so as to sell the compound for 8d. per lb. What quantity of each kind must he take? Ans. 2 lb. at 4d. 2 lb. at 6d. and 6 lb. at 10d. (3) How much tea at 16s. 14s. 9s. and 8s. per lb. will compose a mixture worth 10s. per lb. ? Ans. 1 lb. at 16s. 2 lb. at 14s. 6 lb. at 9s. and 4 lb. at 8s. (4) A farmer would mix as much barley, at 3s. 6d. per bushel, rye at 4s. per bushel, and oats at 2s. per bushel, as will make a mixture worth 2s. 6d. per bushel. How much of each sort? Ans. 6 b. of barley, 6 of rye, and 30 of oats. (5) A tobacconist would mix tobacco at 2s., 1s. 6d., and 1s. 3d. per lb. so that the compound may be worth 1s. 8d. per lb. What quantity of each sort must he take? Ans. 7 lb. at 2s. 4 lb. at 1s. 6d. and 4 lb. at 1s. 3d. CASE 3. ALLIGATION PARTIAL. This is similar to Case 2, except that one of the quantities is limited. RULE. Link the prices, and place the differences as before. Then, as the difference opposite to that whose quantity is given, is to each other difference; so is the given quantity to each required quantity. (1) A tobacconist intends to mix 20 lb. of tobacco at 15d. per lb. with others at 16d. 18d. and 22d. per lb. How many pounds of each sort must he take to make one pound of the mixture worth 17d. ?* (2) How much coffee, at 3s. at 2s. and at 1s. 6d. per lb. with 20 lb. at 5s. will make a mixture worth 2s. 8d. per lb. ? Ans. 35 lb. at 3s. 70 lb. at 2s. and 10 lb. at 1s. 6d. (3) A distiller would mix 10 gallons of French brandy, at 48s. per gallon, with British at 28s. and spirits at 16s. per gallon. What quantity of each sort must he take to afford it for 32s. per gallon? Ans. 8 British, and 8 spirits. (4) What quantity of teas at 12s. 10s. and 6s. must be mixed with 20 b. at 4s. per lb. that the mixture may be worth 8s. per Hb.? Ans. 10 lb. at 6s. 10 lb. at 10s. 20 lb. at 12s. CASE 4. ALLIGATION TOTAL. This is also similar to Case 2, except that the whole quantity of the compound is limited. RULE. Link the prices, and place the differences as before. Then, As the sum of the differences, is to each particular difference; so is the quantity given, to each required quantity. (1) A grocer has four sorts of sugar at 12d. 10d. 6d. and 4d. per lb. and would make a composition of 144 lb. worth 8d. per lb. What quantity of each sort must he take ?* (2) A grocer having 4 sorts of tea at 5s. 6s. 8s. and 9s. per lb. would have a composition of 87 lb. worth 7s. per lb. What quantity must there be of each sort? Ans. 141⁄2 lb. of 5s. 29 lb. of 6s. 29 lb. of 8s. and 141⁄2 lb. of 9s. (3) A vintner having 4 sorts of wine, viz. white wine at 16s. per gallon, Flemish at 24s. per gallon, Malaga at 32s. per gallon, and Canary at 40s. per gallon; would make a mixture of 60 gallons worth 20s. per gallon. What quantity of each sort must he take? Ans. 45 gallons of white wine, 5 of Flemish, 5 of Malaga, and 5 of Canary. (4) A jeweller would melt together four sorts of gold, of 24, 22, 20, and 15 carats fine, so as to produce a compound of 42 oz. of 17 carats fine. How much must he take of each sort? Ans. 4 oz. of 24, 4 oz. of 22, 4 oz. of 20, and 30 oz. of 15 carats fine. COMPARISON OF WEIGHTS AND MEASURES. THIS is merely an application of the Rule of Proportion. (1) If 50 Dutch pence be worth 65 French pence, how many Dutch pence are equal to 350 French pence ?+ (2) If 12 yards at London make 8 ells at Paris, how many ells at Paris, will make 64 yards at London? Ans. 423. Answer. Proof. lb. lb. 4 48 at 12d. = 576 As 12: 4: : 144: 48 24 at 10d. = 240 As 12: 2 :: 144 : 24 12. 810 2 (3) If 30 lb. at London make 28 lb. at Amsterdam, how many lb. at London will be equal to 350 lb. at Amsterdam ? Ans. 375. (4) If 95 lb. Flemish make 100 lb. English, how many lb. English are equal to 275 lb. Flemish ? Ans. 289. PERMUTATION Is the changing or varying of the order of things. RULE. Multiply the numbers 1, 2, 3, 4, &c. continually together, to the given number of terms, and the last product will be the answer. (1) How many changes may be rung upon 12 bells; and in what time would they be rung, at the rate of 10 changes in a minute, and reckoning the year to contain 365 days, 6 hours? 