RULE FOR SIMPLE PROPORTION. Make the number that is of the same kind as the required answer the third term. If the answer should be greater than the third term, make the larger of the other two numbers, upon which the answer depends, the second term, and the smaller the first. If it should be less, make the smaller number the second term, and the larger the first. Multiply the means together, and divide their product by the first extreme. NOTE. - Analysis is the more natural and philosophical method of solving arithmetical questions; but the principles of Proportion are applicable to certain classes of examples. It is recommended to the pupil to perform the following examples in both ways. He should, at least, perform a sufficient number of them by Proportion to fix the method in his mind.

EXAMPLES.

1. If 2 men build 17 rods of wall in a week, how many rods will 100 men build in the same time?

We make 17 rods, which is of the same denomination as the required answer, the third term. As 100 men will build more wall than 2 men, we make 100 the second term, and 2 the first term, and the statement Ans. 850 rods.

is, 2: 100 17: 850.

2. If 9 lbs. of lead make 150 bullets, how many bullets can be made from 105 lbs.? Ans. 1,750 bullets.

3. If 65 pairs of boots can be made from 75 lbs. of calfskin, how many pairs can be made from 850 lbs.? Ans. 7363 prs.

4. How many tons of hay can be made from 750 acres of land, if 13 tons can be made from 3 acres? Ans. 3250 tons.

5. If $2000000 will support an army of 500000 men a day, how many men can be kept for $400000? Ans. 100,000 men.

6. If $500 purchase 200 hats, how many hats can be purchased for $87? Ans. 35 hats.

7. If $800 yield $56 of interest in a certain time, what will $390 yield at the same rate? Ans. $27.30.

8. If 16 horses eat a certain quantity of hay in 13 weeks, how long would the same quantity last 24 horses? Ans. 83 weeks. 9. What time would be required for 5 men to mow an acre of land, if 2 men can mow it in 14 days of 10 hours in length? Ans. 6 hours.

10. If 500 bushels of plaster were sufficient for the dressing of 31 acres of land, what would be required for 17 acres? Ans. 2500 bu.

11. If 95 acres of land were mowed by 20 men in 2 days of 12 hours each, how much time would be required for 3 mowing machines to do the same work (1 machine being equal to 4 men)? Ans. 3 d. 4 h.

12. If a Graham loaf weighs 1 lb. 2 oz. when flour is worth $7 a bbl., what should it weigh, selling at the same price, when flour is $6 per bbl.? Ans. 1 lb. 6 oz.

13. If 400 lbs. of coal are required to run an engine 12 hours, what number of tons will be required to run three similar engines for 30 days constantly?

14. If it requires 13 days of 9 hours each to do a piece of work, how many days of 14 hours each will be required to do the same work? Ans. 8

15. If soda-crackers can be sold at 10 cents a pound when flour is worth $6.50 per bbl., for what can they be sold when flour is worth $9.75 per bbl.?

16. If it requires 30 men to do a piece of work when they work 11 hours a day, what number will be required when they work 15 hours a day?

17. If 30 men, working 11 hours a day, can do a piece of work in a certain time, how many more men must be employed, when it is half done, to finish it in the same number of days, working 10 hours a day? Ans. 3 more.

18. A pole 10 ft. long casts a shadow of 7 ft. 1 in. at 8 o'clock; what is the height of a flagstaff that casts a shadow 58 feet at the same time of day? Ans. 811 ft.

19. If my friend lends me $7000 for 15 days, for what time should I lend him $4500 to requite the favor? Ans. 231 d. 20. If my friend lend me money for 4 months when interest is 10 per cent., for what time should I lend him the same sum when interest is 7 per cent.?

21. A sail vessel has 10 hours the start of a steamer, and sails at the rate of 7 miles an hour, while the steamer sails 16 miles an hour; when will the steamer overtake the sail vessel? Ans.77 h.

22. A deer, 150 rods before a hound, runs 30 rods a minute; the hound follows at the rate of 42 rods a minute; in what time will the deer be overtaken?

23. Two armies are on opposite sides of a river, one being 300 miles east and the other 250 miles west of it, and marching Jowards each other, the first at the rate of 15 and the other of 18 miles in a day; in what number of days will they meet, and where? Ans. 16 d.; 50 m. east. 24. If 2 lbs. 5 oz. of wool make 1 yd. of cloth 32 inches wide, how much will make a yard 14 yards wide of the same quality? Ans. 3 lb. 4 oz.

