RULE. Multiply the greater extreme by the ratio, subtract the less extreme from the product, and divide the remainder by the ratio less 1.

NOTE. Let every decreasing series be inverted, and the first term called the last; then the ratio will be greater than a unit. If the series be infinite, the first term is a cipher.

EXAMPLES.

1. The first term is 2, the last term 512, and the ratio 3; what is the sum of all the terms?

2. The first term is 4, the last term is ratio is 4; what is the sum of the series?

Ans. 767. 262144, and the

Ans. 349524.

3. The first term of a descending series is 162, the last term 2, and the ratio; what is the sum? Ans. 242.

4. What is the value of 1, 2, 15, &c., to infinity? Ans. †. NOTE. In the following examples we first find the last term by the Rule under Case I.

5. What yearly debt can be discharged by monthly payments, the first being $2, the second $6, and the third $18, and so on, in geometrical progression ? Ans. $531440.

6. If a grain of wheat produce 7 grains, and these be sown the second year, each yielding the same increase, how many bushels will be produced at this rate in 12 years, if 1000 grains make a pint? Ans. 252315 bu. 4 qt.

7. Six persons of the Morse family came to this country 200 years ago; suppose that their number has doubled every 20 years since, what would be their number now?

NOTE. The other cases in Progression will be found in the Higher Arithmetic.

PROMISCUOUS EXAMPLES.

1. One half the sum of two numbers is 800, and one half the difference of the same numbers is 200; what are the numbers ?

Ans.

2. What number is that to which, if you add the sum will be 61 ?

3. What part of a day is 3 h. 21 min. 15 sec.?

4. A commission merchant received 70 bags of wheat, each containing 3 bu. 3 pk. 3 qt.; how many bushels did he receive?

5. Four men, A, B, C, and D, are in possession of $1100; A has a certain sum, B has twice as much as A, C has $300, and D has $200 more than C; how many dollars has A? Ans. $100.

6. At a certain election, 3000 votes were cast for three candidates, A, B, and C; B had 200 more votes than A, and C had 800 more than B; how many votes were cast for A?

7. What part of 17 is 31? 8. The difference between

the number?

Ans. 600.
Ans. 8.

and of a number is 10; what is

Ans. 560.

9. A merchant bought a hogshead of rum for $28.35; how much water must be added to reduce the first cost to 35 cents per gallon? Ans. 18 gal. 10. A and B traded with equal sums of money; A gained a sum equal to of his stock; B lost $200, and then he had as much as A; how much was the original stock of each?

Ans. $500.

11. A farmer sold 17 bushels of barley, and 13 bushels of wheat, for $31.55; he received for the wheat 35 cents a bushel more than for the barley; what was the price of each per bushel?

Ans. Barley, $.90; wheat, $1.25.

11

12. What is the interval of time between March 20, 21 minutes past 3 o'clock, P. M., and April 11th, 5 minutes past 7 o'clock, A. M.? Ans. 21 da. 15 h. 44 min. 13. What o'clock is it when the time from noon is of the time to midnight? Ans. 5 o'clock 24 min. P. M. 14. What is the least number of gallons of wine that can be shipped in either hogsheads, tierces, or barrels, just filling the vessels, without deficit or excess? Ans. 126 gal.

15. A ferryman has four boats; one will carry 8 barrels, another 9, another 15, and another 16; what is the smallest number of barrels that will make full freight for any one, and all of the boats?

16. A and B have the same income; A saves of his, but B, by spending $30 a year more than A, at the end of four years finds himself $40 in debt; what is their income, and how much does each spend a year?

Ans.

Income, $160.
A spends $140.
B spends $170.
pounds cost $2.19, how
Ans. $2.40.

17. If a load of plaster weighing 1825 much is that per ton of 2000 pounds? 18. If 2 yards of cloth 12 yards wide cost $3.373, what will be the cost of 36 yards 11 yards wide?

Ans. $52.779.

