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6

arrears.

7. At what rate per cent will $6 amount to $8*7846 in 4 years?
If the first term be 6, the last term 867816, and the number of terms 5, what is the ratio ?
A.

87846 =RT., that is, = 1'4641 the 4th power of the ratio ; and then, by extracting the P. 4th root, we obtain 1'10 for the ratio.

Ans. 10 per cent. 8. In what time will $6 amount to $8-7846, at 10 per cent, compound interest ? A.

87846 RT., that is, =14641 =1'10T.; therefore, if we divide 1'4641 by 1'10, and then P. divide the quotient thence arising by 1'10, and so on, till we obtain a quotient that will not contain 1'10, the number of these divisions will be the number of years.

Ans. 4 years. 9. At 5 per cent. compound interest, in what time will $40 amount to $68-40 ?

Having found the power of the ratio 1'05, as before, which is 1'71, you may look for this number in the table, under the given rate, 5 per cent., and against it you will find the number of years.

Ans. 11 years. 10. At 6 per cent compound interest, in what time will $4 amount to $56352 7

Ans. 5 yrs. Annuities at Compound Interest. 1115,

• It may not be amiss, in this place, briefly to show the application of compound interest, in computing the amount and present worth of annuities.

An Annuity is a sum payable at regular periods of one year each, either for a certain nnm. ber of years, or during the life of the pensioner, or forever. When annuities, rents, &c. are not paid at the time they become due, they are said to be in

The sum of all the annuities, rents, &c., remaining unpaid, together with the interest on each, for the time they have remained due, is called the amount.

1. What is the amount of an annual pension of $100, which has remained unpaid 4 years, allowing 6 per cent. compound interest ?

The last year's pension will be $100, without interest; the last but one will be the amount of $100 for 1 year; the last but two the amount (compound interest) of $100 for 2 years, and so on; and the sum of these several amounts will be the answer. We have then a series of amounts, that is, a geometrical series, (*114,) to find the sum of all the terms.

If the first term be 100, the number of terms 4, and the ratio 1'06, what is the sum of all the terms ? Consult the rule under 1 113, ex. 11. 1'064 - 1

x 100 = 437-45. Ans. $437-45.

06 Hence, when the annuity, the time, and the rate per cent. are given, to find the amount,Raise the ratio (the amount of $1, &c., for 1 year) to a power denoted by the number of years; from this power subtract l; then divide the remainder by the ratio, less 1, and the quotient, multiplied by the annuity, will be the amount.

Note. The powers of the amounts, at 5 and 6 per cent., up to the 24th, may be taken from the table under | 91.

2. What is the amount of an annuity of $50, it being in arrears 20 ycars, allowing 5 per ct. compound interest?

Ans. $1653'29. 3. If the annual rent of a house, which is $150, be in arrears 4 years, what is the amount, ailowing 10 per cent. compound interest ?

Ans. $696*15. 4. To how much would a salary of $500 per arnum amount in 14 years, the money being improved at 6 per cent. compound interest ? in 10 years ? in 20 years in 22 years? in 24 years?

Ans, to the last, $25407-75. 91116. If the annuity is paid in advance, or if it be bought at the beginning of the first year, the sum which ought to be given for it is called the present worth.

5. What is the present worth of an annual pension of $100, to continue 4 years, allowing 6 per cent. compound interest?

The present worth is, evidently, a sum which, at 6 per cent. compound interest, would, in 4 years, produce an amount equal to the amount of the annuity in arrears the same time.

By the last rule, we find the amount - $437-45, and by the directions under T 114, ex. 4, we find the present worth = $346-51.

Ans. $346'51. Hence, to find the present worth of any annuity, - First find its amount in arrears for the whole time: this amount, divided by that power of the ratio denoted by the number of years, will give the present worth.