25. How much cloth that is 2 yd. wide will cover 24 tables 6 ft. long and 3 ft. wide? Ans. 64 yds.

26. If it requires 10 yards of cloth that is 17 yds. wide to make a garment, how much will be required of that which is 13 yds. wide?

27. How many yards of cambric 24 inches wide will be required to line 14 yards of silk which is 22 inches wide?

28. How long must a piece of land be to contain 3 acres, if it is 4 rods wide?

29. If 400 bushels of potatoes were bought for $350.90, and sold for $425.50, what would be gained on 25 bushels?

30. A man failing in trade owes $7,865; his property is sold for $4385.70; what will he be able to pay to a creditor to whom he owes $1500?

31. If it costs $30 to paint the outside of a house 40 ft. by 30 ft. and 25 ft. high, what will it cost to paint one 50 ft. by 40, of the same height? Ans. $38.

32. If a building 13 ft. high casts a shadow of 4 ft., what is the length of a shadow cast by a tree 346 ft. high at the same time?

33. If $110 be paid for 8 T. 12 cwt. 20 lb. of hay, what will be the cost of 6 T. 3 cwt. 3 lbs. ? Ans. $78.597

34. If a hind wheel, which is 83 feet in circumference, turns 800 times in a journey, how many times will the fore wheel which is 6 feet in circumference, turn in the same journey?

35. The weight of a cubic foot of water being 62 lbs., how many pounds' weight will a tank contain which is 24 ft. square at the bottom, and 4 ft. high?

36. If 15 A. 2 R. 20 r. pasture 2 cows a certain time, what will be required to pasture 25 cows the same time?

37. How many yds. of cloth can be bought for 17£. 5 s., if 10 s. were paid for 5 quarters, or 1 Ell English?

38. What is the value of 7 cords 3 cord feet of wood at the rate of $18 for 2 cords 5 cord feet?

of an

acre of land, how sq. rods?

39. If 5g cwt. of leather pay for many pounds would be required to pay for 1

40. If 9 oz. 10 pwt. 3 gr. of gold be worth $150, what will be the value of 7 lb. 5 oz. 5 pwt.?

41. If 15 lbs. of tea pay for 48 lbs. 12 oz. of butter, how many pounds of butter can be purchased with a box of tea which contains 423 lbs.?

42. If .80 acres of land be worth $75.20, what is the value of .3731 acres? Ans. $35.053+.

NOTE.-Perform the following examples by analysis.

43. If a cow and a calf sell for $27, and the value of the calf is that of the cow, what is the value of each?

44. The sum of the ages of a father and son is 48 years, that of the son being of the age of the father; what is the age of each?

45. By one pipe a cistern can be emptied in 2 hours; by another it can be emptied in 3 hours; in what time can it be emptied if both are running?

46. A certain cistern has 3 pipes; the first will empty it in 5 hours, the second in 4 hours, and the third in 10 hours; in what time will all empty it?

47. A certain piece of work can be performed by A in 8 days, by B in 10 days, and by C in 16 days; in what time can all do it working together?

48. In what time can A and B do it together?

49. In what time can A and C do it together? 50. In what time can B and C do it together? 51. A cistern has one pipe which will fill it in 5 another which will empty it in 33 h.; in what time filled with both open?

hours, and will it be Ans. 63 h.

52. If A and B can do a piece of work in 3 mo. 10 d., and A alone can do it in 5 mo., in what time will B do it alone? Ans. 10 mo.

53. If A and B can dig a trench in 63 days, and B can do it alone in 10 days, in what time can A do it alone?

54. If A and B can do it in 5 days, and A, B, and C, can do it in 3 days, in what time can C do it alone?

55. If A and B can do a piece of work in 4 days, and A and C can do it in 5 days, and B and C in 6 days, in what time will all do the work together, and in what time will each do it alone?

Ans. A, B, C, 311d.; A, 8 d.; B, 10 d.; C, 16 d. 56. A man sold a load of coal, containing 3 T., at $.75 for a hundred lbs., and received pay in corn at $.87 per bushel; how many bushels did he receive?

57. Purchased a number of pieces of goods, each containing 22 yards, at $4 for 3 yards, and sold them at $5 for 2 yards, and gained $154 on the lot; how many pieces were purchased? Ans. 6 pieces.

For Dictation Exercises, see Key.

ANALYSIS AND COMPOUND PROPORTION.

375. ILL. Ex. If 10 gas-burners consume 800 feet of gas in 3 hours, how many feet will 12 burners consume in 15 hours?