PROOF.-75 dolls. X 63=500 dolls., the same as above. 26. Multiply 320 by 63. 28. Multiply 837 by 63. 27. Multiply 277 by 663. 29. Multiply 645 by 6663. 30. What will 12 acres of land cost, at 46 dollars per acre? Operation. dolls. 46, price of 1 A. 100 Analysis.-12 acres is of 100 acres; now since 1 acre costs 46 dollars, 100 acres will cost 100 times as much, or 4600 dollars. But we wished to find the cost of only 12 acres, which is of 100 acres. Therefore of the cost of 100 acres, will obviously be the cost of 12 acres. 8)4600 dolls. 575 100 A. 66 12 A. PROOF.-46 dolls. X 12=575 dolls., the same as before. Note.-In like manner, if the multiplier is 37, 62, or 871, we may multiply by 100, and, t, or of the product will be the answer. Hence, 224. To multiply a whole number by 121, 37, 621, or 871. Annex two ciphers to the multiplicand, then take 1, 3, 4, or 7, of the number thus produced, as the case may be, and the result will be the answer required. OBS. The reason of this contraction may be seen from the fact that 12 is 37 is 1, 62 is §, and 87 is of 100. 31. Multiply 275 by 371. 32. Multiply 381 by 124. 33. Multiply 425 by 371. Ans. 103121. 34. Multiply 643 by 621. 225. To multiply a whole number by 13, 163, 166, &c. Annex as many ciphers to the multiplicand as there are integral figures in the multiplier, then of the number thus produced will be the product required. OBS. The reason of this contraction is evident from the fact that 1 is of 10; 163 is of 100; 1663 is of 1000, &c. 36. What will 16 bales of Swiss muslin cost, at 735 dollars per bale? Solution.-Annexing two ciphers to 735 dolls., it becomes 73500 dolls.; and 73500÷6=12250 dolls. Ans. 37. Multiply 767 by 1. 38. Multiply 245 by 163. 39. Multiply 489 by 16. 40. Multiply 563 by 1664. Note.-Specific rules might be added for multiplying by 11, 114, 111, 8, B31, 8331, 61, &c., but they will naturally be suggested to the inquisitive mind, from the contractions already given. DIVISION OF FRACTIONS. CASE I. 226. Dividing a fraction by a whole number. Ex. 1. If 4 yards of calico cost of a dollar, what will 1 yard past? Analysis.-1 is 1 fourth of 4; therefore 1 yard will cost 1 fourth part as much as 4 yards. And 1 fourth of 8 ninths of a dollar, is 2 ninths. Ans. of a dollar. Operation. ÷4 Ans. We divide the numerator of the fraction by 4, and the quotient 2, placed over the denominator, forms the answer required. 2. If 5 bushels of apples cost of a dollar, what will 1 bushel cost? Since we cannot divide the numerator by the divisor 5, without a remainder, we multiply the denomina tor by it, which, in effect, divides the fraction. (Art. 188.) PROOF.- dolls. X 5-dolls., the same as above. Hence, 227. To divide a fraction by a whole number. Divide the numerator by the whole number, when it can be done without a remainder; but when this cannot be done, multiply the denominator by the whole number. 3. What is the quotient of 15 divided by 5? 228. Dividing a fraction by a fraction. 10. At of a dollar a basket, how many baskets of peaches can you buy for of a dollar? Analysis.—Since of a dollar will buy 1 basket, will buy as many baskets as is contained times in contained in , 4 times. Ans. 4 baskets. of a dollar ; and is 11. At of a dollar per yard, how many yards of cloth can be bought for of a dollar? OBS. 1. Reasoning as before, of a dollar will buy as many yards, as is contained times in . But since the fractions have different denominators, it is plain we cannot divide one numerator by the other, as we did in the last example. This difficulty may be remedied by reducing the fractions to a common denominator. (Art. 200.) and First Operation. reduced to a common denominator, become 24 and 18. (Art. 200.) Now ÷4=; and -1. Ans. 1 yards. OBS. 2. It will be perceived that no use is made of the common denominator, after it is obtained. If, therefore, we invert the divisor, and then multiply the 'wo fractions together, we shall have the same result as before. Second Operation. *x(divisor inverted), or 1 yards, the same as above. QUEST.-227. How is a fraction divided by a whole number? 