040 tional quantity. So it is evident that the quantities may be increased or diminished and sold at the same mean rate, while they bear the same proportion to each other. 2. How much gold of 15, 17, 18, and 22 carats fine, must be mixed together, to form a mixture of 80oz. of 20 carats fine ? Ans. 10oz. of :15, of 17, and of 18 carats fine, and 50oz. of 22 carats fine. 3. How many gallons of brandy, at $1,20cts. per gallon, must be mixed with water of no value, so as to fill a cask on 110 gallons, which may be afforded at 80cts. per gallon ? cts. $ cis. gal. gal. gal. gal. 1,20, 80 80: Ans. 73} of brandy. 801 120: 110 :: 40 : 363 of water. 120 gal. 4. A vintner has 3 kinds of wine; one kind at 24d. per gallon, one at 22d., and one at 18d.; how must he fill a cask of 60 gallons with his three kinds of wine, so that he may sell it at 20d. per gallon? S 12gal. at 24d. per gal., 12 Ans. at 22d. and 36 at 18d. QUESTIONS ON ALLIGATION. What is Alligation ? A. It teaches how to mix several simples af different qualities, so that the composition may be of some intermediate quality or rate. How many kinds of Alligation are there? A. Two, Alligation Medial, and Alligation Alternate. Why is alligation me dial so called? A. Because it teaches to find a mean or middle rate or quality. Why is alligation alternate so called? A. Because it generally admits of different answers, when the simples are differently linked together. What rule do you observe in linking the simple rates or qualities together? A. Always unite one that is less than the mean rate with one that is greater than the mean rate, or quality. Why should the numbers which stand opposite the simple rates or qualities form a compound equal to the mean rate? A. Because the gain and loss exactly balance each other, there befng as much loss on that number which exceeds the mean rate, as, there is gain on that which falls short of the mean rate. On what rule does alligation depend? A. On the Rule of Three, from which all its principles are drawn. THE DOUBLE RULE OF THREE. The Double Rule of Three, or, as it is sometimes called, Compound Proportion, teaches to resolve by one statement such questions as require two or more statements by the Single Rrile of Three, and hence the rule has derival its name. In this rule there are always an odd mo. number of termis given, as five, seven, nine, &c. In the questions, generały, under this rule, there are five terms given to find a sixth; the first three terms are a supposition, the last two a demand. RULE.-In stating questions, place the terms of supposition so that the principal cause of gain, loss, or action, possess the first place; that which expresses the space of time, or the distance of place, the second place; and that, which expresses the gain, loss, or action, the third place. Then, place the terms of demand under those of the same kind in the supposition. If the blank place, or term sought, fall under the third term, the portion is direct; then multiply the first and second terms together for a divisor, and the other three for a dividend, and the quotient will be the answer in the same denomination of the term directly above the bank; but if the blank fall under the first or second term, the proportion is inverse; then, multiply the third and fourth terms together for a divisor, and the other three for a dividend, and the quotient will be the answer. NOTE.- Observe the same rule here, as in the Single Rule of Three, when the terms of the question are in different denominations; the reduction, however, ought to be performed before the question is stated. EXAMPLES. 1. If 100 dollars in 12 months, gain 6 dollars; what will 600 dollars gain in 9 months ? [tional terms. 100 : 12 :: 6 The terms of supposition or condi600: 9.... The dernanding terms, or those which 9 move the question. 100 5400 Dem. It is plain, from the nature of 12 6 this question, that this is only a contrac tion of the Single Rule of Three; because 12|00) 324|00(27 Ans. the product of the two terms which com24 pose our divisor here, may be consider ed as a composite number, the compo84 nent parts of which must be the two di84 visors, when the question is solved by two statements, in the single Rule of Three. And it is evident that it can make no dit ference whether we divide directly by a composite number, 'or by the component parts which compose thao .Dumber, as the following work shows By two statements in the Single Rule of Three. mo. mo. 100: 6 :: 600 12 : 36 :: 9 6 $ 9 100)3600(36 12)324 300 Ans. $27. same as above. 600 and no remainder. Thus we see the question solved by the Single Rule of Three in plain and easy statements; for it is evident, if 100 dollars gain 6 dol. lars in one year, that 600 dollars must gain 36 dollars; and it is also obvious, that the gain must be proportional to the time for as 12 months are to the gain for that time, so 9 months are to the gain for that time. Perhaps this rule will appear more simple to the young student, by carrying it still further back, and illustrating it by Multiplication and Division, as follows: cts. year. $3, The gain on $600 for 1 month. 9 Ans. 9:27, The gain on $600 for 9 months. The principles of this rule must appear plain to the student when he discovers that it is only an application of the Single Rule of Three, and from that may be reduced back to Multiplication and Division; because by reducing it back, all the difficulty vanishes. 2. If 8 persons expend 200 dollars in 9 months, how much will serve 18 persons 12 months ? Ans. $6007 pers. mos. NOTE.