2. Three graziers hired a piece of land for $60.50. A put in 5 sheep for 4^ months, B put in 8 for 5 months, and C put in 9 for 6 months; how much must each pay of the rent? Ans. A $11*25, B $20, and C $29-25. 3. Two merchants enter into partnership for 18 months; A put into stock at first $200, and at the end of 8 months he put in $100 more; B put in at first $550, and at the end of 4 months took out $140. Now at the expiration of the time they find they have gained $526; what is each man's just share? Ans. A's $192'95-10 B's 333-04231 T234 1184 4. A, with a capital of $1000 began trade January 1, 1776, and meeting with success in business he took in B as a partner, with a capital of $1500 on the first of March following. Three months after that they admit C as a third partner, who brought into stock $2800, and alter trading together till the first of the next year, they find the gain, since A commenced business, to be $1776-50. How must this be divided among the partners? Ans. A's $457-46364. 466 346 466° ALLIGATION. Alligation teaches how to mix several simples of different qualities, so that the composition may be of a middle quality; and is commonly distinguished into two principal cases, called Alligation medial and Alligation alternate. ALLIGATION MEDIAL. Alligation medial is the method of finding the rate of the compound, from having the rates and quantities of the several simples given. RULE.* Multiply each quantity by its rate; then divide the sum •f the products by the sum of the quantities, or the whole composition, and the quotient will be the rate of the compound required. EXAMPLES. 1. Suppose 15 bushels of wheat at 5s. per bushel, and 12 bushels of rye at 3s. 6d. per bushel were mixed together; how must the compound be sold per bushel without loss or gain? 2. A composition being made of 5lb. of tea at 7s. per pound, 9lb. at 8s. 6d. per pound, and 14£lb. at 5s. 10d. per pound, what is a pound of it worth? Ans. 6s. lO^d. 3. Mixed 4 gallons of wine at 4s. 10d. per gallon, with 7 *The truth of this rule is too evident to need a demonstration. Note. If an ounce or any other quantity of pure gold be reduced into 24 equal parts, these parts are called carats; but gold is often mixed with some baser metal, which is called the alloy, and the mixture is said to be of so many carats fine, according to the proportion of pure gold contained in it; thus, if 22 carats of pure gold and 2 of alloy be mixed together, it is said to be 22 carats fine. If any one of the simples be of little or no value with respect to the rest, its rate is supposed to be nothing, as water mixed with wine, and alloy with gold or silver. gallons at 5s. 3d. per gallon, and 9 gallons at 5s. 8d. per gallon; what is a gallon of this composition worth? Ans. 5s. 4^d. 4. A goldsmith melts 8lb. 5 oz. of gold bullion of 14 carats fine, with 12lb. 84oz. of 18 carats fine ; how many carats fine is this mixture? Ans. 16 carats. 5. A refiner melts 101b. of gold of 20 carats fine with 16lb. of 18 carats fine; how much alloy must he put to it to make it 22 carats fine? Ans. It is not fine enough by 3 carats, so that no alloy must be put to it, but more gold. ALLIGATION ALTERNATE. Alligation alternate is the method of finding what quantity of any number of simples, whose rates are given, will compose a mixture of a given rate; so that it is the reverse of alligation medial, and may be proved by it. RULE 1.* 1. Write the rates of the simples in a column under each other. • Demonstration. By connecting the less rate to the greater, and placing the differences between them and the mean rate alternately, the quantities resulting are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss upon the whole are equal, and are exactly the proposed rate; and the same will be true of any other two simples, managed according to the rule. In like manner, let the number of simples be what it may, and with how many soever each is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole. Q. E. D. It is obvious from the rule, that questions of this sort admit of 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, 3, or 4, &c. the rea 2. Connect or link with a continued line the rate of each simple, which is less than that of the compound, with one or any number of those, that are greater than the compound; and each greater rate with one or any number of the less. g. Write the difference between the mixture rate and that of each of the simples opposite to the rates, with which they are respectively linked. 4. Then if only one difference stand against any rate, it will be the quantity belonging to that rate; but if there be several, their sum will be the quantity. EXAMPLES. 1. A merchant would mix wines at 14s. 19s. 15s. and 22s. per gallon, so that the mixture may be worth 18s. the gallon; what quantity of each must be taken — 2. How much wine at 6s. per gallon and at 4s. per gallon must be mixed together, that the composition may be worth 5s. per gallon? Ans. 12 gallons, or equal quantities 3f each. son of which is evident; for, if two quantities of two simples make a balance of loss and gain, with respect to the mean price, so must also the double or treble, the or part, or any other ratio of these quantities, and so on, ad infinitum. Questions of this kind are called by algebraists indeterminate or unlimited problems, and by an analytical process theorems may be raised, that will give all the possible answers. VOL. I. R Ans. 5s. 41d. 4. A goldsmith melts 8lb. 5 oz. of gold bullion of 14 carats fine, with 12lb. 8oz. of 18 carats fine; how many carats fine is this mixture? Ans. 160 carats. 5. A refiner melts 10lb. of gold of 20 carats fine with 16lb. of 18 carats fine; how much alloy must he put to it to make it 22 carats fine? Ans. It is not fine enough by 3 carats, so that no alloy must be put to it, but more gold. ALLIGATION ALTERNATE. RULE 1.* 1. Write the rates of the simples in a column under each other. * DEMONSTRATION. By connecting the less rate to the greater, and placing the differences between them and the mean rate alternately, the quantities resulting are such, that there is precisely as much gained by one quantity as is lost by the other, and therefore the gain and loss upon the whole are equal, and are exactly the proposed rate; and the same will be true of any other two simples, managed according to the rule. In like manner, let the number of simples be what it may, and with how many soever each is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole. Q. E. D. It is obvious from the rule, that questions of this sort admit of 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, 3, or 4, &c. the rea |