4. A goldsmith melted together 5 lbs. of silver bullion, of 8 oz. fine-10 lbs. of 7 oz. fine-and 15 lbs. of 6 oz. fine; what is the quality of the mixture? Ans. 6 oz. 13 pwt. 8 gr. fine. 5. A wine merchant mixes 15 gallons of wine at 4 s. 2 d. per gallon, with 24 galls. at 6 s. 8 d., and 20 galls. at 6 s. 8 d.; what is a gallon of the composition worth? Ans. 5 s. 10 d. 24 6 qrs. 6. A grocer has several sorts of sugar, viz. one sort at 8 dols. per cwt.-another sort at 9 dols. per cwt.—a third sort at 10 dols. per cwt.—and a fourth sort at 12 dols per cwt.and he would mix an equal quantity of each sort together; what is the price of 34 cwt. of this mixture? Ans. $34.125. ALLIGATION ALTERNATE. 1. A grocer has two sorts of tea, viz. at 9 s. and at 15 s. per lb.; how must he mix them so as to afford the composition for 12 s. per lb. ? Ans. An equal quantity of each sort. 2. A goldsmith would mix gold of 17 carats fine, with some of 19, 21, and 24 carats fine, so that the compound may be 22 carats fine; what quantity of each must he take? Ans. 2 of each of the first three sorts, and 9 of the last. 3. How much water must be mixed with 100 gallons of brandy, worth 7 s. 6 d. per gallon, to reduce it to 6 s. 3 d. per gallon? Ans. 20 galls. 4. How much corn at 2 s. 6 d.—3 s. 8 d.-4 s.--and 4 s. 8 d. per bushel, must be mixed together, that the compound may be worth 3 s. 10 d. per bushel ? Ans. 2 at 2 s. 6 d.; 2 at 3 s. 8 d.; 3 at 4 s.; and 3 at 4 s. 8 d. 5. How much sugar at 4 d.--at 6 d.—and at 11 d. per lb. must be mixed together, so that the composition formed by them may be worth 7 d. per Ib. ? Ans. I lb. or 1 stone, or 1 cwt. or any other equal quantity of each sort. 6. It is required to mix brandy at 12 s.--wine at 10 s. cider at 1 s._and water at 0 per gallon together, so that the mixture may be worth 8 s. per gallon. Ans. 8 gallons of brandy, 7 of wine, 2 of cider, and 4 of water. 6 * SECTION X X X. EXAMPLES INVOLVING PHILOSOPHICAL PRINCI PLES. Specific Gravity. 1. A mineral weighs 960 grains in air, and 739 grains in water : what is its specific gravity ? Ans. 4.343. 2. A Greenland whale sometimes has a surface of 3600 square feet : what pressure would it bear at the depth of 800 fathoms ? Ans. 1080,000,000 lbs. 3. A stone weighs 15 lbs. in the air and 12 lbs. in water : what is its specific gravity ? Ans. 5. Parallelograms. 4. What is the area of a rectangular board, whose length is 121 feet, and breadth 9 inches ? Ans. 93 feet 5. What is the area of a piece of land, whose length is 6.20 chains, and perpendicular height 5.45 ? Ans. 3 A. 1 rood 20 rods. 6. What is the area of a parallelogram, the length being 12.25, and height 8.5? Ans. 104.125. Triangles. 7. What is the area of a triangle, whose base is 49 feet, and height 254 feet ? Ans. 68.7361 ft. 8. What is the area of a triangle, whose base is 150 yds. and perpendicular 60 ft.? Ans. 750 yds. 9. What is the area of a triangular field, whose length is 203 rods, and width 164 rods. Ans. 1698 rods. Circles. 10. What is the area of a circle, whose circumference is 22 ft. and diameter 7 ft. ? Ans. 381 ft. 11. To find the area of a circle, whose diameter is 18, and its circumference 31.416. Ans. 78.54. 12. What is the area of a circular pond of water, whose circumference is 130 rods, and its diameter 43.2 rods ? Ans. 2808 rods. Lever. 13. A body weighs 10 lbs. 9 oz. at one end of a false balance, and 121 lbs. at the other : what is its real weight? Ans. 11 lbs. 6 oz. 14. If a man, weighing 160 lbs., rest on the end of a lever 10 ft. long, what weight will he balance on the other end, supposing the prop is one foot from the weight? Ans. 1440 lbs. 15. A and B are of the same height, and sustain upon their shoulders a weight of 150 lbs. placed on a pole 84 feet long; the weight is placed 31 feet from A ; what is the weight sustained by each person ? Ans. A sustains 88 4 lbs.; B 6119 lbs. Axle. 16. Suppose the diameter of the wheel 60 inches, and that of the axle 6 inches; what weight at the axle will balance 1 lb. at the wheel ? Ans. 10 lbs. 17. Suppose the diameter of the wheel to be 60 inches; what must be the diameter of the axle so as that 1 lb. on the wheel may balance 10 lbs. on the axle ? Ans. 6 in. 18. A power of 5 lbs. keeps in equilibrio a weight of 150 Ibs. by means of a wheel whose diameter is 10 feet; what is the diameter of the axle ? Ans. 4 in. Screw. 