53. 1. Of the trees in an or-jhalf in the water, one third in the chard, bear apples plums and aud and 6 feet above the water, 20 bear cherries; how many trees what is its length? are there? 7. In a school 5 sevenths of the 2. Of the trees in a garden one pupils are learning intellectual ahalf bear apples one fourth peach.-rithmetic, 3 fourteenths are learnes, one sixth plums, 4 bear pearsing geography and the rest, 3 in and 1 cheries, how many trees are number are learning latin, what there? is the number of pupils? 6 ΤΣ 3 1 × 1 + 1 = 5 + 1/2+2=11; 8 Which is most 6 thirds of 20 the remaining apples, or 40 apples? 5,12=12×5—60 then 4+ and it Answer. 3. A boy having spent and ΤΣ 9. What is the difference be- of his money had 10 cents left, how tween I third of 1 half and 1 half 11. John had 1 third of a melon 4. In a school 1 fifth of the pu-and gave of what he had to pils study geography, 2 fifths are James, what part of the melon did learning to write, and the rest, be-James have? what part had John ing 20 in number, are learning in-left? tellectual arithmetic, what is the whole number of scholars? 5. Triple, one half, and 1 fourth of a certain number is 30; what is that number? 6. There is a pole which is one A 12. A boy being asked how many apples lie had, said, if I had as any more as I now have, and half as many more, I should have an hundred; how many had he? 59. 1. A cistern has 2 cocks; would last him 3 and one third the first will fill it in three hours, weeks. the second in six hours: how much 3. If one man could build a of it would each fill in an hour?piece of wall in 4 days, and anhow long would it take them both other man could do it in 6 days, together to fill it? how much of it would each do in 2. A man and his wife found byjone day? How many days would experiment, that when they were it take them both to do it? Ans. together a bushel of wheat would 2 and two-fifths. last only 2 weeks, but when the 4. A cistern has 3 cocks, the man was gone, it would last his first will fill it in 2 hours, the secwife 5 weeks; how much of it did ond in 3 hours, and the third in 4 both together consume in onejhours, what part of the whole will week? what part did the manfeach fill in 1 hour? how long consume in one week? how long would it take to fill it if all were would it last the man alone? Ans. running at once? Both together would consume 5 5. A man being asked the price tenths of a bushel in a week, the of his horse, said, his horse and woman alone 2 tenths in a week, saddle were worth $90, but the the man alone 3 tenths, and a bus horse was worth & times as much as the saddle, what was the worth second one fourth; in the third of each? one sixth, and in the fourth 20 6. A man, having a cow, a hog, bushels; how much grain had he? and a sheep, and being asked the 9. If $40 worth of provisions value of each, said, that the hog will serve 24 men 20 days, how was worth twice as much as the long will $20 worth serve12 men? sheep, and the cow twice as much $60 worth 36 men? $10 worth as the hog, and that all together 18 men? were worth $56; what was the 10. What number is that to value of each? which if its half be added the 7. A boy bought an apple, asum will be 45? If to a number pear and a melon for 18 cents; for half itself be added the sum is the pear he gave twice as much and if are 45. is 30. Ans. as for the apple, and for the mel- 11. What number is that to on 3 times as much as for the pear; which if its third part be added what was the price of each? the sum will be 36? 8. A farmer being asked how 12. A man being asked his age, much grain he had, replied, that replied, that if its 1 fifth part be it was in 4 bins; in the first he added to it the sum would be 54; had one third of his grain, in the what was his age? 60. 1. If I mix a quart of be melted with 2lb. 23 carats fine, cherries, worth 8 cents, with a qt. what will be the fraeness of the of currants worth 6 cents, what mixture? is a quart of the mixture worth? Ans. 7 cents. 7. I have currants worth 5 cents a pint, others worth 3 cents a pint, 2. If a pound of raisins, worth in what proportion must they be 12 cents, be mixed with a pound mixed to make the mixture worth of figs, worth 16 cents, what is 14 cents a pint? Ans. in equal quanpound of the mixture worth? tities. 