at the rate of 10 miles an hour; now, 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 hare 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 greyhound, but is not perceived 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. 15 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, be 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 2 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 time for paying the whole? A. 8 months. 84. How many bricks, 8 inches long, 4 inches wide, and 2 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. 21 A. 105408 brichs. 1 LXXXII. Alligation is the method of mixing several mples of different qualities, so that the compound, or composition, 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 rye, worth 50 cents a bushel, 4 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 be divided by the whole number of bushels, the quotient will be the price of one bushel of the mixture. RULE. Divide the whole cost by the whole number of bushels, &c.; the quotient will be the mean price or cost of the mixture. 2. A grocer mixed 10 cwt. of sugar at $10 per cwt., 4 cwt. at $4 per cwt., and 8 cwt. at $71 per cwt.: what is 1 cwt. of this mixture worth? and what is 5 cwt. worth? A. 1 cwt. is worth $8, and 5 cwt. is worth $40. 3. A composition was made of 5 lbs. of tea, at $1 per lb., 9 lbs. at $1,80 pes B., and 17 lbs. at $14 per Ib.: what is a pound of it worth? A. $1,54677 + 4. If 20 bushels of wheat, at $1,35 per bushel, be mixed with lɔ bushels of ye, at 85 cents per bushel, what will a bushel of this mixture be worth? A. $1,1357 + 5. If 4 lbs. of gold, of 23 carats fine, be melted with 2 lbs. 17 carats fine, what will be the fineness of this mixture? A. 21 carats. ALLIGATION ALTERNATE. ¶ LXXXIII. 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. Reduce the several prices to the same denomination. 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 sev eral, their sum will express the quantity. 2. A merchant has several sorts of tea, some at 10s some at 11 s., some at 13 s., and some at 24s. per lb.; what proportions of each must be taken to make a composition worth 12 s. per lb.?" 3. How much wine, at 5 s. per gallon, and 3 s. per gallon, must be mixed together, that the compound may be worth 4 s. per gallon? A. An equal quantity of each sort. 4. How much corn, at 42 cents, 60 cents, 67 cent, and 78 cents, per bushel, must be mixed together, that the compound may be worth 64 cents per bushel A. 14 bushels at 42 ceris, 3 bushels at 60 cents, bushels at 67 cents, and 2 bushels at 78 cents. 5. A grocer would mix different quantities of sugar; 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 mixture 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, I at 19, 1 at 21, and 4 at 24. We have seen that we can take 3 times, 4 times, 4, 4, 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 EACH PROPORTIONAL QUANTITY :: so is the GIVEN QUANTITY: to the QUANTITIES OF PROPORTIONS of the compound required. 7. A grocer wishes to mix 1 gallon of brandy, worth 15 s. per gallon, with rum worth 8 s., so that the mixture may be worth 10 s. per gallon; how much rum must be taken? By the last rule, the differences are 5 to 2; that is, the proportions are 2 ef brandy to 5 of rum; hence he must take 2 gallons of rum for every gallon a brandy. A. 2 gallons. 8. A person wishes to mix 10 bushels of wheat, at 70 cents per bushel, with rye at 48 cents, corn at 36 cents, and barley at 30 cents per bushel, so that s bushel of this mixture may be worth 38 cents; what quantity of each must be taken? We find by the last rule, that the proportions are 8, 2, 10, and 32. Then, as 8: 9:: 10: 24 bushels of rye. 8: 10: 10: 12 bushels of corn. Ans. 9. How much water must be mixed with 100 gallons of rum, worth 99 cents per gallon, to reduce it to 75 cents per gallon? A. 20 gallons. 10. A grocer mixes teas at $1,20, $1, and 60 cents, with 20 lbs. at 40 cents per lb.; how much of each sort must he take to make the composition worth cents per lb.? 4. 20 at $1,20, 10 at $1, and 10 at 60 cents. As the sum of the PROPORTIONAL QUANTITIES found by linking, as before is to EACH PROPORTIONAL QUANTITI :: so is the WHOLE QUANTITY or compound required to the REQUIRED QUANTITY of each. 6 9. 11. We will now apply this rule in performing the last question. 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 0 per gallon, must be mixed with wine, at 80 cents Der gallon, so as to fill a vessel of 90 gallons, which may be offered at 50 cents per gallon? A. 56 gallons of wine, and 33 gallons of water. 14. How much gold, of 15, 17, 18, and 22 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 22. ↑ LXXXIV make? INVOLUTION, Q. How much does 2, multiplied into itself, or by 2. Q. How much does 2, multiplied into itself, or by 2, and that product by 2, make? Q. When a number is multiplied into itself once or more, in this manner, what is the process called? A. Involution, or the Raising of Powers. Q. What is the number, before it is multiplied into itself, called? A. The first power, or root. Q. What are the several products called? A. Powers. Q. In multiplying 6 by 6, that is, 6 into itself, making 36, we use 6 twino what, then, is 36 called? A. The second power, or square of 6. Q. What is the second power, or square of 8? 10! 19! |