by selling the remainder at 12 cents per pint? and how much will he gain? Ans. $30. 18. My horse and saddle are, together, worth 80 dollars; my horse is worth 7 times as much as my saddle: what is the value of each? Ans. Saddle, $10. The following question is said to be by Sir Isaac New ton. 19. If 12 oxen eat up 31 acres of grass in 4 weeks, and 21 oxen eat up 10 acres in 9 weeks, how many oxen will eat up 24 acres in 18 weeks, the grass being at first equal on every acre, and growing equally? Ans. 36 oxen. This problem requires three statements in compound proportion. Thus, Cause. Effect. Cause. Effect. Oxen, 12 : Weeks,. 4 : 33 :: 21 : [] 9 The result of this proportion gives 131 acres for 21 oxen 9 weeks. But the question gives only 10 acres ; therefore, ten acres become 131 by having 5 weeks to grow, that is, 5 weeks longer than the first quantity. Cause. Effect. Cause. Effect. Acres,. . 10 The result of this proportion gives 21 acres of growth, to be added to the 24 acres.-Total, 45 acres. Lastly, 20. Suppose a meteor so high in the heavens, as to be visible, at the same moment, at BOSTON, longitude 71° 3′ W.; at WASHINGTON, longitude 77° 43′ W.; and at the SANDWICH ISLANDS, longitude 155° W.: and suppose the time of appearance at WASHINGTON to be 5 minutes past 10, P. M. What time is it by the clocks at the other places? At Boston, 10 h. 31 m. 40 s., P.M. At S. Islands, 4 h. 55 m. 52 s., P. M. 21. What time is it, when the past interval from noɔn is of the time, onward to midnight? Ans. { Ans. 5h. 24m. 22. A gentleman bought several gallons of wine for 94 dollars; and, after using 7 gallons himself, found that onefourth of the remainder was worth 20 dollars; how many gallons were there at first? Ans. 47. [Solve questions 21 and 22, by proportion.] 23. Find three numbers in arithmetical progression, such, that the least shall be to the greatest, as 5 to 9, and the sum of the three, 63. Ans. 15, 21, 27. 24. A can produce a certain effect in 3 hours; B in 4 hours, and C in 5 hours; in what time can the three together produce the same effect? Ans. 11 hours. 25. A general, disposing his army into a square, found he had 231 men over and above; but, increasing each side with one soldier, he wanted 44 to fill up the square; how many men had he? Ans. 19000 men. N. B. Problem 25 is very simple, in fact: yet, from some cause or other, it is very troublesome to students— few of whom attempt an unaided solution. 231+44 men are required to place round one side, and one end, of a square; how many men does such a square contain? is the question. 26. A parallelogram is 6 rods longer than it is wide, and its diagonal measures 30 rods; required its area and dimensions. Ans. Area, 432 rods, length 24, breadth 18 rods. 27. Suppose a ball, 9 feet in diameter, cut down to a cube; what would be the length of a side? Ans. 5.19+ 28. How many acres in a square field, the diagonal of which is 20 rods longer than either side? Ans. 14 acres 2 roods 11 poles. N. B. To solve this, assume any square, and find the difference between its side and diagonal; then make a proportion. 29. Suppose the hind-wheels to a wagon to be 12 feet, and the fore-wheels 9 feet in circumference; how far must the wagon run, for the fore-wheels to gain 800 revolutions over the hind-wheels? Ans. 61 miles. 30. A hare starts 40 yards before a grayhound, and is not perceived by him until she has been up 40 seconds; she scuds away at the rate of 10 miles an hour; the dog makes after her at the rate of 18 miles an hour; how long will the hound be in overtaking the hare, and what distance will he run? Ans. 60 seconds, and will run 530 yards. 2 31. A man owing the sum of 6480 dollars to several creditors, orders his clerk to pay the first 40 dollars, and the last 500, in arithmetical progression; required the number of creditors, and the difference between each. Ans. 24 creditors, and difference $20. 