SECTION XXXIII. ALLIGATION.* The operations under this rule show the method of finding the value of a mixture, when the price and quantity of each of its ingredients are given; also, to find the quantity of each ingredient, when its price is given, and it is required to unite them so as to form a mixture of a given value. Case 1. To find the value of the mixture, when the quantity and price of each of the ingredients are given. 1. Mix 15 bushels of oats, at 40 cents per bushel; 12 bushels of barley, at 60 cents; and 24 bushels of corn at 83 cents; what will the mixture be worth per bushel ? It is evident that if you find the value of the whole, and divide the sum by the number of bushels, the quotient will be the value per bushel. 2. Mix 20 pounds of tea, at 43 cents per pound; 18 lbs. at 61 cents; and 11 lbs. at 74 cents per pound; what will the mixture be worth? 3. If 41 lbs. of coffee, at 13 cents per lb. be mixed with 45 lbs. at 9 cents; and 27 lbs. at 15 cents; what will the mixture be worth per pound? Case 2. To find the quantity of each ingredient, when its price and that of the required mixture are given. 4. If I mix oats worth 2s. per bushel, with rye worth 5s., so as to make the mixture worth 3s. per bushel, in what proportion must I mix them? It is evident, that if I put in 1 bushel of oats, I gain 1 shilling. Now I must put in rye enough with this bushel of oats to lose 1 shilling. On every bushel of rye put in, I lose 2 shillings; therefore, in order to lose 1 shilling, I must put in a bushel. I must therefore put in 1 bushel of oats to a *The word Alligation signifies a tying together; and has reference to a particular way of linking numbers together, by means of which operations of this kind have been performed. The name is retained as a matter of convenience; but I have thought it best for the progress of the pupil that he should pursue a strictly analytical method in all the operations. bushel of rye. It is evident that, if I double the quantity thus found of each ingredient, the value of the mixture will be the same; or I may take any equal multiples of the quantities, as 4 bushels of oats, and 2 bushels of rye; 6 bushels of oats, and 3 bushels of rye; 20 bushels of oats, and 10 bushels of rye, &c. 5. If I mix oats, worth 2s. per bushel, with rye, worth 6s., so as to make the mixture worth 3s. per bushel; in what proportion must they be mixed? 6. Mix oats, worth 3s. per bushel, with wheat, worth 7s., so as to make the mixture worth 5s. per bushel; in what proportion must they be mixed? 7. Mix the same ingredients, at the same price, so as to make the mixture worth 6s. per bushel; in what proportion must they be mixed? 8. In what proportion must oats, worth 2s., and wheat, worth 8s., be mixed, to make the mixture worth 4s. per bushel? 9. How can you mix corn, worth 80 cents per bushel, and rye, worth 85 cents, with barley, worth 46 cents, so as to make a mixture worth 60 cents per bushel? Here you have three ingredients. First, mix barley with one of the dearer ingredients, so as to make a mixture of the required value; then mix barley with the other ingredient, and see how much you have taken of each. 10. Mix 3 sorts of tea, at 25 cents, 33 cents, and 40 cents per lb., so as to make a mixture worth 30 cents per lb. 11. Mix tea at 20 cents, with tea at 45 cents, and tea at 54 cents per lb., so as to make a mixture worth 38 cents per lb. 12. If you mix sugar, at 6 cents, 8 cents, 10 cents, and 11 cents per lb., in what quantities may they be taken, so as to make a mixture worth 9 cents per lb.? First, take two of the ingredients, one cheaper and one dearer than the mixture; form a mixture of these; then take the two remaining ingredients in the same way. 13. If three sorts of spirit, worth 60 cents, 75 cents, and 80 cents per gallon, are mixed with water costing nothing, what must be the proportion to make a mixture worth 70 cents per gallon? It is immaterial in what way you select the pairs of ingredients, provided, in each pair, one of the ingredients be cheaper and the other dearer than the required mixture. Thus a great variety of answers may be obtained whenever there is more than one pair of ingredients. In all cases, however, the correctness of the operation may be proved in the following way: Find the total value of all the ingredients; if this is equal to the value of the whole mixture at the required price, the work is right. 14. Mix 5 sorts of grain, at 25 cents, 30 cents, 33 cents, 45 cents, and 50 cents, so as to make a mixture worth 40 cents per bushel. Case 3. When the quantity of one ingredient is given. 