CASE 11. 175. When a given quantity of one of the simples is to be taken. RULE. I. Find the proportional quantities of the simples as in Case I. II. Then say, as the number opposite the simple whose quantity is given, is to the given quantity, so is either proportional quantity to the part of its simple to be taken. EXAMPLES. 1. How much wine at 5s, at 5s 6d, and 6s per gallon, must be mixed with 4 gallons at 4s per gallon, so that the mixture shall be worth 5s 4d per gallon? Ans. 1gal. at 5s, 2 at 5s 6d, and 8 at 6s. PROOF BY ALLEGATION MEDIAL. 4 gal. at 4s per gal. 1 2 8 15 192d. 15)960(64d. price of mixture. 2. A farmer would mix 14 bushels of wheat, at $1,20 per bushel, with rye at 72cts., barley at 48cts., and oats at 36cts. how much must be taken of each sort to make the mixture worth 64 cents per bushel? 14bu. of wheat; 8bu. of rye; 4bu. Ans. of barley; and 28bu. of oats. 3. There is a mixture made of wheat at 4s per bushel, rye at 3s, barley at 2s, with 12 bushels of oats at 18d per bushel how much is taken of each sort when the mixture is worth 3s 6d? 96bu. of wheat; 12bu. of rye; Ans. {12bu. of barley; and 12bu. of oats. 4. A distiller would mix 40gal. of French brandy at 12s per gallon, with English at 7s and spirits at 4s per gallon what quantity must be taken of each sort, that the mixture may be afforded at 8s per gallon? : Ans. {40gal, French; 32gal. English; CASE III. § 176. When the quantity of the compound is given as well as the price. RULE. I. Find the proportional quantities as in Case I. II. Then say, as the sum of the proportional quantities, is to the given quantity, so is each proportional quantity, to the part to be taken of each. EXAMPLES. 2. A grocer has four sorts of sugar worth 12d, 10d, 6d, and 4d per pound; he would make a mixture of 1447b. worth 8d per pound: what quantity must be taken of each sort? 8s 2. A grocer having four sorts of tea worth 5s, 6s, and 9s per lb. wishes a mixture of 871b. worth 7s per lb.: how much must be taken of each sort? 3. A vintner has four sorts of wine, viz., white wine at 4s per gallon, Flemish at 6s per gallon, Malaga at 8s per gallon, and Canary at 10s per gallon: he would make a mixture of 60 gallons to be worth 5s per gallon: what quantity must be taken of each? {45gal. Ans. 45gal. of white wine; 5gal. of Flemish; 5gal. of Malaga; and 5gal. of Canary. 4. A silver-smith has four sorts of gold, viz; of 24 carats fine, of 22 carats fine, of 20 carats fine, and of 15 carats fine he would make a mixture of 42oz. of 17 carats fine: how much must be taken of each sort? S 4 of 24; 4 of 22; 4 of 20; Ans. and 30 of 15 carats fine. Q. How do you find the proportional parts when the price only is given? What is the rule when a given quantity of one of the simples is to be taken? What is the rule when the quantity of the compound, as well as the price, is given? INVOLUTION. $177. If a number be multiplied by itself, the product is called the second power, or square of that number. Thus 4x4 16: the number 16 is the 2nd power or square of 4. If a number be multiplied by itself, and the product arising be again multiplied by the number, the second product is called the 3rd power, or cube of the number. Thus 3x3x3=27: the number 27 is the 3rd power, or cube of 3. The term power designates the product arising from multiplying a number by itself a certain number of times, and the number multiplied is called the root. Thus, in the first example above, 4 is the root, and 16 the square or 2nd power of 4. In the 2nd example, 3 is the root, and 27 the 3rd power or cube of 3. The first power of a number is the number itself. Q. If a number be multiplied by itself once, what is the product called? If it be multiplied by itself twice, what is the product called? What does the term power mean? What is the root? 178. Involution teaches the method of finding the powers of numbers. The number which designates the power to which the root is to be raised, is called the index or exponent of the power. It is generally written on the right, and a little above the root. Thus 42 expresses the second power of 4, or that 4 is to be multiplied by itself once: hence, 44x4 16. For the same reason 33 denotes that 3 is to be raised to the 3rd power, or cubed: hence 33=3×3×3=27: we may therefore write, 4-4 42 4x4 161 4=4X4X4=64 the 1st power of 4. the 2nd power of 4. the 3rd power of 4. 4-4x4x4x4=256 the 4th power of 4. 45 4x4x4x4x4=1024 the 5th power of 4. Q. What is Involution? What is the number called which desig‐ nates the power? Where is it written? Hence, to raise a number to any power, we have the following RULE. Multiply the number continually by itself as many times less 1 as there are units in the exponent: the last product will be the power sought. 5. What is the 5th power of 9? 6. What is the cube of 1? 7. What is the square of ? 8. What is the cube of,1 ? 9. What is the cube of 3? 10. What is the square of,01? 11. What is the square of 2,04? 12. What is the 5th power of 10? 13. What is the cube of 24? Q. How do you raise a number to any power? Ans. 59049. Ans. Ans. 4,1616. Ans. 100000. EVOLUTION. § 179. We have seen (§ 178), that Involution teaches how to find the power when the root is given. Evolution is the reverse of Involution: it teaches how to find the root when the power is known. The root is that number which being multiplied by itself a certain number of times will produce the given power. The square root of a number is that number which being multiplied by itself once will produce the given number. The cube root of a number is that number which being multiplied by itself twice will produce the given number. For example, 6 is the square root of 36; because 6×6 =36; and 3 is the cube root of 27, because 3×3×3= 27. The sign placed before a number denotes that its square root is to be extracted. Thus, √36-6. The sign is called the sign of the square root. When we wish to express that the cube root is to be extracted, we place the figure 3 over the sign of the square root: thus, V8 =2 and 27=3. Q. What is Evolution? What does it teach? What is the square root of a number? What is the cube root of a number? Make the sign denoting the square root? How do you denote the cube root? |