ALLIGATION

784. Alligation treats of mixing or compounding two or more ingredients of different values or qualities.

785. Alligation Medial is the process of finding the mean or average value or quality of several ingredients. 786. Alligation Alternate is the process of finding the proportional quantities to be used in any required mixture.

787. 1. If a grocer mix 8 lb. of tea worth $.60 a pound with 6 lb. at $.70, 2 lb. at $1.10, and 4 lb. at $1.20, what is 1 lb. of the mixture worth?

2. If 20 lb. of sugar at 8 cents be mixed with 24 lb. at 9 cents, and 32 lb. at 11 cents, and the mixture is sold at 10 cents a pound, what is the gain or loss on the whole?

RULE. Find the entire cost or value of the ingredients, and divide it by the sum of the simples.

3. A miller mixes 18 bu. of wheat at $1.44 with 6 bu. at $1.32, 6 bu. at $1.08, and 12 bu. at $.84. What will be his gain per bushel if he sell the mixture at $1.50 ?

4. Bought 24 cheeses, each weighing 25 lb., at 7 a pound; 10, weighing 40 lb. each, at 10; and 4, weighing 50 lb. each, at 13; sold the whole at an average price of 91 a pound. What was the whole gain?

5. A drover bought 84 sheep at $5 a head; 86 at $4.75; and 130 at $51. At what average price per head must he sell them to gain 20%?

788. To find the proportional parts to be used, when the mean price of a mixture and the prices of the simples are given.

1. What relative quantities of timothy seed worth $2 a bushel, and clover seed worth $7 a bushel, must be used to form a mixture worth $5 a bushel?

5

OPERATION.
22

13

Ans.

ANALYSIS.-Since on every ingredient used whose price or quality is less than the mean rate there will be a gain, and on every ingredient whose price or quality is greater than the mean rate there will be a loss, and since the gains and losses must be exactly equal, the relative quantities used of each should be such as represent the unit of value. By selling one bushel of timothy seed worth $2, for $5, there is a gain of $3; and to gain $1 would require of a bushel, which is placed opposite the 2. By selling one bushel of clover seed worth $7, for $5, there is a loss of $2; and to lose $1 would require of a bushel, which is placed opposite the 7.

In every case, to find the unit of value, divide $1 by the gain or loss per bushel or pound, etc. Hence, if every time of a bushel of timothy seed is taken, of a bushel of clover seed is taken, the gain and loss will be exactly equal, and † and 1 will be the proportional quantities required.

To express the proportional numbers in integers, reduce these fractions to a common denominator, and use their numerators, since fractions having a common denominator are to each other as their numerators (241); thus, } and are equal to and, and the proportional quantities are 2 bu. of timothy seed to 3 bu. of clover seed.

2. What proportions of teas worth respectively 3, 4, 7, and 10 shillings a pound, must be taken to form a mixture worth 6 shillings a pound?

10 1

4

4

11

22

3

3

ANALYSIS.-To preserve the equality of gains and losses, always compare two prices or simples, one greater and one less than the mean rate, and treat each pair or couplet as a separate example. In the given example form two couplets, and compare either 3 and 10, 4 and 7, or 3 and 7, 4 and 10.

We find that of a lb. at 3s. must be taken to gain 1 shilling, and of a lb. at 10s. to lose 1 shilling; also of a lb. at 4s. to gain 1 shilling, and 1 lb. at 7s. to lose 1 shilling. These proportional numbers, obtained by comparing the two couplets, are placed in columns 1 and 2. If, now, the numbers in columns 1 and 2 are reduced to a common denominator, and their numerators used, the integral numbers in columns 3 and 4 are obtained, which, being arranged in column 5, give the proportional quantities to be taken of each.

It will be seen that in comparing the simples of any couplet, one of which is greater, and the other less than the mean rate, the proportional number finally obtained for either term is the difference between the mean rate and the other term. Thus, in comparing 3 and 10, the proportional number of the former is 4, which is the difference between 10 and the mean rate 6; and the proportional number of the latter is 3, which is the difference between 3 and the mean rate. The same is true of every other couplet. Hence, when the simples and the mean rate are integers, the intermediate steps taken to obtain the final proportional numbers as in columns 1, 2, 3, and 4, may be omitted, and the same results readily found by taking the difference between each simple and the mean rate, and placing it opposite the one with which it is compared.

3. In what proportions must sugars worth 10 cents, 11 cents, and 14 cents a pound be used, to form a mixture worth 12 cents a pound?

4. A farmer has sheep worth $4, $5, $6, and $8 per head. What number may he sell of each and realize an average price of $5 per head?

RULE.-I. Write the several prices or qualities in a column, and the mean price or quality of the mixture at the left.

II. Form couplets by comparing any price or quality less, with one that is greater than the mean rate, placing the part which must be used to gain 1 of the mean rate opposite the less simple, and the part that must be used to lose 1 opposite the greater simple, and do the same for each simple in every couplet.

III. If the proportional numbers are fractional, they may be reduced to integers, and if two or more stand in the same horizontal line, they must be added; the final results will be the proportional quantities required.

1. If the numbers in any couplet or column have a common factor, it may be rejected.

2. We may also multiply the numbers in any couplet or column by any multiplier we choose, without affecting the equality of the gains and losses, and thus obtain an indefinite number of results, any one of which being taken will give a correct final result.

5. What amount of flour worth $5, $6, and $7 per barrel, must be sold to realize an average price of $61 per barrel?

6. In what proportions can wine worth $1.20, $1.80, and $2.30 per gallon be mixed with water so as to form a mixture worth $1.50 per gallon?

789. When the quantity of one of the simples is limited.

1. A farmer has oats worth $.30, corn worth $.45, and barley worth $.84 a bushel. To make a mixture worth $.60 a bushel, and which shall contain 48 bu. of corn, hov ́ many bushels of oats and barley must he use ?

of barley. But since 48 bu. of corn is 6 times the proportional number 8, to preserve the equality of gain and loss take 6 times the proportional quantity of each of the other simples, or 6 × 4 = 24 bu. of oats, and 6 × 10 = 60 bu. of barley. Hence, etc.

2. A dairyman bought 10 cows at $40 a head. How many must he buy at $32, $36, and $48 a head, so that the whole may average $44 a head?

RULE.-Find the proportional quantities as in (788). Divide the given quantity by the proportional quantity of the same ingredient, and multiply each of the other proportional quantities by the quotient thus obtained.

3. A grocer having teas worth $.80, $1.20, $1.50, and $1.80 per pound, wishes to form a mixture worth $1.60 a pound, and use 20 lb. of that worth $1.50 a pound.

4. Bought 12 yd. of cloth for $30. How many yards must I buy at $34 and $1 a yard, that the average price of the whole may be $2 a yard?

5. How many acres of land worth $70 an acre must be added to a farm of 75 A., worth $100 an acre, that the average value may be $80 an acre ?