2 mo.; the third, 3 cows for 3 mo.; the fourth, 2 cows for 4 mo. How much should each pay? 5. Jones, Hunt, and Gray, having been in partnership for one year, under an agreement to divide the profit proportionately to their respective shares of capital, have made $2403. On the first day of the year, each put in $10000; but Hunt in 4 mo. withdrew 20% of his share, and Gray at the end of 6 mo. put in $2000 more. each partner's share of the profit. Find Ans. Jones's share, $810, etc. CHAPTER Xxx. ALLIGATION. 619. Alligation is the process of solving questions that relate to the mixing of ingredients of different values. There are two kinds of Alligation, Medial and Alternate. Alligation means connecting, and the process is so called from connecting the prices of the ingredients together, as shown in Art. 623. Alligation Medial. 620. Alligation Medial is the process of finding the average value of a mixture, when the value and quantity of each ingredient are known. Ex.-A liquor-dealer mixed 5 gal. of whiskey worth $2 a gal., with 4 gal. worth $1.50, 8 gal. worth $1.25, and 2 gal. of water. What was the mixture worth per gallon? 621. RULE.--To find the average value of a mixture, divide the total value of the ingredients by the sum of the quantities. The principle of this rule applies to many questions that involve the finding of an average, besides those relating to values or prices. 622.-EXAMPLES FOR PRACTICE. 1. If 100 lb. of coffee worth 16 c. a pound, are mixed with 40 lb. worth 20 c., 200 lb. worth 15 c., and 25 lb. of damaged worth only 10 c., what is the value of the mixture per pound? 2. A grain-dealer mixes a cental of damaged wheat that cost 75 c. a bushel, with 150 bu. at $1.30, 80 bu. at $1.35, and 50 bu. at $1.40. At what price per bushel must he sell the mixture, to make 10%? 3. A dishonest grocer adulterated 100 lb. of powdered sugar worth 14 c. a pound, with 5 lb. of flour worth 4 c. a pound, and then added 25 lb. of sugar at 13 c. If he sold the whole at 15 c. a pound, what % advance on the value did he get ? Ans. 11%. 4. With 10 gal. of wine costing $5 a gallon, are mixed 6 gal. that cost $3, and 2 gal. of water. At what price must the mixture be sold, to yield a profit of 25 c. a gallon? 5. A goldsmith melted up together 6 oz. of silver .900 fine, 3 lb. 2 pwt. .800 fine, and 8 oz. of alloy. How many thousandths fine was the mixture? Ans. .684+. 6. If a bushel of chaff is mixed with 10 bu. of oats that cost 60 c. a bushel, and 20 bu. at 50 c., and the whole is sold at 10% below cost, what is the selling price? Alligation Alternate. 623. Alligation Alternate is the process of finding the quantities to be taken of two or more ingredients, of given values, to make a mixture of given value. Ex. 1.-In what relative quantities must teas worth respectively 75 c., 95 c., $1.10, and $1.20, be mixed, that the mean value may be $1 a pound? That the mean value ($1) may be preserved, the gain on tea worth less than $1 taken for the mixture must balance the loss on tea worth more than $1. Accordingly, we consider a price less than the mean in connection with 75 c. 20 4 95 c. 10 2 $1 $1.10 5 1 $1.20 25 5 one greater. On every pound worth 75 c. and valued in the mixture at $1, there is a gain of 25 c.; and, on every pound worth $1.20 and valued at $1, there is a loss of 20 c. Therefore, as the gain and loss on 1 lb. of each of these ingre dients are as 25 to 20, the pounds taken must be as 20 to 25, or 4 to 5; that is, 4 lb. and 5 lb. In like manner, comparing 1 lb. at 95 c. with 1 lb. at $1.10, we find that there is a gain of 5 c. against a loss of 10 c.; hence the quantities taken of these two ingredients must be as 10 to 5, or 2 to 1. The relative quantities, therefore, of the four ingredients are 4 lb. at 75 c., 2 lb. at 95 c., 1 lb. at $1.10, and 5 lb. at $1.20. To prove, find by Alligation Medial whether the mean value of these quantities, at their several prices, is $1. In practice, the values may be linked in pairs, as in the margin, one less than the mean with one greater. The intermediate reasoning may be omitted, and the difference between the mean and each value in succession may be written opposite the value with which it is .75 20 4 .95 10 2 $1 1.10 5 1 1.20 25 5 linked, as representing the quantity to be taken of that ingredient. $1 { .75 10 2 .95 1.10 25 5 1.20 The terms may be compared and linked in a different way from that shown above (see margin), provided one less than the mean is connected with one greater. The answers, of course, differ according to the linking.-Prove these answers by Alligation Medial. As any set of answers obtained shows merely the relative quantities, we may multiply or divide these numbers by any number whatever, and thus produce an infinite variety of answers. 624. The quantity to be taken of one of the ingredients may be given. For instance, in Example 1, if 8 lb. of the 75-cent tea are to be put in the mixture, how many pounds of the other kinds must be taken? The results of the last linking show the quantities of the other kinds required for every 2lb. of the 75-cent kind; for8 lb. of the 75-ceut kind, 4 times as much of each kind must be taken. 2 X 4 = 8 lb. 4 X 4 16 lb. = Ans. The results of the first linking show the quantities of the other 4 X 2 8 lb. 2 X 2 = 4 lb. Ans. kinds required for every 4 lb. of the 75cent kind; for 8 lb. of the 75-cent kind, twice as much of each kind must be taken.-Prove both sets of answers by Alligation Medial. 625. The quantity of the mixture may be given. For instance, in Example 1, it may be asked how many pounds of each kind must be taken to make a mixture of 15 lb. Adding the relative quantities obtained in Art. 623 by either linking, as great; and the quantity of each ingredient must, therefore, be in creased times.-Prove each set of answers by Alligation Medial. 626. RULES.-I. Write the values in a column, and the mean value on the left. Link each value less than the mean with one greater, and each greater with one less. Write the differences between the mean and the several values opposite the values they are respectively linked with. These differences are the relative quantities of the ingredients taken in the order in which their values stand. II. If the quantity of one ingredient is given, to find the corresponding quantities of the others, multiply their differences by the ratio of the given quantity to the difference of the ingredient it represents. III. If the quantity of the mixture is given, to find the quantity of the ingredients, multiply their differences by the ratio of the given quantity to the sum of the differences. Ex. 2.-A dishonest grocer adulterated three kinds of sugar, costing respectively 6, 7, and 8 cents a pound, with sand sufficient to reduce the cost of the mixture to 5 c. a pound. How much of each did he take? The sand is represented by 0. 6 5 7. 5 5 8 5 1+2+3 6. = As there are three values greater than the mean, and but one less, we have to link the three with the one. There will, therefore, be three differences opposite the 0, and their sum will represent the relative quantity of sand. He must have taken 6 lb. of sand to 5 lb. of each kind of sugar. 627.-EXAMPLES FOR PRACTICE. Prove the results obtained, by Alligation Medial. 1. Find the relative quantities that must be taken of different grades of corn, bought for 80, 75, 70, and 68 |