6. What is the present worth of an annual salary of $100, to continue 20 years, allowing 5 per cent. ?

Ans. 81246.22. The operations under this rule being somewhat tedious, we subjoin a

TABLE, Showing the present worth of $1, or 1 l. annuity, at 5 and 6 per cent. compound interest, for

any number of years from 1 to 34. Yrs. 5 per ct.6 per ct.|| Yrs. 15 per ct. 6 per ct. Yrs. (5 per ct. 16 per ct. Yrs. 5 per ct. 6 per ct.

0.952381 094339 10 7721731 736008! 19 12.08532 11'15811 27 14 643031321053 2 1'85941 1'83339 11 8-30641 7'88687|| 20 12'46221 11:46992 | 28 14'8981313140616

2723251 2'67301 12 8-86325) 8-38334 21 12 82115 11676407 29 15 14107/13'59072 3'54595 3'4651 13 9 39357 8'85268 22 13-163 12'04158 30 15'37245 13.76483 4.32948 4'21236 14 9 89864 9.294981 23 13'4880712-30338 31 15'59281/13-92908

5'07569 4 91732 15 10'37966 9.71225 24 1379864 12'55035 32 15.80268 14'08398 7 5.78637 5 58238 16 10 83777 10.1058911 25 14.09.941278335 33 16*011255 1422917 8 646321 6-209791 17 11 27407 10-47726 26 14.3751013600316|| 34 (113* 1929 114'36613

7-10782) 6'80169|| 18 |1168958 10'8276 It is evideut that the present worth of $2 annuity sa 2 times as much as wiat of $1 ; the pre

1

3

5 6

sent worth of $3 will be 3 times as much, &c. Hence, to find the present worth of any annuity al 5 or 6 per cent., -Find, in this table, the present worth of $1 annuity, and multiply it by the given annuity, and the product will be the present worth.

7. What ready money will purchase an annuity of $150, to continue 30 years, at 5 per cent. compound interest ?

The present worth of $1 annuity, by the table, for 30 years, is $15'37245; therefore $15+37245 x 150 = $2305*867, Ans. 8. What is the present worth of a yearly pension of $40, to continae 10 years, at 6 per cent. compound interest ? at 5 per cent. ? to continue 15 years ? 20 years? 25 years? 31 years?

Ans to last, $647'716. When annuities do not commence till a certain period of time has'elapsed, or till some partienlar event has taken place, they are said to be in reversion.

9. What is the present worth of $100 annuity, to be continued 4 years, but not to commence till 2 years hence, allowing 6 per cent. compound interest ?

The present worth is evilenily a snm which, at 6 per cent. compound interest, would in 2 years produce an amount equal to the present worth of the annuity, were it to commence im. mediately. By the last rule, we find the present worth of the annuity, to commence immedi. ately, to be $346-51, and, by directions under $ 114, ex. 4, we find the present worth of $316-51 for 2 years, to be $308-393.

Ans. $308-393. Hence, to find the present worth of any annuity taken in reversion, at compound interest, -First, find the present worth, lo commence immediately, and this sum, divided by the power of the ratio, denoted by the time in reversion, will give the answer.

10. What ready money will purchase the reversion of a lease of $60 per annum, to continue 6 years, but not to commence lill the end of 3 years, allowing 6 per cent. compound interes: to the purchaser ? The present worth, to commence immediately, we find to be $295-039, and 256 039 = 247-72.

Ans. $217-72. It is plain, the same result will be obtainer by finding the present worth of the annuity, to commence immediately, and to continue to the end of the time, that is, 3 +6=9 years, and then subtracling from this sum the present worth of the annuity, continuing for the time of reversion, 3 years Or, we may find the present worth of $1 for the two times by the table, and multiply their difference by the given annuity. Thus, by the table,

The whole time,

9 years, 6.80169, The time in reversion, 3 years, - 2-67301

Difference,

- 4'12858

60

$247-72030 Ans. 11. What is the present worth of a lease of $100, to continue 20 years, but not to commence till the end of 4 years, allowing 5 per cent. ? what, if it be 6 years in reversion ? 8 years? 10 years ? - 14 years?