32. Three men, A, B, and C, have a sum of money to divide among themselves; A is to have 9 dollars, B is to have as much as A and of C's, and C is to have as much as A and B both; what sum is to be divided, and how much is the share of each? Whole sum, 72 dollars. 33. Bodies fall by gravity, about 16 feet the first second of time, and the distance or extent of fall increases as the square of the time in seconds; how far, then, will a body fall in 7 seconds? Ans. 49 X 16=784 feet. 34. A body was observed to fall during 6 seconds; how far did it fall during the last second? Ans. 176 feet. 35. A body occupied 8 seconds in falling through a certain space; how far did it fall during the last half second? Ans. 132 feet. 36. The ditch of a fortification is 1000 feet long, 9 feet deep, 20 feet broad at bottom, and 22 at top; how much water will fill the ditch? Ans. 1158127 gals. nearly. 37. Seven men bought a grind-stone, of 60 inches diameter, each paying part of the expense; what part of the diameter must each grind down for his share? Ans. The 1st, 4,4508, 2d, 4,8400, 3d, 5,3535, 4th, 6,0765, 5th, 7,2079, 6th, 9,3935, 7th, 22,6778 inches. 38. How many rods less will it take to fence in an acre, if it be laid out in the form of a circle, than in the form of a square? Ans. 53+rods. 39. How many pieces of cloth, at 20,8 dollars per piece, are equal in value to 240 pieces, at 12,6 dollars per piece? Ans. 145,38+pieces. 40. If, when the price of wheat is 74,6 cents per bushel, the penny roll weighs 5,2 ounces, what should it be per bushel when the penny roll weighs 3,5 ounces? Ans. $1,108. 41. If a globe of stone, 9 inches in diameter, weigh 36 pounds, how much will another globe of the same kind of stone weigh, whose diameter is 15 inches? Ans. 166,6+lbs. 42. Mix 6 bushels of oats, worth 20 cents per bushel, with 8 bushels of oats, worth 25 cents per bushel, to rye, worth 70 cents per bushel, and wheat worth 80 centsand sell the mixture at 75 cents per bushel; how much rye and wheat must there be in the mixture? Ans. Rye, 14 bush.; wheat 160 bush. 43. Seven men not agreeing with the owner of a boarding house, about the price of boarding, offer to give 100 dollars each, for as long time as they can seat themselves every day differently at dinner; this offer being accepted, how long may they stay? Ans. 5040 days, or 13 years 295 days. 44. What will be the expense of paving a rectangular yard, whose length is 63 feet, and breadth 45 feet, in which there is laid a foot-path 5 feet 3 inches broad, running the whole length, with broad stones, at 36 cents a yard; the rest being paved with pebbles, at 30 cents a yard? Ans. $96,70. 45. In exchanging 20 yards of cloth, of 11 yards wide, for some of the same quality, of of a yard wide, what quantity of the latter makes an equal barter? - Ans. 34 yards. 46. Suppose a large wheel, in mill-work, to contain 72 cogs, and a smaller wheel, working in it, to contain 50 cogs; in how many revolutions of the greater wheel, will the lesser one gain 100 revolutions? Ans. 227T APPENDIX.. MECHANICAL POWERS. THE mechanical powers properly belong to Natural Philosophy, and not to Arithmetic. The problems pertaining to them, in a numerical point of view, are usually very trifling; but the principles on which the computations are based; require exact thought, and should not be passed over in a careless manner. We make these observations to guard the pupil against imbibing the error of measuring the importance of a thing by the difficulty or ease of numerical computations. Properly speaking, there are but two fundamental mechanical powers, viz.: the lever and inclined plane. From the lever, we derive the pulley, and wheel and axle. From the inclined plane, we derive the screw and the wedge. Theoretically, the lever is an imponderable and inflexible bar, supported on a fixed point, called the fulcrum. Let the line AB represent it, resting on the point C, call ed the fulcrum, and weights A that balance each other at A and B. C -B The figure below represents the application of the power of the lever as used in prying. |