3. If 2 quarts of rum, worth 14 8. I have berries worth 5 cents cents a quart, be mixed with 1a pint, and others worth 8 cents a quart of rum, worth 18 cents a pint, in what proportion must I quart, what is a quart of the mix-mix them that the compound may ture worth? 2 quarts at 14 cts. be worth 7 cents a pint? Each pint are 28 cents, and 1 quart at 18 is at 8 cents will be worth 1 cent 18 cents, and 18plus28-46 cts.jmore than the mean, and each at 3qts. then cost 46 cts. and 46÷35 cents will be worth two cents 15 cents. Ans. less than the mean, they must. 4. If 3lb. of sugar, worth 10cts. therefore be mixed in proportion a pound, be mixed with 21b. worth as 2 pts. at 8 cents to 1 pint at 5 13 cents a pound, what is a pound cents, to make the mean worth 7 of the mixture worth? cents a pint. 5. If 4 bushels of oats, worth 9. If 21b. of silver 20 carats 25 cents a bushel, be mixed with fine, be mixed with 4lb. of pure 4 bushels of rye, worth 40 cents a silver, what will be the fineness of bushel, what is a bushel of the the mixture? A carat is the 24th mixture worth? Ans, 32 cents. part of any thing, or quantity, and 6. If 3 lb. of gold, 20 carats fine, when gold or silver is said to be so 12. What fineness is the stand 13. What is generally used for alloy? 14. For what purpose is alloy many carats fine, it is understood that if the whole mass be divided ard for our coms? into 24 equal parts, so many of them would be pure gold, or sil ver, and the rest alloy. Pure gold would be said to be 24 carats fine.used? The standard for our gold and sil- 15. What are the gold coins of ver coins, is 1 parts fine, to 1 part the United States? (33) the silalloy, or in other words, 22 carats ver coins? the copper coins? fine. Copper is commonly used as Uncoined gold, 22 carats fine, is alloy in gold and silver, and is em-worth, at the mint of the United ployed to render coins more hard States, $209.77 per lb. Troy, and and durable. Juncoined silver, of the same fine10. What is meant by a carat?ness, is worth $9.92 per pound. 11. How many carats fine is pure gold? 61. 1. What is a square? Ans. root of a number?-by the second A square is a A. figure having-4 equal sides, and 4 equal angles or corners, as A B C D. C B power of a number?--by the square of a number? 7. What is the square root of 9? of 4? 16? 25? of 10? of 100? 9. Why? Because if the length D of the side of a square be only 1 2. How many square feet in ajfoot, the content of the square can square measuring 3 ft. on each be only 1 square foot, side? (36.) 10. What is a cube? 3. How many square feet in a cube is a square measuring 4 feet on each solid body, side? 5 feet? 6 feet? 8 feet? which has A Ans. A B 4. What is the square of 7 ft. six equal F The square of 7 is the area, or sides, all of content, of a square, which meas-which are nres 7 ft. on each side, and is found squares, as by multiplying 7 by 7-49 sq. ft. A B C D The product of a number multi-E F. plied by itself is also called its second power; thus 49 is the 11. How many solid feet in a square or second power of 7, and cubic block, which measures 3 ft. 7 is called the square root of 49. on every side? E D The square root of a number, then 12. How many feet in a block is the length of the side of a square which measures 2 feet on every of which the number expresses the side? 4 feet on every side? If the area. The length A B is the root square of a given number he mulof the square A B C D. tiplied by the given number, the 5. What is the second power of given number is said to be cubed, 5 of 8? of 10? as 3 times 3 are 9, and 3 times 9 6. What is meant by the squareļare 27. The 27 is called the cube or third power of 3, and 3 is called a foot on each side. 1×1=1 the cube root of 27. sq. ft. Ans. (121.) a 13. What is the cube root of 816. What would be the content 14. How would you proceed tojof a square which measures find the cube of a given number?--quarter of a foot on each sidǝ how to find the cube root of a giv- by 11 sq. ft. Ans. (121.) 17. What is the square root of 15. What would be the content one-fourth? of one-sixteenth? of of a square, which measures ball two-twentyfifths? en number? 62. 1. What is the difference] 11. What is the difference bebetween 2 feet square and 2 square tween a cube 2 inches in height, feet? lone 3 inches? 