1 × 2 × 3 × 4×5×6×7×8×9×10×11×12 = 479001600 changes, which ÷ 10= 47900160 minutes = 91 years, 26 days, 6 hours. (2) A young scholar, coming to town for the convenience of a good library, made a bargain with the person with whom he lodged, to give him £40. for his board and lodging, during so long a time as he could place the family (consisting of 6 persons besides himself) in different positions, every day at dinner. How long might he stay for his £40. ? Ans. 5040 days. VULGAR FRACTIONS. DEFINITIONS. 1. A Fraction is a part or parts of a unit, or of any whole number or quantity; and is expressed by two numbers, called the terms, with a line between them. 2. The upper term is called the Numerator, and the lower term, the Denominator. The Denominator shows into how many equal parts unity is divided; and the Numerator is the number of those equal parts signified by the Fraction.* 3. Every Fraction may be understood to represent Division; the Numerator being the dividend, and the Denominator, the divisor.† Fractions are distinguished as follows: 4. A SIMPLE FRACTION Consists of one numerator and one denominator: as, 1, &c. 5. A COMPOUND FRACTION, or fraction of a fraction, consists of two or more fractions connected by the word of: as of of, &c. This properly denotes the product of the several fractions. 6. A PROPER FRACTION, is one which has the numerator less than the denominator: as, 2, 1, 11, &c. ‡ 4' 7. An IMPROPER FRACTION is one which has the numerator either equal to, or greater than the denominator: as, 4, 4, 14, &c.‡ 5 8 8. A MIXED NUMBER is composed of a whole number and a fraction, as 12, 174, 874, &c. 9. A COMPLEX FRACTION has a fractional numerator or denominator: but this denotes Division of Fractions. Thus, two-thirds divided by five-sixths, eight divided by one and two thirds. 8 In the fraction five-twelfils (5), the Denominator 12 shows that the unit or whole quantity is supposed to be divided into 12 equal parts: so that if it be one shilling, each part will be one-twelfth of ls. or one penny. The Numerator shows that 5 is the number of those twelfth parts intended to be taken: so of a shilling are the same as 5 pence; of a foot, the same as 5 inches. + The fraction signifies not only of a unit, but 5 units divided into 12 parts, or a twelfth part of 5: and it is obvious that five t-elfth parts of one shilling (or five pence) is the same as one twelfth part of five shillings. This mode of considering Fractions removes many of the student's difficulties. A proper fraction is always less than unity: thus & wants one fourth, and wants one-twelfth of being equal to 1. But an improper fraction is equal to unity when the terms are equal, and greater than unity when the numerator is the greater. Thus, or, or, is each 1; and §=14, f=2, §}=3,5%. = 10. A COMMON MEASURE (or DIVISOR) is a number that will exactly divide both the terms. When it is the greatest number by which they are both divisible, it is called the GREATEST COMMON MEASURE. NOTE. A prime number has no factor, except itself and unity. A multiple signifies any product of a number; and is therefore divisible by the number of which it is a multiple: thus 14, 21, 28, &c. are multiples of seven. Also 14 is a common multiple of 2 and 7; 21, of 3 and 7, &c. REDUCTION Is the method of changing the form of fractional numbers or quantities, without altering the value. Case 1. To reduce a fraction to its lowest terms. RULE. Divide both the terms by any common measure that can be discovered by inspection; which will produce an equivalent fraction in lower terms. Treat the new fraction in a similar manner; repeating the operation till the lowest terms are obtained.* When the object cannot be accomplished by this process, divide the greater term by the less, and that divisor by the remainder, and so on till nothing remains. The last divisor will be the greatest common measure; by which divide both terms of the fraction, and the quotients will be the lowest terms. (4) Reduce to the least terms. (6) Reduce 19 to its lowest terms. (5) Abbreviate & as much as possible. This first method of abbreviating fractions is, when practicable, always to be preferred: and in the application of it the following observations will be found exceedingly useful. An even number is divisible by 2. A number is divisible by 4, when the tens and units are so; and by 8, when the hundreds, tens, and units are divisible by 8. A number is a multiple of 3, or of 9, when the sum of its digits is a multiple of 3, or of 9. A 5 or a 0 in the units' place, admits of division by 5; one cipher admits of division by 10, two, by 100, &c. |