lost by his death, it is required to divide the whole sum properly between the other two.

Ans. A's part is $ 57,142, and B's $42,8574.

6. A father devised of his estate to one of his sons, and of the residue to the other, and the remainder to his wife. The difference of his sons' legacies was found to be 257£. 3s. 4d. What money did he leave for his widow? Ans. 635£. Os. 1032d.

7. In the walls of Balbec, in Turkey, the ancient Heliopolis, there are three stones laid end to end, now in sight, that measure 61 yards in length, one of which is 63 feet long, 12 feet thick, and 12 feet broad; what is its weight, supposing its specific gravity to be 3 times that of water? Ans. 850 tons.

8. A burden of 2001b., suspended on a pole 4ft. in length, the point of suspension being 6in. from the middle, is carried by two men, the ends of the pole resting on their shoulders; how much of this load is borne by each man? Ans. 125lb. and 75lb.

9. The court-house in Boston has eight pillars of granite, each 25ft. 4in. in length, 4ft. 5in. in diameter at one end, and 3ft. 5in. in diameter at the other end. How many cubic feet do they contain, and what is their weight, allowing a cubic foot to weigh 3000 ounces? Ans. 2455.03 cub. ft.; 230.15+tons. 10. A father, dying, left his son a legacy, of which he spent in 8 months; of the remainder lasted him 12 months longer, after which he had only $410 left. What amount did his father bequeath him? Ans. $956.663.

11. A merchant sold goods to a certain amount, on a commission of 4 per cent., and, having remitted the net proceeds to the owner, received per cent. for prompt payment, which amounted to $15.60. What was the amount of his commission? Ans. $260.

12. A, of Boston, remits to B, of New York, a bill of exchange on London, the avails of which he wishes to be invested in goods on his account. B, having disposed of the bill at 7 per cent. advance, received $9675; and having reserved for himself per cent. on the sale of the bill, and 2 per cent. for commission, he invests the remainder What is the amount invested, and for how much was the bill drawn? Ans. Investment, $ 9461.58; the bill was £ 2025. 13. Bunker Hill Monument is 30ft. square at its base, and 15ft. square at its top; its height is 220 feet. From the bottom to the top, through its centre, is a conical avenue 15ft. in diameter at the bottom, and 11ft. at the top. How many cubic feet are there in the monuAns. 86,068.518+ ft. 14. A hired a house for one year for $ 300; at the end of 4 months he takes in M as a partner, and at the end of 8 months he takes in P. At the end of the year, what rent must each pay ?

ment?

Ans. A pays $1833; M pays $831; P pays $ 331.

15. A merchant receives on commission three kinds of flour; from A he receives 20 barrels, from B 25 barrels, and from C 40 barrels. He finds that A's flour is 10 per cent better than B's, and that B's is 20 per cent. better than C's. He sells the whole at $6 per barrel. What in justice should each man receive?

Ans. A receives $13911; B, $15817; C, 211111.

1939

16. Bought 100 barrels of flour, at $5 per barrel, and immediately sold it on a credit of six months. The note which I received for pay I got discounted at the Suffolk Bank, and, on examining my money, I found that I had gained 20 per cent. on my purchase. What did I receive per barrel for the flour? Ans. $6.181898. 17. Required the greatest possible number of hills of corn that can be planted on a square acre, the hills to occupy only a mathematical point, and no two hills to be nearer than three and a half feet. Ans. 4165. 18. Lent a friend $ 700, which he kept 20 months. Some years after I borrowed of him $ 300; how long should I keep it to balance the favor? Ans. 46 months. of the remainder to

19. John Lee gave of his estate to his wife, his oldest son, and of the residue to his oldest daughter, and of what then remained, which was $1500, was to be equally distributed among his other children, who received $150 each; required the number of his children, and the value of his estate.

20. A and B set out to travel round a certain island, which is 80 miles in circumference. A travels 5 miles a day, and B 7 miles a day. How far must B travel to overtake A? Ans. 280 miles.

21. If 24.4 cubic inches of lead weigh 16 pounds, required the number of feet of lead pipe that can be made from 80 pounds of lead, the caliber of the pipe to be 1 inch, and the thickness of it of an inch. Ans. 10.35 feet.

22. How long a tube can be made from a cylinder of lead 8 inches long and 2 inches in diameter, and through the centre of which is a hole of an inch in diameter; the tube or pipe to be of an inch in caliber, and of an inch in thickness? Ans. 16.29+ in. 23. Four men, A, B, C, and D, bought a stack of hay, containing

8 tons, for $100. A is to have 12 per cent. more of the hay than B, B is to have 10 per cent. more than C, and C is to have 8 per cent. more than D. Each man is to pay in proportion to the quantity he receives. The stack is 20 feet high, and 12 feet square at its base, it being an exact pyramid; and it is agreed that A shall take his share first from the top of the stack, B is to take his share the next, and then C and D. How many feet of the perpendicular height of the stack shall each take, and what sum shall each pay? Ans. A takes 13.22+ft., and pays $ 28 9312217; B takes 3.14+ft., and pays $25.8311; C takes 2.06+ft., and pays $ 23.48181}}; D takes 1.58ft, and pays $21.74.

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24. A, B, and C bought a grindstone, for which they paid $10.60. B paid 20 per cent. more than A, and 10 per cent. less than C. The diameter of the stone was 65 inches, and the diameter of the place for the shaft 3 inches. What sum did each pay, and how much must each grind off from the semidiameter to obtain his proper share of the stone?