2. What is the difference be- 12. How many feet in a cubical tween 3 feet square and 3 square pile of wood which measures 5 feet feet? A surface 3, feet square on every side? Is there a cord? measures 3 feet on every side and how many feet does it lack? contains 9 square feet. (36, fig.) All similar solids are to one ano3 square feet are 3 squares, each ther as the cubes of three similar of which measures a foot on every diameters, or sides. side. 13. What proportion have simi 14. I have two lead balls, the di 3. What is the difference be-lar solids to one another? tween 4 miles square and 4 square miles? 5 miles square and 5 square ameter of one is one inch, of the miles? other 2 inches, what proportion do 4. How many square miles in a they bear to each other? township, which is 6 miles square? 1 to 8. Ans. as 5. What is meant by a solid 15 If the weight of tho smallest foot? be 2 lbs. what will be the weight 16. If a bullet 2 inches diameter 6. What is the difference be-of the other? tween half a solid foot and a solid half foot? Ans. Half a solid foot weigh 4 lbs. what will be the is a block a foot square and half a weight of a bullet 3 inches diamefoot high. A solid half foot is alter? block a half foot square and half a 17. There are 2 globes, one is 1 foot high-then half a solid foot is foot diameter and the other 4 feet equal to 4 solid half feet and the diameter, how many of the smaller difference is 3 solid half feet, (36 globes would be required to make fig.) one of the larger? 7. Which is most, a solid foot, or 18. There is a cubical box one 2 solid half feet? what is the dif-side of which is 1 foot, what is the ference? side of a box which will hold 3 times as much? 27 times as much? 64 times as much? 8. What part of a foot is a solid half foot? 2 solid half feet? 4? 9. What part of a solid foot are half a solid foot and a solid half foot? 10. I have two cubes, one is an inch high and the other 2 inches 19. In a certain company, the number of men in rank and file is the same, being 20 in each, what is the whole number of men? 20. How many square rods in a high. how many solid inches in piece of land 6 rods square? 60 one more than in the other? Trods square? GEOMETRICAL DEFENITIONS AND PROBLEMS. 63. A point is a position without magnitude. It is sometimes represented by a dot. A line is length, without breadth, or thickness. Sometimes represented by a mark, as A A. B. fig. 2. Marks are of great use in reasoning, of the properties of lines and figures, and when used for that purpose are called lines. Lines are either straight orc curved. A Straight (or right) line is the shortest distance between two points. A curve line continually changes its direction, as C. D. Parallel lines are always at the same perpendicular distance; and they never meet though continued ever so far. E. F. Oblique lines change their distance, and, if produced, would meet on the side of the least distance, fig. 4. An angle is the opening between two lines having different directions and meeting at a point called the apex, fig 4. When an angle is read by three letters, the letter which stands at the apex of the angle is always placed between the other two, as A B C, or C B A, fig. 4. E A B One line is perpendicular to A another when the angles on both sides of it are equal, fig 5. The angles made by one line falling perpendicular upon another are called right angles as AB C, or A B D, fig. 5. Angles either greater or less A than a right angle are called oblique angles. If an oblique angle be less than a right angle it is called an accute angle, if greater, an obtuse angle, fig 6. 64. A surface, or superficies is a figure having length & breadth without thickness. 10 A 16 2 B 3 F 5 -D What is a point How is it represented? Why is not a dot a point? What is a line? How represented? Why is not a mark a line? What is their use? What is a straight line? a curve line! Make a straight line -a curve line? Is A B a straight or curve line? What are parallel lines? Will parallels meet if produced? Draw two parallels. What are oblique lines? Will oblique lines meet if pro(duced? on which side? Draw two lines oblique to each otber. What is an angle? What is the point of meeting called? Make an angle. How is an angle read? Give an ex. ample. When is a line perpendicular toanother? What is a right angle? Make one. What is an ob lique angle? What is an acute angle? What is an obtuse angle? Make an acute angle ?an obtuse one? What is a surface! What is the meaning of the word aE cute? What is the meaning of the word op tuse! |