19. A lever 6 feet long is fixed at right angles in a screw, whose threads are one inch apart, so that the lever turns just once round in raising or depressing the screw one inch. If then this lever be urged by a force of 50 lbs., with what force will the screw press ? Ans. 2219.448. 20. Suppose the lever to be 30 inches, the threads of the screw 1 inch apart, and the power 11.88 lbs.-required the weight to be raised. Ans. 2240 lbs. nearly. 21. Suppose the weight is 2240 lbs., the power 11.88 lbs., and the lever 30 inches : required the distance between the threads? Ans. 1 inch. Cube. 22. Required the solid content of a cube, whose side is 24 inches. Ans. 13824. 23. How many cubic feet are in a block of marble, its length being 3 feet 2 inches, breadth 2 feet 6 inches, and thickness 2 feet 6 inches? Ans. 214. 24. How many gallons of water will a cistern contain, whose dimensions are the same as in the last example, when 282 cubic inches are contained in one gallon ? Ans. 12947. Cylinder. 25. What is the content of a cylinder, whose length is 20 feet, and circumference 5 feet 6 inches ? Ans. 48.1459 ft. 26. What is the content of a cylinder, whose length is 15 feet, and circumference 4 feet? 27. Required the content of a cylinder, whose length is 5 feet, and circumference 6 inches. Pyramid and Cone. 28. What is the solidity of a square pyramid, each side of its base being 30 feet, and its perpendicular height 25 feet? Ans. 7500. 29. Required the content of a triangular pyramid, whose perpendicular height is 30, and each side of the base 3. Ans. 38.97117. 30. What is the content of a pentagonal pyramid, its height being 12 feet, and each side of its base 2 feet? Ans. 27.5276. 31. Required the content of a cone, its height being 103 feet and the circumference of its base 9 feet. Ans. 22.56093. Sphere. 32. What is the content of a sphere whose axis is 12? Ans. 904.7808. 33. Find the solid content of the earth, supposing its diameter to be 7918.7, and consequently its circumference 24877.4 miles. Ans. 260002677535. MISCELLANEOUS EXAMPLES. 1. What number is that which being multiplied by 21, the product will be 1365? Ans. 65. 2. The remainder of a division is 325, the quotient 467, and the divisor is 43 more than the sum of both ; what is the dividend ? Ans. 390270. 3. How many minutes from the creation to the commencement of the christian era, it being 4004 years ? Ans. 2104840032. 4. What is the difference between 126 d. 12 h. 40 m. 02 sec. and 86 d. 18 h. 48 m. 58 sec. ? Ans. 39 d. 17 h. 51 m. 4 sec. 5. Reduce 4 of a farthing to the fraction of a pound. Ans. 1920 6. Reduce 7 s. 6 d. to the fraction of a pound. Ans. 38. 7. What is the interest of 750 dollars for 2 years and 6 months, at 6 per cent? Ans, 112.50. 8. What is the compound interest of 12 £. 17 s. 6 d. for 6 years, at 6 cent? 9. If 8 men can do a piece of work in 24 days, in what time will 16 men do it? Ans. 12 days. 10. If a man's income be two dollars per day, what is it per annum ? Ans. 730 dollars. 11. Reduce 24 pounds to farthings. Ans. 23040 qrs. 12. What number is that to which if of ģ be added the sum will be 1 ? Ans. 63. per QUESTIONS.--What are the conditions of sum 3d ? What is the first thing to be done in obtaining a true answer ? What is the second thing? How do you reason on the 9th sum ?-Ans. If 8 men can accomplish the work in 24 days, it must take one man 8 times 24 days to perform it = 192 days' work. It will take 16. men one sixteenth part as long 12 days. |