Ans. to last, $629'426. 1 117. 12. What is the worth of a freehold estate, of which the ycarly rent is $60, al. lowing to the purchaser 6 per cent. ?

In this case, the annuity continues forever, and the estate is evidently worth a sum of which the yearly interest is equal to the yearly rent of the estate. The principal multiplied by the rate gives the interest; therefore, the interesi divided by the raie will give the principal; 50 = '06 = 1000.

Ans. $1000, llence, to find the present worth of an annuity, continuing forever,-Divide the annuity by the rate per cent., and the quotient will be the present worth.

Note. The worth will be the same, whether we reckon simple or compound interest ; for, since a year's interest of the price is the annuity, the profits arising from that price can peither be more nor less than the profits arising from the annuity, whether they be employed at simple or compound interest. 13. What is the worth of $100 annuity, to continue forever, allowing to the

e purchaser 4 per cent. ?

zallowing 5 per cent. ? <-8 per cent. ? -10 per centra cent. ?

Ans. to last, $500. 14. Suppose a freehold estate of $50 per annum, to commence 2 years hence, be put on sale; what is its value, allowing the purchaser 6 per cent. ?

Its present worth is a sum which, at 6 per cent. compound interest, would, in 2 years, produce an amount equal to the worth of the estate if entered on immediately. 60

81000 06 $1000 = the worth, if entered on immediately, and

T062

$389*996, the present worth. The same result may be obtained by subtracting from the worth of the estate, to commence immediately, the present worth of the annuity 60, for 2 years, the time of reversion. Thus, by the table, the present worth of $1 for 2 years is 1'83339 X 60 = 110-0034 - present worth of $60 for 2 years, and $1000 - $110'0034= $88969966, Ans. as before.

15. What is the present worth of a perpetual annuity of $100, 10 commence 6 years hence, allowing the purchaser 5 per cent, compound interest ? what, if 8 years in reversion 1

-- 10 years ? 4 years ? - 15 years ? 30 years? Ans. to last, $ 462 755. The foregoing examples in compound interest have been confined to yearly payments ; it the payments are half yearly, we inay take half the principal or annuity, half the rate per cent., and twice the number of years, and work as before, and so for any other part of a year.

QUESTIONS. 1. What is a geometrical progression or series? 2. What is the ratio ? 3. When the first term, the ratio, and the number of terms are given, how do you find the last term? 4. When the extremes and ratio are given, how ao you find the sum of all the terms? 5. When the

15 per

first term, the ratio, and the number of terms are given, how do you find the amount of the series? 6. When the ratio is a fraction, how do you proceed?' 7. What is compound interest ? 8. How does it appear that the amounts, arising by compound interest, format geometrical series? 9. What is the ratio, in compound interest ?

the number of terms? - the first term ? the last term? 10. When ihe rate, the time, and the priucipal are given, how do you find the amount? 11. When A. R. and T. are given, how do you find P.? 12. When A. P. and T. are given, how do you find R.? 13. When A. P. and R. are given, how do you find T. ? 14. What is an annuity? 15. When are annuities said to be in arrears? 16. What is the amount? 17. In a geometrical series, to what is the amount of an annuity equivalent ? 18. How do you find the amount of an annuity at compound interest ? 19. What is the present worth of an annuity ? -- how computed at compound interest ?

how found by the table? 20. What is understood by the term reversion? 21. How do you find the present worth of an annuity, taken in reversion? by the table? 22. How do you find the present worth of a freehold estate, or a perpetual annuity ?

the same taken in reversion ? by the table?

PERMUTATION, 1 118. Permutation is the method of finding how many different ways the order of any number of things may be varied or changed.

1. Four gentlemen agreed to dine together so long as they could sit, every day, in a different order or position; how many days did they dine together ?

