Exercises. 37. " " 38. 40. " Ans. 32 f. Ans. 198 d. Ans. £. Ans. £. 36. Convert of a pound to the fraction of a farthing, Ans. 576 f. of a shilling to the fraction of a farthing, of a guinea to the fraction of a penny, of a penny to the fraction of a pound, of a guinea to the fraction of a pound, " of a farthing to the fraction of a pound, of a pole to the fraction of an acre, " of a grain to the fraction of a lb., "of a cwt. to the fraction of a lb., " of a lb. to the fraction of a ton, 41. 42. " VIII. TO CONVERT A COMPOUND NUMBER (As 14s. 6d.) To THE FRACTION OF ANY DENOMINATION OF THE SAME KIND. RULE.-Convert the given sum into its lowest denomination, and place it as the numerator; then convert one of the proposed denomination into the same denomination as the other, and place it as the denominator: the resulting fraction may then be reduced to its lowest terms. Example.-Convert 5s. 34d. to the fraction of a pound. 5s. 34d., reduced to farthings, its lowest denomination = 253 This rule is the same in principle as the last: for 5s. 34d. = 253 farthings 253 farthings; and to change this fraction to the fraction of a pound, I divide by 4, 12, and 20; therefore, 5s. 34d. = 253 253 £. 1x4x12×20 960 = Exercises. 46. Convert 14s. 7 d. to the fraction of a pound, 23 lb. 13 oz. to the fraction of a cwt., Ans. 953 5 hr. 48 min. 48 sec. to the fraction of a day, 16s. 8d. to the fraction of a pound, IX. TO FIND THE VALUE OF A FRACTION OF A GIVEN DENOMINATION. RULE.-Reckon the numerator of the fraction as so many of the given denomination, and then divide it by the denominator, as in Compound Division. Note. The pupil ought to remember that a fraction only indicates the division of the numerator by the denominator (p. 81); when this indicated division is actually performed, the fraction is said to be valued. Example.-What is the value of of a pound? £2 7)40 Here the 2 is reckoned as £2, and divided by 7, as in Compound Division. 58. 8d. 4. Ans. Exercises. 55. Find the value of of a shilling, Ans. 8d. 70. The height of Arthur's Seat is 821 feet: if a person has ascended five-sevenths of this height, through what space must he still ascend before he arrive at the top? Ans. 234 feet. 71. A person's debts amounted to £654, 13s. 4d.; the elder of two sons agrees to pay of the amount, the second son the other g How much did each son pay? Ans. The elder pays £409, 3s. 4d.; the younger, £245, 10s. ADDITION OF VULGAR FRACTIONS. RULES.-I. When the fractions have a common denominator; add together the numerators, and under the sum write the common denominator. If this sum is an improper fraction, reduce it to a whole or a mixed number. II. When the fractions have not a common denominator; convert them to equivalent fractions having a common denominator, then add the numerators as before. III. When mixed numbers are to be added; first add the fractions, then the integers. IV. When the fractions are not of the same denomination; find their values (by Rule IX., p. 89), and add as in Compound Addition. Or, convert the fractions to equivalent ones of the same denomination; then convert the resulting fractions to a common denominator, and proceed as before. THE REASON of the rule is obvious from the following examples: Example 1.-Add together 3, 4, and 7. In this example the fractions have a common denominator, the numerators therefore express units of the same kind, and consequently can be added together. Hence ++ z = ¥ = 13. Example 2.-Find the sum of the fractions, %, and. X X X = 웃음 103 Pupil. In this example the fractions have not a common denominator, and consequently must be changed to fractions having a common denominator, that they may be numbers of the same name. Converting them, therefore, 48 2 to a common denominator, and adding the numerators together, I find the sum is equal to 103; the answer being an improper fraction, is converted into the mixed number, 2775. Example 3.-Add together 15%, 211, and 271. Or thus: I convert s. to the fraction of a pound, and find it to be equal to £; then g. to the fraction of a pound, and find it equal to £. I now change the fractions,, and, to a common denominator, then add them together, and find the sum to be £13, which, valued, gives £1, 3s. 1d., the same as before. SUBTRACTION OF VULGAR FRACTIONS. RULES.-I. When the fractions have a common denominator; subtract the less numerator from the greater, and under the difference write the common denominator. The resulting fraction may then be reduced to its lowest terms. common II. When the fractions have not a common denominator; convert them to equivalent fractions having a denominator, then subtract as before. III. When mixed numbers are given; convert the fractions, when necessary, to equivalent fractions having a common denominator, subtract, if possible, the lower numerator from the upper, and under the difference write the common denominator. Find the difference of the integers, as in Simple Subtraction. But if the lower exceeds the upper numerator, add the upper numerator and denominator together; from the sum subtract *This is merely an easy way of borrowing 1 integer and converting it to the denominator of the given fraction, on the same principle as in Compound Subtraction. The 1 is carried to the integer in the lower line for the same reason. G the lower numerator, and write the common denominator below the remainder. Then carry 1 to the integer in the lower line, and subtract the sum from the integer in the upper line. IV. When the fractions are of different denominations, find their values, then proceed as in Compound Subtraction. THE REASON of the rule is evident from the illustrations given under Simple Subtraction, and Addition of Fractions. RULE.-Multiply all the numerators of the given fractions together, for the numerator, and all the denominators together for the denominator of the product; when necessary, reduce the resulting fraction to its lowest terms, Cancel when possible. Compound and complex fractions must be changed to simple ones, and mixed numbers to their fractional form, before the rule can be applied. If a fraction be multiplied by its denominator, the product is the numerator; thus, multiplied by 13 becomes ; and cancelling by 13, the product is 7, the numerator of the fraction. 7 X 13 |