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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.

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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.

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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.

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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.

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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.

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