and 100 cts. per pound; how many pounds of each must he use with 25 lbs. of that worth 40 cts. to form a mixture worth 88 cts.? 25 lbs. lb. at 40 cts.=1200 times the proportional quantities. 150 lbs. ANALYSIS.-1. Arrange the prices and find the proportional quantities as in §308. They are lb., lb., 13 lb., 1 lb., and lb. 38 2. If the proportional quantity for the mixture at 40 cts. is lb., as many times the other proportional quantities will be required in the mixture containing 25 lbs. at 40 cts. as pound is contained times in 25 lbs., which is 1200 times. 38 of a 3. 1200 times lb.=311 lbs.; 1200 times lb.=92,4 lbs. ; 1200 times lb.=2574 lbs.; 1200 times lb.=150 lbs. Therefore, according to the condition of the problem there may be used in the mixture with 25 lbs. at 40 cts., 311 lbs. at 50 cts., 92 lbs. at 75 cts., 2574 lbs. at 95 cts., and 150 lbs. at 100 cts. RULE.-I. Find the proportional quantities as in § 308. II. Divide the given quantity by the proportional quantity of the required price, and multiply the quotient by each of the other proportional quantities. EXAMPLES FOR PRACTICE. 7. I have sugar at 11 cts., 13 cts., 15 cts., and 17 cts. per pound; how much of each must I mix with 10 pounds of the 13 cent sugar to make a mixture worth 14 cts. per pound? *NOTE.-The difference between the total loss and the total gain is 1 ct. which added equally to the losses makes 1 cts. each. 8. A farmer wishes to mix 20 bushels of corn worth 50 cts. per bushel, with rye worth 75 cts., barley worth 30 cts., and oats worth 38 cts.; how much of each must he take to make a mixture worth 45 cents? 9. A merchant has molasses worth 40 cts., 50 cts., 60 cts., and 70 cts. per gallon; how much of each must he mix with 10 gallons of the 50 cent molasses to make a mixture worth 55 cts.? To make one worth 62 cents? QUESTIONS.-What is alligation medial? (307.) What is alligation alternate? (308.) Give the rule for finding the mean price, when the price or quality of a number of simples is given? (309.) Give the rule for finding the proportional quantity of a number of simples, when the quantity of one is limited. (310.) LESSON IV. 311. To find what quantity of each simple is required to form a mixture at a given price when the quantity of the mixture is limited. I have sugar at 10 cts., 12 cts., 14 cts., and 15 cts. per pound, and wish to make a mixture of 100 pounds worth 11 cts. per pound; how much of each must I use? ANALYSIS.-1. Arrange the prices, and find the proportional quantities as in § 308. They are 3 lbs., 1 lb., lb., and lb. 2. Since the sum of the proportional quantities makes but 4,7 lbs. of the required quantity of the mixture, it will take as *NOTE.-The difference between the loss and gain is 2 cents, which added to the gain makes 3 cents. many times each of the proportional quantities as 47 lbs. are contained times in 100 lbs., which are 21. 3. 21 times 3 lbs. are 655 lbs. ; 21 times 1 lb. are 21 lbs. ; 21 times lb. are 7,3 lbs. ; 21 times lb. are 5,5 lbs. Therefore, according to the condition of the question 65,5 lbs. at 10 cts., 21 lbs. at 12 cts., 73 lbs. at 14 cts., and 5,5 lbs. at 15 cts. are required to make a mixture of 100 lbs. worth 11 cents per pound. RULE.-I. Find the proportional quantities as in § 308. II. Divide the given quantity by the sum of the proportional quantities, and multiply each of the proportional quantities by the quotient thus obtained. EXAMPLES FOR PRACTICE. 10. A farmer has a box that will hold 50 bushels: he has corn worth 75 cts., oats worth 37 cts., and barley worth 25 cts.; how much must he take of each, to fill the box with a mixture worth 30 cents? 11. I have a hogshead holding 65 gallons: I wish to fill it with brandy worth $2 per gallon, alcohol worth $1.50 per gallon, and water, so that the mixture may be worth $1.25; how much of each must I take? 12. A merchant has a barrel which will hold 200 pounds of sugar, and he wishes to fill it with sugars worth respect-. ively 10 cts., 12 cts., 14 cts., and 20 cts. per pound, so that the mixture may be worth 15 cts.; how much of each must he take? What is QUESTIONS.-Give the rule for finding the proportional quantity of each simple when the entire quantity is limited. (311.) Addition? (31.) Subtraction? (39.) Multiplication? (46.) ion? (54.) Divis SECTION XIII. LESSON I. RATIO. 312. Ratio is the relation in respect to magnitude which one number has to another of the same kind. 313. The Terms of a ratio are antecedent and consequent, and when taken together they are called a coup`let. 314. The Antecedent is the first term of the couplet, and is the standard by which the other number is measured. 315. The Consequent is the second term of the couplet, and is the quantity measured. 316. Ratio may be expressed in two ways: 1st. By two dots between the terms of the couplet; as, 3:12. 2nd. In the form of a fraction; as, 2, or 12 divided by 3. (a.) NOTE.-All numbers are finally compared with unity as a standard, hence when we say that the ratio 4:8 is 2, it is understood that the ratio 4:8 is equal to the ratio 1:2; and the ratio 4:3 is 3 or 4:3 is the same ratio as 1:1. MENTAL EXERCISES. What part of 12 is 4? ILLUSTRATION.-4 is of 12; or the ratio 12:4==}; that is, 12 has the same ratio to 4 that 1 has to }. 1. What part of 20 is 2? is 5? is 7? is 8? is 6? is 9? 2. What part of 24 is 3? is 9? is 24? is 30? 4. What is the ratio of 6 to 7? ILLUSTRATION.-The ratio 6:7=,=1}; or the ratio 6:7 is 340 to 8} is =4%; or the ILLUSTRATION.-The ratio of ratio : 3 is the same as the ratio 1:44. 8. What is the ratio of to? of to? of to ? of 3 to? 9. What is the ratio of 5 qts. to 3 pts.? of 1 gal. to 3 qts. 1 pt.? 10. What is the ratio of 7 oz. to 3 oz.?. of 4 lbs. to 1 lb. 6 oz.? 11. If the consequent is 4 and the ratio is 3, what is the antecedent? ANALYSIS.-If the ratio is 3, or 1:3, then, as the antecedent is as large as the consequent 3, the antecedent sought must be as much as the given consequent, and 3 of 4 is 13, which is the antecedent required. 12. If the consequent is 3 the ratio 9, what is the antecedent? 13. If the antecedent is 5 and the ratio 3, what is the consequent? QUESTIONS.-What is the equation of payments? (292.) What is the term of credit? (293.) What is the equated time? (294.) What is an account? (295.) What is a balance? (296.) Give the rule for finding the equated time, when the items have the same date. (297., a.) Give the rule, when the items have different dates. (298., a.) |