Note. When the cube root of a vulgar fraction is required, reduce it to its lowest terms and extract the cube root of the numerator for a new numerator, and the cube root of the denominator for a new denomi nator; when the numerator and denominator or both have remainders reduce the fraction to a decimal, and extract the cube root. 2. When a mixed number is given, reduce the fraction to a decimal and extract the cube root. Questions. How do you prepare a given sum for the extraction of the cube root? What is to be noted when a whole number and decimal, or decimal only is given? What is next to be done after dividing the given number into periods of three figures ? When you have found the greatest root contained in the left hand period and placed its cube under the first period, what is to be done next? How do you proceed after having squared the as certained root and multiplied it by three? When you have tried how often the defective divisor is contained in the dividend omitting the left hand period, &c. what is to be done next? When you have multiplied the last figure by all the figures of the ascertained root and by 30, what is next to be done? When the cube root of a vulgar fraction is required, how do you find it? When the cube root of a mixed number is required, how do you find it? Defective div. and square of 6 67536)421416 +2700 complete divisor 70236 421516 2. What is the cube root of 7532641? Ans. 196.02+ 3. What is the cube root of 12.1138475? Ans. 2.299+ 4. What is the cube root of 5382674? Ans. 175.3+ 5. What is the cube root of .37862135? Ans. .723+ 6. What is the cube root of 46.295363543 ? Ans. 3.5904 Ans. .585+ Ans. 3.96+ ALLIGATION. Alligation is a rule which enables us to resolve questions concerning the mixture of several simples into one compound quantity. Case 1. When the quantity and rates of the simples are given to find the rate of a mixture compounded of these simples. Rule. 1. Find the value of each quantity according to their respective costs. 2. As the whole of the quantities is to one of its parts, So is the total amount of their value ი Case 2. When the prices of several simples are given to find how much of each at their respective rates will be required to make a mixture at any proposed price. Rule. 1. Place all the rates of the simples under each other, and link each rate which is less than the mean rate with one or more that is greater. 2. Take the difference between each rate and the mean price placed opposite the respective rate with which it is linked which will be the quantity required. Note. If all the given prices are greater or less than the mean price they must be linked to a cipher. Different modes of linking produce different answers. Case 3. When the prices of all the simples, the quantity of one of them and the mean price of the whole mixture are given to find the quantities of the rest. Rule. 1. Place the mean rate and the several prices, link them and take their differences as in the preceding case. 2. As the difference of the same name with the quantity given, Is to the differences respectively, To the several required quantities. Case 4. When the prices of the several simples, the quantity to be compounded, and the mean price are given to find the quantity of each simple. Rule. 1. Link the several prices, and take their differences as before, 2. As the sum of the differences What is Alligation? Questions.. When the quantities and the rates of the simples are given to find the rate of a mixture compounded of these simples, how do you work? By what rule do you work when the price of several simples are given to find how much of each at their respective rates will be required to make a mixture at any proposed price? How do you proceed when the price of all the simples the quantity of one of them and the mean price of the whole are given to find the quantities of the rest? How do you proceed when the prices of the several simples the quantity to be compounded, and the mean price are given to find the quantity of each simple? Case 1. 1. If a person have 4 lb. of tea at 90 cents per lb., 81b. at 75 cents, and 67b. at 110 cents, to be mixed together, what will a pound of the mixture be worth? 2. A grocer has 2 Cwt. of coffee at 25 dollars per Cwt. 4 Cwt. at 20 dollars 50 cents per Cwt. and 7 Cwt at 18 dollars 62 cents per Cwt. which he will mix together, what will one Cwt. of this mixture be worth? Ans. $20.18. Case 2. 1. What quantity of Sugar at 11 cents per lb., at 6 cents per lb. and at 8 cents per lb. will make a mixture worth 7 cents per b.? Ans. 1lb. at 11 cents, 17b. at 2. How much wheat at 110 cents per bushel, rye at 86 cents per bushel, oats at 34 cents per bushel and barley at 42 cents per bushel, will it take to make a composition worth 50 cents per bushel? Ans. 8 bu. at 110 cts. 16bu. at 86 cents, 60bu. at 42 cts. 36bu. an 34 cts. Case 3. 1. What quantity of coffee at 20 cents, and at 16 cents per lb. must be mixed with 35lb. at 14 cents to make a mixture worth 18ents per lb.? 14 Mean rate 18 16 20 2 Then as 2:35::2: 35 at 16 2. How much tea at 86 cents, at 94 cents, and at 105 cents per lb. ought to be mixed with 6 lb. at 75 cents per Ib. for a mixture to sell at 92 cents per lb.? Ans. 187b. at 105 cents, 51b. at 94 cents, 39b. at 86 cts. Case 4. 1. A grocer has 3 sorts of sugar, viz. 10 cents. 11 cents, and 8 cents per pound, and he would have a composition of 40lb. worth 9 cents per lb.. how much of each sort must he take? Ans. Sum of differences 5 5 3: 40: 24 at 8 2. A vintner has wine at 130 cents, at 160 cents, and at 180 cents per gallon, and he would have 32 gallons worth 145 cents per gallon, I demand how much of each sort he must have? Ans. 20 gal. at 130 cents, 6 gal. at 160 cents, and 6 gal. at 180 cents. |