229. Hence, to divide a fraction by a fraction. I. If the given fractions have a common denominator, divide the numerator of the dividend by the numerator of the divisor. II. When the fractions have not a common denominator, invert the divisor, and proceed as in multiplication of fractions. (Art. 219.) OBS. 1. When two fractions have a common denominator, it is plain one numerator can be divided by the other, as well as one whole number by another; for, the parts of the two fractions are of the same denomination. 2. When the fractions do not have a common denominator, the reason that inverting the divisor and proceeding as in multiplication, will produce the true answer, is because this process, in effect, reduces the two fractions to a common denominator, and then the numerator of the dividend is divided by the numerator of the divisor. Thus, reducing the two fractions to a common denominator, we multiply the numerator of the dividend by the denominator of the divisor, and the numerator of the divisor by the denominator of the dividend; (Art. 200;) and, then dividing the former product by the latter, we have the same combination of the same numbers as in the rule above, which will consequently produce the same result. We do not multiply the two denominators together for a common denominator; for, in dividing, no use is made of a common denominator when found, therefore it is unnecessary to obtain it. (Art. 228. Obs. 2.) The object of inverting the divisor is simply for convenience in multiplying. 3. Compound fractions occurring in the divisor or dividend, must be reduced to simple ones, and mixed numbers to improper fractions. 230. The principle of dividing a fraction by a fraction may also be illustrated in the following manner. example, Thus, in the last Operation. 7÷2= Dividing the dividend by 2, the quotient is. (Art. 188.) But it is required to divide it by 1 third of two; consequently the is 3 times too small for the true quotient; therefore multiplying by 3, will give the quotient required; and X3, or 1. And 1 Ans. Note. By examination the learner will perceive that this process is precisely QUEST.-229. How is one fraction divided by another when they have a common denominator? How, when they have not common denominators? Obs. When the fractions have a common denominator, how does it appear that dividing one numerator by the other will give the true answer? When the fractions have not a common denominator, how does it appear that inverting the divisor and proceeding as in multiplication will give the true answer? What is the object of inverting the divisor? How proceed when the divisor or dividend are compound fractions, or mixed numbers? the same in effect as the preceding; for, in both cases the denominator of the dividend is multiplied by the numerator of the divisor, and the numerator of the dividend, by the denominator of the divisor. 231. The process of dividing fractions may often be contracted by canceling equal factors in the divisor and dividend; (Art. 146;) or, after the divisor is inverted, by canceling factors which are common to the numerators and denominators. (Art. 191) 18. Divide of of by 4 of 3 of 4. Operation. $1 14 11 45 23 17 11/5 Ans. rule above. Hence, For convenience we arrange the numerators, (which answer to dividends,) on the right of a perpendicular line, and the denominators, (which answer to divisors,) on the left; then canceling the factors, 2, 3, 4, and 7, which are common to both sides, (Art. 151,) we multiply the remaining factors in the numerators together, and those remaining in the denominators, as in the 232. To divide fractions by CANCELATION. Having inverted the divisor, cancel all the factors common both to the numerators and denominators, and the product of those remaining on the right of the line placed over the product of those remaining on the left, will be the answer required. OBS. 1. Before arranging the terms of the divisor for cancelation, it is always necessary to invert them, or suppose them to be inverted. 2. The reason of this contraction is evident from the principle, that if the numerator and denominator of a fraction are both divided by the same number, the value of the fraction is not altered. (Arts. 148, 191.) 19. Divide 183 by 63. Answer 3. QUEST.-232. How divide fractions by cancelation? How arrange the terms of the given fractions? Obs. What must be done to the divisor before arranging its terms? How does it appear that this contraction will give the true answer? |