ALLIGATION. 475. Alligation, or Average, treats of the mixing of dif ferent ingredients. 476. Alligation Medial is the process of determining the average or mean value of given quantities of different values. 477. Alligation Alternate is the process of determining what quantities of different values may be so combined that the mixture shall be of a given value. NOTE. The word alligation means a tying together, and is applied to these processes because, in the solutions of many examples, the amounts or prices of articles are linked or tied together. Average is perhaps the better name to use, as it applies to all the examples. ALLIGATION MEDIAL. 478. ILL. EX. Let it be required to mix 10 lbs. of sugar at 7 cents per lb. with 7 lbs. at 9 cents, and 8 lbs. at 11 cents; what will be the value of the mixture? worth of 221 cents 82 cents, Ans. Hence we deduce the following RULE. To find the mean value of given quantities of different values: Divide the sum of the values of the several quantities by the sum of the quantities. EXAMPLES. 1. If 10 lbs. of raisins worth 10 cents per lb. be mixed with 4 lbs. worth 15 cents per lb., what is the value of the mixture per pound? Ans. 113 cents. 2. There are in a certain school, 10 pupils 14 years old; 9 pupils 12 years old; 5, 11 years; 8, 9 years, and 17, 10 years old; what is their average age? 3. A family spent, during the year, as follows: in January $89.75, in February $70.16, in March $85.32, in April $90.21, in May $87.00, in June $66.14, in July $69.42, in August $72.68, in September $80.65, in October $90.45, in November $98.54, in December $109.63; what was their average expense per month? 4. In Philadelphia, during the year 1861, rain or snow fell as follows in January on 13 days, in February on 9 days, in March on 9, in April on 9, in May on 13, in June on 15, in July on 14, in August on 12, in September on 6, in October on 10, in November on 11, in December on 4; what was the average number of days per month when rain or snow fell? 5. In Massachusetts, during the year 1850, the value of home manufactures was $205,333. During the year 1860, it was $245,886. What was the average rate of increase per year during the 10 years? 6. A flour merchant sold 50 bbls. flour at $7.50 per bbl., 60 bbls. at $9.00 per bbl., 25 bbls. at $8.50, 40 at $8.75, and 100 at $9.50; what did his sales average per barrel? 7. A baker made wedding-cake of the following ingredients: 5 lbs. flour worth 5 cents per lb., 5 lbs. sugar at 11 cents per lb., 5 lbs. of butter at 22 cents per lb., 6 lbs. raisins at 17 cents per lb., 12 lbs. currants at 20 cents per lb., 2 lbs. citron at 50 cents per lb., 50 eggs, 1 lbs. to the dozen, 18 cents per dozen, pint wine at 37 cents per pint, 3 oz. cinnamon at 56 cents per lb., 3 oz. nutmegs at $1.00 per lb., 14 oz. mace at $1.00 per lb. ALlowing $2.00 for labor and fuel, lb. for the weight of the wine, and 1 oz. in every lb. for loss of weight in baking, what was the cost of the cake lb.? per Ans. $.241488. ALLIGATION ALTERNATE. 479. ILL. Ex. A merchant has teas of the following values per lb., 42, 68, 75, and 84 cents, with which he wishes to make a mixture worth 70 cents per pound. How many pounds of each kind shall he take? cents per We first compare the various prices of the tea with the price of the mixture. If that which is worth 42 cents is sold at 70 cents, there is a gain of 28 cents on one pound, which we indicate by writing +28 opposite 42. In the same way we find there is a gain of 2 cents per lb. on the 68-cent tea, a loss of 5 lb. on the 75-cent tea, and a loss of 14 cents per lb. on the 84cent tea. We indicate the gain and losses by their proper signs, and proceed to take, two by two, such kinds of tea and of such quantities that the gains shall balance the losses. Comparing the first with the fourth, we find that the gain on 1 lb. of the first equals the loss on 2 lbs. of the fourth. We also find that the gain on 5 lbs. of the second equals the loss on 2 lbs. of the third. We therefore take 1 lb. of the first, of the second, 2 of the third, and 2 of the fourth; or, we may take any quantities of the first and fourth that are in the ratio of 1 to 2, and of the second and third that are in the ratio of 5 to 2. Instead of comparing the first with the fourth, and the second with the third, we may compare the first and third and the second and fourth together, thus:— Other comparisons might be made, and thus an indefinite number of answers be obtained. But it is best to compare those gains and losses together that have the greatest common factors; for in such comparisons, whatever factors are common can be disregarded, and the remaining factors of each gain or loss will show the required quantity of the other article. From the above operations we derive the following RULE. To find what quantities of different values shall be taken to make a mixture of a given value: Write the different values in a column with the medium value at the left. Compare each given value with the value of the mixture. Write what it requires to equal that of the mixture in a column at the right with the sign + prefixed, or what it exceeds that of the mixture with the sign Take such quantities of each ingredient that the gains and losses shall be equal. 480. PROOF. Examples in Alligation Alternate may be proved by finding the mean value of the several ingredients as given in the answer, and comparing it with the given mean value. 481. The following simple method of solving this class of examples is sometimes given, which is preferable whenever it does not give fractional portions of the given quantities. ILL. EX. Let it be required to mix sugar at 8, 9, 11, 13, and 15 cents per lb., that the mixture may be worth 10 cents per lb. First answer, 7 lbs. at 9 cents, and 1 lb. of each of the others. 482. EXAMPLES. 1. How shall corn at 50 cents a bushel, be mixed with grain at 80 cents a bushel, that the mixture may be worth 75 cents per bushel? Ans. 1 bu. at $.50 to 5 bu. at $.80. 2. How shall oil at 80, 95, and $1.50 per gallon, be propor tioned that the mixture may be worth $1.00 per gallon? 3. How shall tea at 62, 75, 68, 90, and 98 cents, be proportioned that the mixture may be worth 80 cents per lb.? 4. A grocer makes a mixture of syrup, worth 62 cents per gall., from syrups worth 45, 60, 75, and 80 cents per gall.; how many gallons of each may he use? 5. A grocer has cider at 28 and 30 cents per gall., which he wishes to mix with vinegar at 27 cents per gall., and water, so that the mixture may be worth 25 cents per gall.; what proportions may he use? Ans. 1 gal. of each of the other ingredients to gal. water, etc. 483. When one of the quantities is limited, find the entire gain or loss on that quantity, and take such quantities of the other ingredients that their gains and losses shall balance each other and the gain or loss on the limited quantity. When more than one quantity is limited, find the resulting loss or gain from taking the limited quantities, and balance as before. ILL. EX. How much tea at 60, 75, and 87 cents per lb., may be mixed with 30 lbs. of tea at 95 cents per lb., that the mixture shall be worth 85 cents per lb.? Ans. 12 lbs. at 60 cents, 1 at 75 cents, and 5 at 87 cents. EXAMPLES. 6. How many lbs. split peas at 5 cents per lb., must be put with 40 lbs. coffee at 21 cents per lb., that the mixture shall be worth 14 cents per lb.? Ans. 314 lbs. 7. A goldsmith has gold 16 carats fine, which he wishes to mix with 4 oz. gold 17 carats fine, 5 oz. 20 carats fine, 2 oz. 22 carats fine, and 3 oz. 24 carats fine, that the mixture may be 18 carats fine; how many oz. of it shall he use? |