15. Mix brandy, at 74 cents per gallon, with 24 gallons of brandy, at 1 dollar per gallon, so that the mixture may be worth 80 cents per gallon. Here you observe that the quantity of one of the ingredients is given. We will first make a mixture of the two, without regard to this circumstance. If I put in 1 gallon at 1 dollar, I lose 20 cents. For every gallon at 74 cents, which is put in, I gain 6 cents. In order to gain 20 cents, I must, therefore, put in 3 gallons. The quantities stand, then, 1 gallon at 1 dollar, 3 gallons at 80 cents. But I wish to put in 24 gallons at 1 dollar. To balance this, I must, therefore, put in 24 times 34 gallons, at 74 cents; that is 80 gallons. 16. Mix sugar, at 8 cents, 11 cents, and 12 cents, with 100 lbs. of sugar, at 7 cents, so as to make the mixture worth 10 cents per lb. Case 4. When the quantity of the required mixture is given. 17. Mix oats, at 40 cents, with corn, at 60 cents, so as to form a mixture of 100 bushels, worth 48 cents per bushel. If I put in 1 bushel, at 40 cents, I gain 8 cents; if I put in 1 bushel, at 60 cents, I lose 12 cents. To lose 8 cents, therefore, I must put in only 3 of a bushel. The quantities are, then, 1 bushel at 40 cents, of a bushel at 60 cents; making, when added, 14 bushels. But 100 bushels is the quantity required. 1005 60. Each ingredient, therefore, must be multiplied by 60; 60×1=60; 60×3=40. The quantities, then, are 60 bushels at 40 cents, and 40 bushels at 60 cents. If A owes B several sums of money to be paid at different times, he may desire to pay the whole at once, and consequently to know at what time the whole becomes due. This time is found by making an equation of the payments, multiplied by the time, as follows. 1. A owes B 200 dollars; 100 due Jan. 1, 1844; 100 due Jan. 1, 1846; he wishes to pay it all at once. At what time should he pay it? Now, on Jan. 1, 1844, A is entitled to the use of 100 dollars for 2 years longer; 100×2=200; equal to the use of 1 dollar for 200 years. If he is to pay the whole together, he must keep the 200 dollars long enough to balance that claim; 200)200(1 year, - the answer. The whole should be paid one year from Jan. 1, 1844. 2. A owes B 100 dollars, due in 6 months; 200 dollars due in 12 months; in how many months should the whole be paid together? 100 X 6 600 300: 300)3000 10 months; the answer. The above is the method usually employed, and is sufficiently exact for the necessities of business; but it gives a result a little in favor of the debtor; that is, it makes the equated time a little later than it should be. To find the exact equated time is a problem far too difficult to be used in ordinary business. Rule. Multiply each payment by the length of time before it becomes due. Divide the sum of the products by the sum of all the payments; the quotient will express the length of time in which the whole is due. 3. A owes B several sums, due at different times, as follows; $600 in 2 months, $150 in 3 months, $75 in 6 months; what is the equated time for the whole? 4. A man owes $1000; of which, 200 are to be paid in 3 months, 400 in 6 months, and the remainder in 8 months; what is the equated time for the payment of the whole? 5. If I owe $1200, one half to be paid in 3 months, one third in 6 months, and the remainder in 9 months; in what time should the whole be paid? 6. A owes B $640; 150 due in 30 days, 200 due in 60 days, and the remainder in 90 days; what is the equated time for the whole? 7. A merchant buys goods to the amount of $1800; one third to be paid in 30 days, one third in 45 days, and the remainder in 90 days; what is the equated time for the whole? 8. If I owe $1000, half to be paid in 60 days, and half in 120 days, and if I pay $100 down, what will be the equated time for the remainder? SECTION XXXV. SQUARE MEASURE. [See Section XV. Part I.] 1. There is a field in the form of a square, 15 rods on a side; how many square rods does it contain? 2. If the square be 15 rods on a side, how many square rods will it contain? 3. How many square rods are there in a square field measuring 17 rods on a side? 4. If the field measure 17 rods on a side, how many square rods will it contain? 5. What is the contents of a square field measuring 21 rods on a side? 6. What is the area of a rectangular field, its length being 64 rods, and its breadth, 12 rods? 7. There is a rectangular field, its dimensions being 24 rods, and 76 rods; what is the area? 8. How many acres are there in a rectangular field, its dimensions being 94 rods and 76 rods? 9. There is a rectangular field containing 7 acres; its length is 35 rods; what is its breadth? |