at the rate of 10 miles an hour ; non, if the hunter does not change his place, how far will the hare get from the hunter in 45 seconds ? A. 52 rods. 78. If a dog, by running 16 miles in one hour, gain on a hara 6 miles every hour, how long will it take him to overtake her, provided she has 52 rods the start ? A. 97} seconds. • 79. A hare starts 12 rods before a greylound, but is not per-, ceived by him till she has been up 45 seconds; she scuds away at the rate of 10 miles an hour, and the dog after her at the rate of 16 miles an hour; what space will the dog run before he overtakes the hare? A. 138 rods, 3 yards, 2 feet. 80. A gentleman has an annuity of $2000 per annum; I wish to know how much he may spend daily, that, at the year's end, he may lay up 90 guineas, and give 20 cents per day to the poor of his own neighborhood? A. $4,128. 81. What is the interest of $600 for 120 days ?-12. For 2 days?-20. For 10 years, 10 mo. and 10 days?-391. For 5 years, 5 mo. and 5 days ?-19550. For 6 years, 6: mo. and 6 days ?-23460. For 4 years, 4 mo. and 4 days ?-15640. A. Total, $989,70. 82. What is the present worth of $3000, due 21 years hence, discounting at 6 per cent. per annum ? A. $2608,695+. 83. Suppose A owes B $1000, payable as follows; $200 in 4 . mo., $400 in 8 mo., and the rest in 12 mo. ; what is the equated cime for paying the whole ? A. 8 months. 84. How many bricks, 8 inches long, 4 inches wide, and 24 inches thick, will it take to build a house 84 feet long, 40 feet wide, 20 feet high, and the walls to be 1 foot thick ? The pupil will perceive that he must deduct the width of the wall, that is, 1 foot, from the length of each side, because the inner sides are 1 foot less in length than the outer sides. A. 105408 bricks. APPENDIX. ALLIGATION. I LXXXII. Alligation is the method of mixing several Amples of different qualities, so that the compound, or com position, may be of a mean or middle quality. When the quantities and prices of the several things or simples are given, to find the mean price or mixture compounded of them, the process is called ALLIGATION MEDIAL. 1. A farmer mixed together 2 bushels of ryn, worth 50 cents a bushel, • bushels of corn, worth 60 cents a bushel, and 4 bushels of oats, worth 30 cents • bushel: what is a bushel of this mixture worth? In this example, it is plain, that, if the cost of the whole he divided by the whole number of bushels, the quotient will be the price of one bushel of the pixture. 2 bushels at $,50 cost $1,00 $4,60 -10 = 46 cts., Ans. $4,60 RULE. Divide the whole cost by the whole number of bushels, &c.; the quoticnt will be the mean price or cost of the mixturc. 2. A grocer mixed 10 cwt. of sugar at $10 per cwt., 4 cwt. at $4 per cwt.. and 8 cwt. at 87 per cwt. : what is I cwt, of this nixture worth? and what is 5 cwt. worth: A. I cwt. is worth $8, and 5 cwt. is worth $40. 3. A composition was made of 5 lbs. of tea, at $li per lb., 9 lbs. at $1,80 per 8., and 17 lbs. at $14 per lb. : what is a pound of it worth?' A. $1,546, to 4. If 20 bushels of wheat, at $1,35 per bushel, be mixed with ly bushels of gye, at 85 certs per bushel, what will a bushel of this misture be worth? A. $1,135, to 5. If? 13. of gold, of 23 carats fine, be melted with 2 lbs. 17 carats fine, when will be the fipeness of this mixture? A. 21 carats. ALLIGATION ALTERNATE. ILXXXIII. The process of finding the proportional quantity of each simple, from having the mean price or rate, and the mean prices or rates of the several simples given, is called Alligation Alternate ; consequently, it is the reverse of Alligation Medial, and inay be proved by it. Hence we derive the following RULE, Connect, by a line, each price that is less than the mean rate, with one or more that is greater, and each price greater than the mean rate with one or more that is less. Place the difference between the mean rate and that of each of the simples opposite the price with which they are connected Then, if only one difference stands against any price, it expresses the quantity of that price; but if there be sepo eral, their sum will express the quantity. 