Had there been but two of them, a and b, they could sit only in 2 times 1 (1 X 2 = 2) different positions, thus, a b, and b a. Had there been three, a, b, and c, they could sit in 1 x 2 X 3 = 6 different positions ; for, beginning the order with a, there will he 2 positions, viz: a b c, and a cb; next, beginning with b, there will be 2 positions, ba c, and b c a; lastly, beginning with c, we have c a b, and cba, that is, in all, 1 X 2 X 3= 6 different positions. In the same manner, if there be four, the different positions will be l x 2 X 3 X 4=24;,

Ans, 24 days. Hence, to find the number of different changes or permutations, of which any number of different things are capable, -Multiply continually together all the terms of the natural series of numbers, from 1 up to the given number, and the last product will be the answer. 2. How many variations may there be in the position of the nine digits ? Ans. 362880.

3. A man bought 25 cows, agreeing to pay for them 1 cent for every different order in which they could all be placed ; how much did the cows cost him ? Ans. $155112100433309859840000. 4. Christ Church, iu Boston, has 8 bells; how many changes may be rung upon them?

Ans. 40320,

Ans, 30.

Ans. 34.

MISCELLANEOUS EXAMPLES, 1 119. 1. 4 726 X7-1 = 60. A live, or vinculum, drawn over several numbers, signifies that the numbers under it are to Oe taken jointly, or as one whole number.

2. 9-8 + 4 x 8+4-6= how many ? 3. 7+ 4-2+3+40 x 5= how many ?

Ans. 230. 4.3 + 6 -2 X 4-2

= how many ? 2 X 2 5. There are two numbers; the greater is 25 times 78, and their difference is 9 times 15; their sum and product are required.

Ans. 3765 is their sum; 3539250 their product. 6. What is the difference between thrice five and thirty and thrice thirty-five ?

35 X 3–5 X3 +30=60, Ans. 7. What is the difference between six dozen dozen and half a dozen dozen 7 Ans. 792 3. What number divided by 7 will make 6488 ? 9. What number multiplied by 6 will make 2058 ?

10. A gentleman went io sea at 17 years of age; 8 years after he had a son horn, who died at the age of 35; after whom the father lived twice 20 years; how old was the father at his death?

Ans. 100 years. 11. What number is that which, being multiplied by 15, the product will be 4 ? 1

15 – 20 Ans. 12. What decimal is that, which, being multiplied by 15, the product will be 75?

75 - 15 = '05, Ans. 13. What is the decimal equivalent to ?

Ans. "0285714,

13 14. What fraction is that, to which if you add to the sum will be ?

Ans.

30 15. What number is that, from which if you take , the remainder will be j? Ans. 29 16. What number is that which, being divided by 4, the quotient will be 21 ?

Ans. 15% 17. What number is that which, multiplied by , produces ?

Ans 18. What number is that, from which if you take of itself, the remainder will be 12?

Ans. 20. 19. What number is that, to which if you add sort of itself, the whole will be 202 Ans. 12 20. What number is that of which 9 is the part?

Ans. 13.

21. A farmer carried a load of produce to market; he sold 780 lbs. of pork, at 6 cents per lb.; 250 lbs, of cheese, at 8 cents per Ib.; 151 lbs. of butter, at 15 cents per lb.; in pay he received 60 lbs. of sugar, at 10 cents per lb.; 15 gallons of molasses, at 42 cents per gallon ; f barrel of mackerel, at $3.75; 5 bushels of salt, at $125 per bushel; and the balance in money; how much money did he receive ?

Ans. $68-85 22. A farmer carried his grain to market, and sold

75 bushels of wheat, at $1.45 per bushel.
6+ ............. rye,

$95 ....
142...

... corn, ... $ 50.... In exchange he received sundry articles :

3 pieces of cloth, each containing 31 yds., at $1'75 per yd. 2 quintals of fish,

$2'30 per quin. 8 hhds. of salt,

$4'30 per hhd. and the balance in money. How much money did he receive ?

