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. 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. $4,122. 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 24 years hence, discounting at 6 per cent. per annum ? A. $2608,695+ 83. Suppose A owes B $1000, payable as follows; $200 in 4 ino., $400 in 8 mo., and the rest in 12 mno.; 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 house $4 feet long, 40 feet wide, 20 feet high, and the walls to be I foot thick ? The pupil will perceive that he must deduct the width of the wall, that is, I foot, from the length of each side, because the inner sides are 1 foot less in length than the outer sides. 21 A. 105408 bricks. 242 APPENDIX. ALLIGATION. ¶ LXXXII. Alligation is the method of mixing several simples 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 a 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 $4,60 $4,60 1046 cts., Ans. RULE. Divide the whole cost by the whole number of shels, &c.; the quotient will be the mean price or cost of e mixture. 2. A grocer mixed 10 cwt. of sugar at $10 per cwt., 4 cwt. at $4 per cwt., and 8 cwt. at $7 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 $ per lb., 9 lbs. at $1,80 per lb., and 17 lbs. at $1 per lb.: what is a pound of it worth? A. $1,546 4. If 20 bushels of wheat, at $1,35 per bushel, be mixed with 15 bushels of rye, at 85 cents per bushel, what will a bushel of this mixture be worth? A. $1,135 5. ff 4 lbs. of gold, of 23 carats fine, be melted with 2 lbs. 17 carats fine, what will be the fineness of this mixture? A. 21 carats. ALLIGATION ALTERNATE. ¶ LXXXIII. The process of finding the proportional quantity of each simple, from having the mean price or rate, and the mean prices or rates of the several simples given, called Alligation Alternate; consequently, it is the reverse of Alligation Medial, and may be proved by it. 1. A farmer has oats, worth 25 cents a bushel, which he wishes to mix with corn, worth 50 cents per bushel, so that the mixture may be worth 30 cents per bushel; what proportions or quantities of each must he take? In this example, it is plain, that, if the price of the corn had been 35 cents, that is, had it exceeded the price of the mixture, (30 cents,) just as much as it falls short, he must have taken equal quantities of each sort; but, since the difference between the price of the corn and the mixture price is 4 times as much as the difference between the price of the oats and the mixture price, consequently, 4 times as much oats as corn must be taken, that is, 4 to I, or 4 bushels of oats to 1 of corn. But since we determine this proportion by the differences, hence these differences will represent the same proportion. These are 20 and 5, that is, 20 bushels of oats to 5 of corn, which are the quantities or proportions required. In determining these differences, it will be found convenient to write them down in the following manner: OPERATION. Cts. Bushels. 30,50$,25-1-20 Ans. 55 It will be recollected, that the difference between 50 and 30 is 20, that is, 20 bushels of oats, which must, of course, stand at the right of the 25, the price of the oats, or, in other words, opposite the price that is connected or linked with the 50; likewise the difference between 25 and 305, that is, 5 bushels of corn, opposite the 50, (the price of the corn.) The answer, then, is 20 bushels of oats to 5 bushels of corn, or in that proportion. By this mode of operation, it will be perceived that there is precisely as much gained by one quantity as there is lost by another, and, therefore, the gain or loss on the whole is equal. The same will be true of any two ingredients mixed together in the same way. In like manner the proportional quantities of any number of simples may be determined; for, if a less be linked with a greater than the mean price, there will be an equal balance of loss and gain between every two, consequently an equal balance on the whole. It is obvious, that this principle of operation will allow a great variety of answers; for, having found one answer, we may find as many more as we please, by only multiplying or dividing each of the quantities found by 2, or 3, or 4, &c.; for, if 2 quantities of 2 simples make a balance of loss and gain, as it respects the mean price, so will also the double or treble, the, or part, or any other ratio of these quantities, and so on to any extent whatever. PROOF. We will now ascertain the correctness of the foregoing operation by the last rule, thus: 20 bushels of oats, at 25 cents per bushel, 5 corn, at 50 $5,00 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 several, their sum will express the quantity. 2. A merchant nas several sorts of tea, some a 10 s., some d: 11 s., some at 13 s., and some at 24 s. 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 cents, and 78 cents, per bushel, must be mixed together, that the compound may be worth 64 cents per bushel? A. 14 bushels at 42 cents, 3 bushels at 60 cents, 4 bushels at €7 cents, and 22 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 at 23 cents, and 2 at 26 cents. 6. A jeweller wishes to procure gold of 20 carats fine, from gold of 16, 19, 21, and 24 carats fine; what quantity of each must he 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 times, 1, 1, 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 of brandy to 5 of rum; hence he must take 2 gallons of rum for every gallon of 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 a 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: 2:10: 2 bushels of rye. 8 10 10 12 bushels of corn. Ans 9. How much water must be mixed with 100 gallons of rum, worth 90 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 50 cents per lb. A. 20 at $1,20, 10 at $1 and 10 at 60 cents. 11. A grocer has currants at 4 cents, 6 cents, 9 cents, and 11 cents per fb.; 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 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. 4 We will now apply this rule in performing the last question. 6 11 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 per 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. 5oz. of 15, of 17, of 18, and 25 oz. of 22. INVOLUTION. ¶ LXXXIV. Q. How much does 2, multiplied into itself, or by 2, make? 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 twice; what, then, is 36 called? A. The second power, or square of 6. Q. What is the second power, or square of 8? 10? 12? |