CASE II. When the rates of all the ingredients, the quantity of but one of them, and the mean rate of the whole mixture are given, to find the several quantities of the rest, in proportion to the quantity given. RULE. Take the difference between each price, and the mean rate, and place them alternately as in Case I. Then, as the difference standing against that simple, whose quantity is given, is to that quantity, so is each of the other differences, severally, to the several quantities required. EXAMPLES. 1. A merchaut has 20 lbs, of tea at $1 04 a pound, which he would mix with some at 98 cents, some at 92 cenis, and some at 80 cents a pound; how much of each sort must he take to mix with the 20 pounds, that he may sell the mixture at 96 cents a pound? 104 4 stand against the given quantity. 98%+ 16 96 92 8 80 2 lbs. lbs. cts. 16 : 80 at 78 2 : 10 66 80 2. Bought a pipe of brandy containing 120 gallons, at 1.30 a gallon; how much water must be mixed with it to reduce the first cost to $1.10 a gallon ? Ans. 2141 galls. 3. How much gold of 16, 20 and 24 carats fine, and how much alloy, must be mixed with 10 oz. of 18 carats fine, that the composition may be 22 carats fine ? 10 oz. of 16 carats fine. 10 oz. of 20 carats fine. 170 oz, of 24 carats fine. 10 oz, of alloy. Ans. I 302. When the rates of all the ingredients, the quantity of but one of them, and the mean rate of the whole mixture are given ; what is the rule for finding the several quantities of the rest, in proportion to the given quantity ? CASE III. When the rates of the several ingredients, the quantity to be compoun ded, and the mean rate of the whole mixture are given, to find how much of each sort will make up the quantity. . RULE. Write the difference between the mean rate, and the several prices alternately, as in Cael; then, as the sum of the quantities or differences thus determined, is to the given quantity, or whole composition ; so is the difference of each rate, to the required quantity of each rate. EXAMPLES. 1. A merchant haviog sugars at 12 dollars, 10 dollars, and 8 dollars a cwt would make a mixture of 30 cwt. worth 9 dollars a cwt.; what quantity of each must be taken ? cwt. dols. 1: 5 at 12 9 10% 1 As 6 : 30 : : 1: 5 " 10 Ans. 8 1+3=4 4 : 20", 8 2. A goldsmith has several sorts of gold; viz. of 15, 17, 20 and 22 carats fine, and would melt together of all these sorts, so much as may make a mass of 40 oz. of 18 carats fine; how much of each sort is required ? 16 oz. of 15 carats fine. 8 oz. of 17 carats fine. Ans. 4 02, of 20) carats fine. 12 oz. of 22 carats fine. POSITION. Position is a rule, by which any true or required numbers are found by means of assumed or supposed numbers. It is of two kinds, Single and Double. 303. When the rates of the several ingredients, the quantity to be compounded and the mean rate of the whole mixture are given ; what is the rule for finding how much of each sort will make up the quantity ? . 304. What is Position ? SINGLE POSITION Is the working of one supposed number, as if it were the true one, to find the true number. RULE. 1. Take any number and perform the same operations with it as are described to be performed in the question. 2. Then say as the sum of the errors is to the given sum, so is the supposed number to the true one required.* Proof. Add the several parts of the sum together, and if it agree with the sum, it is right. EXAMPLES. 1. A school master, being asked how many scholars he had, said, if I had as many more as I now have, three quarters as many, half as many, ope fourth and one eighth as many, I should then have 435:Of wbat number did his school consist ? Suppose he bad 80 As 290 : 435 : : 80 As many=80 80 as many=60 120 as many-_-40 290)3480|0(120 Ans. 120 as many=20 29 90 as many=10 60 30 15 435 Proof. 2. A person lent his friend a sum of money unknown, to receive interest for the same at 6 per cent. per annum, simple interest, and at the end of 12 years, received for principal and ioterest $360: What was the sum lent ? Ans. $500. * The operations contained in the question being performed upon the answer or numher to be found, will give the result contained in the question. The same operations, performed on any other nunber, will give a certain result. When the results are proportional to their supposed numbers, it is manifest that the result of the operations performed on the supposed number, must be to the supposed number, as the result in the question is to the true number or answser. In any case, when the results are not proportional to their supposed numbers, the answer cannot be found by this rule. 305. What is Single Position ?-----306. What is the rule for working questions in Single Position ? 3. A, B, and C joined their stocks, and gained $353 127c. of which A took up a certain sum, B took up four times as much as A, and C, five times as much as B: What share of the gain had each? $14 121c. A's share. Ans. 56 50 B's share. 282 50 C's share. 4. A and B, talking of their ages, B said his age was once and an half the age of A; C said his was twice and one tenth the age of both, and that the sum of their ages was 93 : wbat was the age of each? Ans. A's 12, B's 18, and C-s 63 years. 5. Seven eights of a certain number exceeds four fifths by 6 : what is the number? Ans. 80. 6. What oumber is that, which, being increased by }, }, and of itself, the sum will be 2348? Ans. 90. DOUBLE POSITION. Double Position teaches to resolve questions by making two suppositions of some convenient numbers. Those questions, in which the results are not proportional, to their positions, belong to this rule : such are those, in which the number sought is increased or diminished by some given number, which is no known part of the number required. RULE.* 1. Take any two convenient numbers, and proceed with each according to the conditions of the question. 2. Place the result or errours against their positions or suppos Pos. Err. 30 12 ed numbers, thus, X and if the errour be too great, mark it 20 6 with t; and if too small, with * The rule is founded on this supposition, that the first error is to the second, as the difference between the true and first supposed number is to the difference between the true and second supposed number: When that is not the case, the exact answer to the question cannot be found by this rule. 307. What is Double Position ?-308. And what is the rule ? 3. Multiply them crosswise ; that is, the first position by the last errour, and the last position by the first errour. 4. If the errours be alike, that is, both too small or both too great, divide the difference of the products by the difference of the errours, and the quotient will be the answer. 5. If the errours be unlike ; that is, one too small, and the other too great, divide the sum of the products by the sum of the errours, and the quotient will be the answer. Note.-When the errours are the same in quantity, and unlike in quality, half the sum of the suppositions is the number sought. E.XAMPLRI. 1. A lady bought damask for a gown, at 8s. per yard, and lining for it at 3s. per yard ; the gown and lining contained 15 yds., and the price of the whole was £3 10$. : How many yards were there of each ? Suppose 6 yards damask, value 489. Sum of their values=753, Then she must have 11 yards of lining, value 33s. Sum of their values=65$. £ s. d. Then X 20 30 20 3 10 Oproof. Sum of errors=5+5=10)50 Ans. 5 yds. damask, and 15-5=10 yards Or, 6+4;-2=5 as before. [lining. AA |