given fractions to others of the same kind, for instance to the fraction of a £, and then adding them all together thus: 53. If the fractions have the same denominator, their dif ference is equal to the difference of their numerators, written above this denominator. 7 Ex. 1. 2 5 9 this is manifest; for two quantities of any kind subtracted from 7 of the same, will leave 5 of that kind. If the fractions have different denominators, they must be reduced to other equivalent ones, having the same denominator, before the subtraction can take place. Ex. 4. Required the difference of 87 and 511. Having reduced the fractional parts of the quantities, to fractions of the same denominator, we find that the fractional part of the minuend exceeds that of the subtrahend. In order, therefore, to effect the subtraction, we add a unit, or subtracted from this, there remains 24 45 ; Now, as the minu end was increased by unity, the subtrahend must also be in creased by unity, that the difference may remain the same, (Principle 3, No. 18.) We shall therefore have 6 to subtract from 8, which leaves a remainder of 2. Hence the difference of the two given quantities is 223. 24 This example may be done in a way similar to that in which the last example of addition is solved. 54. The rule for the multiplication of a fraction by an integer has been already stated, (No. 41.) To multiply one fraction by another is to take such a part or parts of the one as the other expresses. Thus, for example, to multiply by, means that the of is to be taken twice, or, which is the same thing, that is to be multiplied by 2 and divided by 3. Hence the process is the same as that of reducing a compound fraction to a simple one. The product of by is therefore = 28 of = 5 2 X 5 10 7 3x7 21 53 = 8 1X2X9 53 2×3×2 Ex. 4. Required the price of 33 yards at 16s. 8d. per ell. Here we may find the price of a yard, and then multiply it by 3; or, we may reduce 33 yards to the fraction of an ell, and then multiply the 16s. 10d. by it. Under the impression that it is more profitable for the student to understand one example thoroughly, than to run through a multitude carelessly, we shall solve the question by the two methods alluded to. Since a yard contains 4 quarters, and an ell 5, a yard is, of course, of an ell; and, therefore, the price of a yard will be of 16s. 8d. 13s. 4d. or 13}. (No. 49.) 55. The manner of dividing a fraction by an integer has been already explained, (No. 42.) When the divisor is a fraction, the rule is to invert the terms of the divisor, and then multiply the dividend by this inverted divisor, as in Multiplication. The reason of this rule will be perceived by the following examples. 7 8 ÷2=1, (No. 42.); now, as the divisor is not 2, but, or 16' of 2, (No. 35.), will therefore be contained three times as 3 7 5 often in 1 as 2 is; hence x3=21: =1 is the quotient, 8 which agrees with the rule." 16 16 16 The quotient may be obtained in another way, though less convenient than the preceding. 16 contain just as often as 21 contains 16, which is In division of fractions there are sometimes results which wear the air of a paradox; but when the student reflects that the quotient expresses the number of times that the dividend contains the divisor, these difficulties will disappear. 2 for example, should evidently give 4 as a quotient, since is 1 and will therefore contain, 4 times. By the rule÷ 4. ===2, meaning, that 12=2, meaning, that contains 2 30 6+6x=15, that is, 6 contains, 15 times. 118 1. Reduce EXERCISES IN FRACTIONS. to its lowest terms. Ans.. 2. Reduce 9 to a fraction whose denominator shall be 5. 4. Reduce 45 to a mixed number. 5. Reduce of of to a simple fraction. Ans. 45. Ans.. Ans. 47. Ans.. Ans. 1. Ans.. Ans. 8s. 9d. 6. Reduce of a guinea to the fraction of a £. 7. Reduce 12s. 6d. to the fraction of a £. 8. Required the value of of a guinea. 9. Reduce,, and to equivalent fractions having a Ans. 1, 1, 11. common denominator. 10. Reduce,, and to equivalent fractions having the least common denominator. Ans. 1, 1, 1. 180 224 18 21 11. Add together the fractions 1, §, and 11. Ans. 21, 198 12. Add 17 of a pound, § of a guinea, and of a shilling together. Ans. 1, 9s. 9 d. 13. Add of 7 and of 8 together. 23. From a piece of cloth, a merchant cut off, and then of the whole; after which there remained 35 yards. Required the length of the original piece. Answ. 100 yds. 24. A can do a piece of work in 8 days, and B in 10 days. In what time will A and B, working together, do it? Ans. 4 days, 25. A, B, and C can do a piece of work in 4 days. A by himself, can do it in 12 days, and B in 15; in what time will C do it? Ans. 10 days. 26. After paying away of my money, and then of the remainder, I had 18 guineas left; how much money had I at first? Ans. 30 guineas. 27. A, B, and C have a joint stock concern, of which belong to A, to B, and the remaining part to C. C having occasion to sell out, sold of his share to A, and the remainder to B. What fraction of the whole will now belong to A and B respectively? Ans. A 1, and B 11. CONTINUED FRACTIONS. 56. The object of Continued Fractions is to find such approximate values of an irreducible fraction, as shall be expressed in less terms. The nature and properties of these fractions will be most clearly unfolded by an example. Suppose that we have the fraction, which, though it consists of several figures, does not admit of being reduced to lower terms. The value of this fraction will not be altered, if both numerator and denominator are divided by the same quantity. Let the divisor be the numerator 216. The result of the division will be 1 The 4216 23 |