Ans. 838 80 23. A man exchanges 760 gallons of molasses, at 374, cents per gallon, for 664 cwt. of cheese, at $4 per cwt.; how much will be the balance in his favor?

Ans. $19 24. Bought 31 yards of cloth, at $1'25 per yard; how much did it come to ? How many bushels of wheat, at $150 per bushel, will it take to pay for it? Ans. to the last, 70 bushels.

25. A man sold 312 pounds of beef, at 6 cents per pound, and received his pay in nolasses, at 374 cents per gallon; how many gallons did he receive?

Ans. 54'72 gallons. 26. A mau exchanged 70 Dushels of rye, at $ 92 per hushel, for 40 bushels of wheat, ar $1374 per bushel, and received the balance in oats, at $-40 per bushel; how many bushels or oats did he receive ?

Ans. 234 27. How many bushels of potatoes, at I s. 6 d. per bushel, must be given for 32 bushels of barley, at 2 s. 6 d. per bushel"?

Ans. 53bushels 28. How much salt, at $1'50 per bushel, must be given in exchange for 15 bushels of oats, at 2 s. 3 d. per bushel ?

Note.It will be recollected that, when the price and cost are given, to find the quantity, they must both be reduced to the same denomination before dividing. Ans. 3 bushels.

29. How much wine, at $2475 per gallon, must be given in exchange for 40 yards of cloth, at 7 s. 6 d. per yard?

Ans. 1811 gallons. 30. A had 11 cwt. of hops, at 30 s. per cwt., for which В gave him 20 l. in money," and the rest in prunes, at 5 d. per lb.; how many prunes did A receive ? Ans. 17 cwt 3 qrs. 4 lbs.

31. A has linen cloth worth $ 30 per yard, but in bartering, he will have $ 35 per yard; B has broadcloth worth $3675 ready money; at what price ought the broadcloth to le rates in bartering with A ? 630 : 35 : : 3.75 : $1375, Ans.

Or,

35 The two operations will be seen to be exactly alike.

30

of 3.75= $4'374, Ans 32. If cloth, worth 2 s. per yard, cash, be rated in barter at 2 s. 60., how should wheat, worth 8 s, cash, be rated in exchanging for the cloth ?

Ans. 10 s., or $146663. 33. If 4 bushels of corn cost $2, what is it per bushel ?

Ang. $ 50. 34. If 9 bushels of wheat cost $13.50, what is that per bushel ?

Ans. $1.50 35. If 40 sheep cost $100, what is that per head?

Ans. $250. 36. If3 bushels of oats cost 7 s. 6 d., how much are they per bushel ? Ans. 2 s. 6 d. = 9:414 37. If 22 yds. of broadcloth cost 21 i. 9 s., what is the price per yd. Ans. 19 s. 6 d. = $3:25 38. At $ 50 per bushel, how much corn can be bought for 82 00 ?

Ans. 4 bushels. 39. A man having $100, would lay it out in sheep at $250 apiece ; how many can he buy!

Ans. 40 40. If 20 cows cost $300, what is the price of 1 cow ? of 2 cows?

of 5 cows 7 - of 15 cows?

Ans. to the last, $225. 41. If 7 men consume 24 lbs. of meat in one week, how much would 1 man consume in the same time? - 2 men ? 5 men? - 10 men ? Ans. to the last, 34 lbs.

Note. Let the pupil also perform these questions by the rule of proportion. 42. If I pay $6 for the use of $100, how much musi I pay for the use of $75? Ans. $4'50. 43. What premium must I pay for the insurance of my house against loss by fire, at the rate off per cent., that is, $ dollar on a hundred dollars, if my house be valued at $2175 ?

Ans. $12-375. 44. What will be the insurance, per annum, of a store and contents, valued at $9876'40, at 14 per centum ?

Ans. $148 146. 45. What commission must I receive for selling $478 worth of books, at 8 per cent,"?