19. I lend my neighbor $200 for 6 months; how long ought he to lend me $1000 to balance the favor?

20. Bought railroad stock to the amount of $2356.80, and found that the sum invested was 40 per cent of what I had left; what sum had I at first? Ans. $8248.80. 21. 20 per cent. of of a number is what per cent. of g of it? Ans. 12.

22. Divide a prize of $10200 among 60 privates, 6 subaltern officers, 3 lieutenants, and a commander, giving to each subaltern double the share of a private, each lieutenant 3 times as much as the subaltern, and to the commander double that of a lieutenant; how much is each man's share? Ans. Com. $1200; each man, $100.

23. A is 51 miles in advance of B, who is in pursuit of him; A travels 16 miles per hour, and B 19; in how many hours will B overtake A?

24. How much wool, at 20, 30, and 54 cents per pound, must be mixed with 95 pounds at 50 cents, to make the whole mixture worth 40 cents per pound?

Ans. 133 lb. at 20; 95 lb. at 30; 190 lb. at 54 cents. 25. If 240 bushels of wheat are purchased at the rate of 18 bushels for $22, and sold at the rate of 22 bushels for $33.75, what is the profit on the whole? Ans. $60.

26. My horse, wagon, and harness together are worth $169; the wagon is worth 4 times the harness, and the horse is worth double the wagon; what is the value of each ?

Horse, $104. Ans. Wagon, $ 52. Harness, $13.

27. The shadow of a tree measures 42 feet; a staff 40 inches in length casts a shadow 18 inches at the same time; what is the height of the tree?

Ans. 93 ft. 28. If a piece of land 40 rods long and 4 rods wide make an acre, how wide must it be to contain the same if it be but 25 rods long? Ans. 63 rods.

13

29. A, B, and C are employed to do a piece of work for $26.45; A and B together are supposed to do of it, A and C, and B and C 18, and paid proportionally; how much must each receive? 30. If 12 ounces of wool make 21 yards of cloth that is 6 quarters wide, how many pounds of wool will it take for 150 yards of cloth 4 quarters wide?

31. Six persons, A, B, C, D, E, and F, are to share among them $6300: A is to have of it, B, C, D is to have as much as A and C together, and the remainder is to be divided between E and F in the proportion of 3 to 5; how much does each one receive? 32. What is the amount of $200 for 8 years at 6 per cent. compound interest? Ans. $318.769.

33. A garrison, consisting of 360 men, was provisioned for 6 months; but at the end of 5 months they dismissed so many of the men that the remaining provision lasted 5 months longer; how many men were sent away?

84. A certain principal, at compound interest for 5 years, at 6 per cent., will amount to $669.118; in what time will the same principal amount to the same sum, at 6 per cent. simple interest? Ans. 5 yr. 7 mo. 19.8+-da.

35. Paid $148.352 for 9728 feet of pine lumber; how much was that per thousand?

36. Comparing two numbers, 483 was found to be their least common multiple, and 23 their greatest common divisor; what is the product of the numbers compared ? Ans. 11,109.

37. Eight workmen, laboring 7 hours a day for 15 days, were able to execute of a job; in how many days can they complete the residue, by working 9 hours a day, if 4 workmen are added to their number? Ans. 155 days.

38. If a hall 36 feet long and 9 feet wide require 36 yards of carpeting 1 yard wide to cover the floor, how many yards 14 yards wide will cover a floor 60 feet long and 27 feet wide?

Ans. 144 yards.

39. A, B, and C traded in company; A put in $1 as often as B put in $3, and B put in $2 as often as C put in $5; B's money was in twice as long as C's, and A's twice as long as B's; they gained $52.50; how much was each man's share of the gain? A's, $12. Ans. B's, $18. C's, $22.50.

40. A and B found a watch worth $45, and agreed to divide the value of it in the ratio of to; how much was each one's share?

{$20,

Ans. $20, A's. 41. A man received $33.25 interest on a sum of money, loaned 5 years previous, at 7 per cent.; what was the sum lent?