Ans. A paid $3, B $ 3.60, and C $4. A grinds off 5 inches; B 71⁄2 inches, and C 18 inches.

25. A servant draws off a gallon on each day, for 20 days, from a cask containing 10 gallons of wine, each time supplying the deficiency by the addition of a gallon of water; and then, to escape detection, he again draws off 20 gallons, supplying the deficiency each time by a gallon of wine. How much water still remains in the cask?

Ans. 1.0679577 gallons, or more than a gallon and half a pint. 26. The dimensions of a bushel measure are 18 inches wide, and 8 inches deep; what should be the dimensions of a similar measure that would contain 8 bushels? Ans. 37in. wide, 16in. deep.

27. What is the weight of a hollow spherical iron shell 5 inches in diameter, the thickness of the metal being 1 inch, and a cubic inch of iron weighing of a pound? Ans. 13.2387lb.

28. At a certain time between 2 and 3 o'clock, the minute-hand was between 3 and 4. Within an hour after, the hour-hand and minute-hand had exactly changed places with each other. What was the precise time when the hands were in the first position? Ans. 2h. 15m. 562s. 29. Required the contents of the largest cube that can be inscribed in a sphere 20 inches in diameter. Ans. 1539.58+ cu. in.

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30. If in a pair of scales a body weigh 90 pounds in one scale, and only 40 pounds in the other, required the true weight, and the proportion of the lengths of the two arms of the balance-beam on each side of the point of suspension.

Ans. Weight 60lb., and the proportions 3 to 2.

31. In turning a one-horse chaise within a ring of a certain diameter, it was observed that the outer wheel made two turns, while the inner wheel made but one; the wheels were each 4 feet high; and supposing them fixed at the distance of 5 feet asunder on the axletree, what was the circumference of the track described by the outer wheel? Ans. 62.83 feet.

32. The ball on the top of St. Paul's Church is 6 feet in diameter. What did the gilding of it cost, at 34d. per square inch?

33. There is a conical glass, 6 inches high, 5 inches wide at the top, and which is part filled with water. What must be the diameter of a ball, let fall into the water, that shall be immersed by it?

Ans. 2.445

inches.

34. A certain lady, the mother of three daughters, had a farm of 500 acres, in a circular form, with her dwelling-house in the center. Being desirous of having her daughters settled near her, she gave to them three equal parcels, as large as could be made in three equal circles included within the periphery of her farm, one to each, with a dwelling-house in the center of each; that is, there were to be three equal circles, as large as could be made within a circle that contained 500 acres. How many acres did the farm of each daughter contain, how many acres did the mother retain, how far apart were the dwelling-houses of the daughters, and how far was the dwelling-house of each daughter from that of the mother?

Ans. Each daughter's farm contained 107 acres 2 roods 31.22+ rods. The mother retained 176 acres 3 roods 26.34+ rods. The distance from one daughter's house to the other was 148.119817+ rods. The mother's dwelling-house was distant from her daughters' 85.51+ rods.

35. James Page has a circular garden, 10 rods in diameter; how many trees can be set in it, so that no two shall be within ten feet of each other, and no tree within two and a half feet of the fence enclosing the garden? Ans. 241.

36. A and B engaged to reap a field for 90 shillings; and as A could reap it in 9 days, they promised to complete it in 5 days. They found, however, that they were obliged to call in C, an inferior workman, to assist them for the last two days, in consequence of which B received 3s. 9d. less than he otherwise would have done. In what time could B and C each reap the field?

Ans. B could reap it in 15 days, and C in 18 days. 37. A merchant tailor bought 40 yards of broadcloth, 24 yards wide; but on sponging it, it shrunk in length upon every 4 yards half a quarter, and in width, one nail and a half upon every 11 yards. To line this cloth, he bought flannel 5 quarters wide, which, being wet, shrunk the whole width on every 20 yards in length, and in width it shrunk half a nail. Required the number of yards of flannel used in lining the cloth. Ans. 71 yards.

38. I have a garden in the form of an equilateral triangle, whose sides are 200 feet. At each corner stands a tower; the height of the first is 30 feet, the second 40, and the third 50. At what distance from the base of each tower must a ladder be placed, that it may just reach the top of each? And what is the length of the ladder, the garden being a horizontal plane?

Ans. The foot of the ladder from the base of the first tower 118.811+ feet; second tower, 115.827+ feet; third tower, 111.875 feet. Length of the ladder, 122.535+ feet.

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THE METRIC SYSTEM.

682. THE METRIC SYSTEM of Weights and Measures is based upon the decimal system of notation.

It was authorized to be used in the United States, and its use introduced into some departments of public service, in 1866, by an Act of Congress.

683. The Principal Units, or those from which the others are derived, are:

The METER, which was intended to be, and is nearly, one ten-millionth of the distance on the earth's surface from the equator to the pole.

The SQUARE METER, which is the square whose side is one meter; and the ARE (pronounced air), which is the square whose side is ten meters.

The CUBIC METER, or STERE (pronounced stair), which is the cube whose edge is one meter; and the LITER (pronounced leeter), which is one thousandth of a cubic meter.

The GRAM, which is the weight, in vacuum, of a cubic centimeter of distilled water, at the greatest density.

684. The HIGHER DENOMINATIONS are expressed by prefixing to the name of the principal unit,

DEKA, 10; HECTO, 100; KILO, 1000; MYRIA, 10,000; and the LOWER DENOMINATIONS by prefixing

DECI, 10th; CENTI, 100th; MILLI, 1000th.

NOTE. Kilo is pronounced kilo, and Deci, dès'é.

685. MEASURES OF Length.