Ans. $38 24. 46. A merchant bought a quantity of goods for $734, and sold them so as to gain 21 per ct.; how much did he gain, and for how much did he sell his goods? Ans. to the last, $889'14.

47. A merchant bought a quantity of goods, at Boston, for $500, and paid $13 for their transportation ; he sold them so as to gain 24 per cent on the whole cost; for how much did he sell them ?

Ans. 8673-32. 48. Bought a quantity of books for $64, but for cash a discount of 12 per cent. was made; what did the books cost?

Ans. $56-32 49. Bought a book, the price of which was marked $4'50, but for cash the bookseller will sell it at 335 per cent discount; what is the cash price?

Ans. $3.09. 50. A merchant bought a cask of molasses, containing 120 gallons, for $42; for how inuch must he sell it to gain 15 per cent.? how much per gallon ?

Ans. to last, $104. 51. A merchant bought a cask of sugar, containing 746 pounds, for $59-20; how musi he sell it per pound to gain 25 per cent. ?

Ans. $19. 52. What is the interest, at 6 per cent, of $71'02, for 17 months 12 days ? Ans. $6178 + 53. What is the interest of $187.003 for 18 months ?

Ans. $43'83 54. What is the interest of $8.50 for 7 months ?

Ans. $* 2974 55. What is the interest of $1000 for 5 days?

Ans. $'8339 56. What is the interest of $50 for 10 years?

Ans. $ 30. 57. What is the interest of $9124 for 15 months and 7 days, at 7 per cent.? Ans. $7'456 +

58. What is the interest of $154'01 for 2 yrs. 4 months and 3 days, at 5 per ct? Ans. $19-032. 59. What sum, put to interest at 6 per cent., will, in 2 years quid 6 nuonths, amaril 10 $150 ? Note. See 65.

18. $130 431 + 60. I owe a man 3175'50, to be paid in 16 months without interest; what is the present worth of that debt, the use of the money being worth 6 per cent.?

Ans. 3110277 61. What is the present worth of $1000, payable in 4 years and 2 months, discounting at the rate of 6 per cent.?

AN. 800 62. A merchant bought articles to the amount of $300, and sold them for $575 ; how much did he gain? What per cent. was his gain? that is, how many Jollars did he gain on each $100 which he laid out? If $500 gain $75, what does $100 gain? Ans. 15 per cent

63. A merchant bought cloth at $3450 per yard, and sold it at $1.25 per yard; how much did he gain per centum ?

ins. 214 per cent 61. A man bought a cask of wine, containing 126 gallons, for $283-50), and sold it out at the rate of 82 75 per gallon; how much was his whole gain? How much per gallon? how much per cent.?. Ans. his whole gain, $53.00 ; per gallon, $ 50; which is 222 per centum. 65. Ir $100 gain $6 in 12 months, in what time will it gain $1?" 3107 $11?

Ans, to the last, 28 months. 66. In what time will $51'50, at 6 per cent. gain $2:18 ?

An&. 8 months. 67. 20 men built a certain bridge in 60 days, but, it being carried away in a freshet, it is required how many men can rebuild it in 50 days. 50 days : 60 days : : 20 men : 24 men, Ans.

68. If a field will feed 7 horses 8 weeks, how long will it feed 23 horses ? Ans. 2 weeks.

69. If a field, 20 rods in length, must be 8 rods in width to contain an acre, how much in width must be a field, 16 rods length, to contain the same?

Ans. 10 rods. 70. If I purchase for a cloak 12 yards of plaid of a yard wide, how much bocking 14 yards wide must I have to line it?

Ans. 5 yards. 71. If a man earn $75 in 5 months, how long must he work to earn $160 ?

Ans. 303 mo: 72. A owes B 8540, but A not being worth so much money, B agrees to take $75 on ä dollar; what sum must B receive for the debt?