Ans. $95. 42. The diameter of a ball weighing 32 pounds is 6 inches; what is the diameter of a ball weighing 4 pounds? Ans. 3 inches. 43. Divide $360 in the proportion of 2, 3, and 4.

Ans. $80, $120, $160. 44. If by working 68 hours a day a man can accomplish a job in 12 days, how many days will be required if he work 8 hours per day? Ans. 9 days. 45. An open court contains 40 square yards; how many stones, 9 inches square, will be required to pave it?

10

Ans. 640.

46. A drover paid $76 for calves and sheep, paying $3 for calves, and $2 for sheep; he sold of his calves and of his sheep for $23, and in so doing lost 8 per cent. on their cost; how many of each did he purchase? Ans. 12 calves; 20 sheep. 47. If a cistern, 171 feet long, 101 broad, and 13 deep, hold 546 barrels, how many barrels will that cistern hold that is 16 feet long, 7 broad, and 15 deep? Ans. 384 bbls.

48. If 12 men, working 9 hours a day, for 155 days, were able to execute of a job, how many men may be withdrawn, and the residue be finished in 15 days more, if the laborers are employed only 7 hours a day? Ans. 4 men.

49. A general formed his men into a square, that is, an equal number in rank and file, and found that he had 59 men over; and increasing the number in both rank and file by 1 man, he wanted 84 men to complete the square; how many men had he? Ans. 5100.

50. Bought wheat at $1.50 per bushel, corn at $.75 per bushel, and barley at $.60 per bushel; the wheat cost twice as much as the corn, and the corn twice as much as the barley; of the sum paid, $243 and of the whole was for wheat, and $153 and of the whole was for the corn; how many bushels of grain did I purchase? Ans. 756.

51. Divide $630 among 3 persons, so that the second shall have as much as the first, and the third as much as the other two; what is the share of each i

1st, $240. Ans. 2d, $180. 3d, $210.

52. Bought a hogshead of molasses for $28, and 7 gallons leaked out; at what rate per gallon must the remainder be sold to gain 20 per cent.?

53. 20 per cent. of of a number is how many per cent. of 2 times of 1 times the number?

Ans. 7. 54. B and C, trading together, find their stock to be worth $3500, of which C owns $2100; they have gained 40 per cent. on their first capital; what did each put in ? Ans.C, $1500. SB, $1000.

55. If the ridge of a building be 8 feet above the beams, and the building be 32 feet wide, what must be the length of rafters? 56. If 12 workmen, in 12 days, working 12 hours a day, can make up 75 yards of cloth, of a yard wide, into articles of clothing: how many yards, 1 yard wide, can be made up into like articles, by 10 men, working 9 days, 8 hours each day? Ans. 23.

57. A grocer sells a farmer 100 pounds of sugar, at 12 cents a pound, and makes a profit of 9 per cent.; the farmer sells him 100 pounds of beef, at 6 cents a pound, and makes a profit of 10 per cent.; who gains the more by the trade, and how much?

Ans. The grocer gains $.445+ more. 58. In 1 yr. 4 mo. $311.50 amounted to $336.42, at simple interest; what was the rate per cent. ? Ans. 6.

59. Three persons engage to do a piece of work for $20; A and B estimate that they do of it, A and C that they do of it, and B and C that they do 3 of it; according to this estimate, what part of the $20 should each man receive? Ans. A's, $11; B's, $5§; C's, $24. 60. Paid $375, at the rate of 2 per cent., for insurance on a cotton factory and the machinery; for what amount was the policy given ?

61. A merchant bought goods in Boston to the amount of $1000, and gave his note, dated Jan. 1, 1857, on interest after 3 months; six months after the note was given he paid $560, and 5 months after the first payment he paid $406; what was due Aug 23, 1859 ? Ans. $66.63+ of B's, and of B's be equal of D's, and D has $45 more A, $378; C, $360; Ans. B, $336; D, $405.

62. If of A's money be equal to to of C's, and of C's be equal to than C, how much has each ?