-Of the three conditional terms, 8: 9 :: 200 or terms of supposition, the student 18 : 12 will discover, that the eight persons must be the cause of expense, and con sequently put for the first term; and 9 8 3600 months, the time, and therefore put in 12 the second place; and 200 dollars is what was expended in that time, and conse72)43200(600 quently it is put in the third place; the 432 other terms regularly fall under those of the same name. 00 3. If 10 men spend 9 dollars in 12 weeks, how much will 20 men spend in 24 weeks? Ans. $36. 4. If $100, in 12 months, gain 7 dollars interest, what will $600 gain in 8 months ? Ans. $28. 5. If 20 bushels of oats be sufficient for 18 horses 20 days, how many bushels will serve 60 horses 36 days? Ans. 120 bushels. 6. If $100, in one year, gain 7 dollars interest, what sum will gain $38 50cts. in 15 months ? Ans. $440. 200 $ cts. Nors. In this question, the proportion is 100 : 12 :: 7,00 inverse; therefore, the third and fourth terms 15 :: 38,50 other terms for a dividend. are multiplied together for a divisor, and the 7. If $100 gain 7 dollars in one year, in what time will 440 dollars gain $38,50cts. ? Ans. 15 months. 8. An usurer put out $650, to receive interest for the same; at the end of 6 months he received for principal and interest, $672,75cts.; at what rate per cent did he receive interest ? Ans. 7 per cent. 9. If 100 men, in 6 days of 10 hours each, can dig a trench 200 yards long, 3 wide, and 2 deep; in how many days of 8 hours long, will 180 men dig a trench of 360 yards long, 4 wide, and 3 deep? Áns. 15 days. QUESTIONS ON THE DOUBLE RULE OF THREE. What does the Double Rule of Three teach ? A. It teaches to resolve, by one statement, such questions as require two or more statements, when worked by the Single Rule of Three. What number of terms is generally given in the Double Rule of Three ? A. Five.What rule do you observe, in stating questions, in this rule ? A. Write that term which is the principal cause of gain, loss, or action, for the first term; time or distance for the second ; and, gain, loss, or action, for the third : and then write the two remaining terms under their corresponding terms, that is, terms of the same name. When the blank falls under the third term, how do you proceed ? A. Multiply the first and second terms together for a divisor, and the other three for a dividend, and the quotient will be the answer. But when the blank falls under the first or second terms, multiply the third and 4th terms for a divisor, and the other three for a dividend, and the quotient will be the answer. What term must be of the same kind with the answer ? A. The term directly above the blank. EQUATION OF PAYMENTS, Teaches how to reduce several stated times, at which money is pay. able, to one mean or equated time, for the payment of the whole. RULE.-Multiply, each payment by its time, and add the several products together; then divide the sum of the products by the whole debt, and the quotient will be the equated time, or answer. PROOF.-The interest of the whole sum to the equated time, at any given rate, will equal the interest of the several payments, for their respective times, at the same rate. NOTE.--This rule is founded on the supposition that what is gained, by keeping some of the debts or payments after they are due, is lost by Daying other debts (or payments, before they are due. EXAMPLES. 1. A, owes B $800, to be paid as follows, viz. $400 in 4 months, and $400 in 8 months; what is the equated time for the payment of the whole debt ? Ans. 6 mo's. $400X4=1600 DEM. It is evident, if B 400X8=3200 Ans: wait on A, two months, after one half of his debt is due, Divisor, 800) 4800(6 mo's. that B should receive the other 4800 half, two months before it is due. And it is also evident, tnat the product of each payment, divided by the payment, gives the time it is due; then the sum of the products divided by the sum of the payments, must give the equated time. 2. B, owes C, $400, of which $200 are to be paid in two months, and $200 in four months; but they agree that the whole shall be paid at one time; at what time must it be paid? Ans. in 3 months. 3. A, owes B, $760, to be paid as follows ; $200 in 6 months, 240 in 7 months, and 320 in •10 months; what is the equated time for the payment of the whole debt ? Ans. 8 months. 4. A, holds B's note for $1200, which is to be paid in the following manner ; 1 in 6 months, } in 8 months, and the remainder in 10 months; what is the equated time for the payment of the whole ? Ans. 7 months. QUESTIONS ON EQUATION OF PAYMENTS. What does Equation of Payments teach ? A. It teaches to find a mean time for the payment of a deht, which is made payable by instalments. How do you proceed in the work? A. First, multiply cach payment by its time, then divide the sum of the products, by the whole debt, and the quatient wilt be the answer. Oh what supposi tion is this rule founded ? A. On the supposition that what is gained by putting off some payments after they are due, is lost by paying others, before they are due. INVOLUTION, Teaches how to find the powers of numbers. A power is a product or number produced by multiplying any given number, called the root, a certain number of times, continually by itself. Thus, 2 is the root or Ist power of 2; 4 is the 2d power, or square of 2, produced thus, 2X2=4; 8 is the 3d power, or cube of 2, produced thus, 2X2X2=8; 16 is the 4th power, or biquadrate of 2, produced thus, 2X2X2X2=16, and so on. The power is often denoted by a figure placed at the right hand of the number, and a little above it, which figure is called the index or exponent of that power. This index or exponent is always one more than the number of multiplications, to produce the power; or it is equal to the number of times the given number is taken as a factor, in producing the power; thus, 3 is used twice to produce the square,' or |