2. A merchant has several sorts of tea, some at 10 s . some at 11 s., somo a 13 s., and some at 24 s. per Ib. ; what proportions of each must be taken to makon composition worth 12 s. per lb. OPERATIONS Ibs. 110OR, 110 -2+1=3) = 1 (14 3. How much wine, at 5 . per gallon, and 3 8. per gallon, must be mixed together, that the coinpound may be worth 48. per gallon? A. An equal quantity of each sort. 4. How much corn, at 42 cents, 60 cents, 67 cents, and 78 cents, per bushel, must be mixed together, that the compound may be worth 64 cents per bushell A. 14 bushels at 42 ceris, 3 bushels at 60 cents, bushels at 07 cents, and 2 bushels at 78 cents, 3. A grocer would mix different quantities of sogar ; viz. one at 20, one at 23, and one at 26 cents per lb.; what quantity of each sort must be taken to make a mixturo worth 22 cents per lb.? A. 5 at 20 cents, 2 &: 23 cents, and 2 at 26 cents. 6. A jeweller wishes to procure gold of 20'carats fixe, from gold of 16, 19, 21, and 24 carats fine; what quantity of each must ne take? A. 4 at 16, 1 at 19, 1 at 21, and 4 at 24. We have seen that we can take 3 times, 4 tiines, t, d, or any proportion of each quantity, to form a mixture. Hence, when the quantity of one simple is given, to find the proportional quantities of any compound whatever, after having found the proportional quantities by the last rule, we have the following RULE. As the PROPORTIONAL QUANTITY of that price whose quantity is given : is to EACI PROPORTIONAL QUANTITI :: so is the GIVEN QUANTITY : to the QUANTITIES OF PROPORTION$ of the compound required. 7. A grocer wishes to mix 1 gallon of brandy, worth 15 6. per gallon, with rum worth 8 s., so that the mixture may be worth 10 s. per gallon; bow much rum must be taken? By the last rule, the differences are 5 to 2; that is, the proportions are ad brandy to 5 of ruin ; hence he must take 2 gallons of rum for every gallon & brandy. A. 2 gallons. 8. A person wishes to mix 10 bushels of wheat, at 70 cento per bushel, with rre at 48 cents, corn at 36 cents, and barley as 30 conts per bushel, so that I bushel of this mixture may be worth 38 cenu; what quantity of each must be raken? We find by the last rule, that the proportions are 8, 2, 10, and 22. Then, as 8 :'3:: 10 ; 21 bushels of rye. , 8: 10 :: 10 : 12 buahele of corn. Ans. 8:32.:: 10 : 40 bushels of barley.) 9. How much water must be mixed with 100 gallons or rum, worth con por gallon, to reduce it to 75 cents per gallon? A. 20 gallons. 10. A grocer mixes tons at 310, $i, and 60 centa, with 20 lbs. at 40 cand per lb. : how much of each sort must he take to make the corposition word ho cents por Xb.? A. 20 at $1,20, 10 at 91, and 10 at 60 contu. 11. A grocer has currants at 4 cents, 6 cents, 9 cents, and 11 cents per lb. and he wishes to make a mixture of 240 lbs., worth 8 cents per lb.: how many currants of each kind must he take?-In this example, we can find the propor tional quantities by linking, as before ; then it is plain that their sum will be in the same proportion to any part of their sum, as the whole compound is to any part of the compound, which exactly accords with the principle of Fellowship. Hence we have the following SE RULE. As the sum of the PROPORTIONAL QUANTITIES found by linking, as before : is to EACH PROPORTIONAL QUANTITY :: so is the wHOLE QUANTITY or compound required : to the REQUIRED QUANTITY of each. We will now apply this rule in performing the last question. 10 : 3 :: 240 : 72 lbs., at 4 cts.) J 10:1:: 240 : 24 lbs., at 6 cts. Les Then, 10 : 2 :: 240 : 48 lbs., at 9 cts. (10 : 4 :: 240 : 96 lbs., at 11 cts.) 12. A grocer, having sugars at 8 cents, 12 cents, and 16 cents per pound, wishes to make a composition of 120 lbs., worth 13 cents per pound, without gain or loss; what quantity of each must be taken? A. 30 lbs, at 8, 30 lbs. at 12, and 60 lbs. at 16. 13. How much water, at O per gallon, must be mixed with wipe, at 80 cento Der gallon, so as to fill í vessel of 90 gallons, which may be offered at 50 cenu por gallon? A. 567 gallons of wine, and 335 gallons of water. 14. How much gold, of 15, 17, 18, and 2 carats fine, must be mixed together, to form a composition of 40 ounces of 20 carats fine? A. 5 oz. of 15, of 17, of 18, and 25 oz. of 2. INVOLUTION I LXXXIV Q. How much does 2, multiplied into itself, or by 2 mako? Q. How much does 2, multiplied into itself, or by 2, and that product bý? aske? Q. When a number is multiplied into itself once or more, in this manden, what is the process called ? A. Involution, or the Raising of Powers. Q. In multiplying 6 by 6, that is, 6 into itself, making 36, wo we 6 hrobowe what, then, is 36 oalled ?" Å. The second power, or square of 6. |