Ans. $105. 73. A cistern, whose capacity is 400 gallons, is supplied by a pipe which lets in 7 gallons in 5 minutes; but there is a leak in the bottom of the cistern which leis out 2 gallons in 6 ininutes; supposing the cistern empty, in what time would it be filled ?

In 1 minute of a gal. is admitted, but in the same time of a gal. leaked out. Ans. 6 h. 15 m.

74. A ship has a leak which will fill it so as to make it sink in 10 hours; it has also a pump which will clear it in 15 hours; now, if they begin to pump when it begins to leak, in what time will it , sink?: In hour, the ship would be io filled by the leak, but in the same time it would be 13 emptied by the pump.

Ans. 30 hours. 75. A cistern is supplied by a pipe whi will fill it in 40 minutes ; how many pipes of the same bigness will fill it in 5 minutes ?

Ans. 3. 76. Suppose I lend a friend $500 for 4 mo., he promising to do me a like favor: sone time afterward I have need of $300; how long may I keep it to balance the former favor? Ans.68 mo.

77. Suppose 800 soldiers were in a garrisou, with provisions sufficient for 2 months ; how many soldiers must depart that the provisions may serve them 5 months ?

Ans. 480. 78. if my horse and saddle are worth $84, and my horse be worth 6 times as much as my saddle, pray what is the value of my horse ?

Ans. $72. 79. Bought 45 barrels of beef, at $3.50 per barrel, among which are 16 barrels, whereof 1 are worth no more than 3 of the others; how much must I pay?

Ans. $1 13:50 80. Bought 126 gallons of rum for $110; how much water must be added to reduce the first cost to $75 per gallon ? Note. If$75 buy 1 gallon, how many gallon: will $110 buy?

Ans. 20% gallons. 81. A thiet, having 24 miles start of the officer, holds his way at the rate of 6 miles an hour; the officer pressing on after him at the rate of 8 miles an hour, how much does he gain in 1 hour ? how long before he will overtake the thief ?

Ans. 12 hours. 82. A hare starts 12 rods before a hound, but is not perceived by him till she has been up 14 minutes; she scuds away at the rate of 36 roils a minute, and the dog, on view, makes after, at the rate of 10 rods a minute; how long will the course hold, and what distance will the dog run?

Ans. 144 minutes, and he will run 570 rods. 83. The hour and minute hands of a watch are exactly together at 12 o'clock; when are they next together?-In 1 hour the minute hand passes over 12 spaces, and the hour hand over ! space; that is, the minute hand gains upon the hour hand I spaces iu 1 hour; and it mus. gain 12 spaces to coincide with it.

Ans. I h. 5 m. 27

? 84. There is an island 20 miles in circumference, and three men start together to travel the same way about it; A goes 2 miles per hour, B 4 miles per hour, and Co miles per hour; in what time will they come together again ?

Ans. 10 hours. 85. There is an island 20 miles in circumference, and 2 men start together to travel around it ; A travels 2 miles per hour, and B 6 miles per hour; how long before they will again come together 2-B gains 4 miles per hour, and inust gain 20 miles to overtake A; A and B will therefore be together once in every 5 hours.

86. In a river, supposing two boais start at the same time from places 300 miles apart; the one proceedling up stream is retarded by the current 2 miles per hour, while that moving down stream is accelerated the same; if both be propelled by a steam engine, which would move them 8 miles per hour in still water, how far from each starting place will the boats meet?

Ans. 1124 miles from the lower place, and 137 miles from the upper place. 87. A man bought a pipe (19 gallons) of wine for $275; he wishes to fill 10 bottles, 4 of which contain 2 quarts, and 6 of them 3 pints each, and to sell the remainder so as to make 30 per cent. on the first cost; at what rate per gallon must he sell it? Ans. $2936 + 88. Thomas sold 150 pine apples at $33. apiece, and received as much money as

Harry received for a certain number of watermelons at $25 apiece; how much money did each re. ceive, and how many melous had Harry ?

Ans. $50, and 200 melons

Bec.

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