If the numerator is greater than the denominator, the value is greater than one. Thus, 42, 5=1. If the numerator is less than the denominator, the value is less than one. Thus, 1 third of 1, 4-4 fifths of 1. 193. Fractions may be added, subtracted, multiplied, and divided, as well as whole numbers. But, in order to perform these operations, it is often necessary to make certain changes in the terms of the fractions. 8 OBS. It is evident that any changes may be made in the terms of a fraction, which do not alter the quotient of the numerator divided by the denominator; for, if the quotient is not altered, the value remains the same. Thus, the terms of the fraction may be changed into 2, 4, 16, &c., without altering its value ; for in each case the quotient of the numerator divided by the denominator is 2. Hence, for any given fraction, we may substitute any other fraction, which will give the same quotient. REDUCTION OF FRACTIONS. 194. The process of changing the terms of a fraction into others, without altering its value, is called REDUCTION OF FRAC obtain, whose terms are the lowest to which the given fraction If we divide both terms by 10, their greatest common divisor, (Art. 170,) the given fraction will be reduced to its lowest Hence, terms by a single division. QUEST.-When the numerator is larger than the denominator, what? When smaller, what? Obs. What changes may be made in the terms of a fraction? 194. What is meant by reduction of fractions? 195. How is a fraction reduced to its lowest terms? 195. To reduce a fraction to its lowest terms. Divide the numerator and denominator by any number whic will divide them both without a remainder; and thus continue the operation, till there is no number greater than 1 that will divide them exactly. Or, divide both the numerator and denominator by their greatest common divisor; the two quotients thence arising will be the lowest erms to which the given fraction can be reduced. (Art. 170.) OBS. 1. Since halves are larger than twentieths, it may be asked, how the fraction 1, can be said to be in lower terms than 18. It should be observed, the expression lowest term, has reference to the number of parts into which the unit or thing is divided, and not to the value or size of the parts. Thus, in 1, there are fewer parts than in 10; in 1, there are fewer parts than in 3, &c. Hence, a fraction is said to be reduced to its lowest terms, when its numerator and denominator are expressed in the smallest numbers possible. 4 2. The value of a fraction is not altered by reducing it to its lowest terms; for, the numerator and denominator are both divided by the same number. 3. When the terms of the fraction are small, the former method will generally be found to be the shorter and more convenient; but when the terms are large, it is often difficult to determine whether the fraction is in its simplest form, without finding the greatest common divisor of its terms. 2. Reduce to its lowest terms. 18 Ans. 1. 19. Reduce 23 to a whole or mixed number. Analysis.-The object in this example, is to find a whole, or mixed number, whose value is equal to the given fraction. Now, since 7 7 3 3 Operation. 7)23 34 Ans, QUEST.-Obs. What is meant by the expression, lowest terms? When is a fraction said to be reduced to its lowest terms? Is the value of a fraction altered by reducing, it to its lowest terms? Why not. sevenths inake 1 whole one, 23 sevenths will make as many whole ones as 7 is contained times in 23. And 23-7=34. But the value of a fraction is the quotient of the numerator divided by the denominator. (Art. 185.) Hence, 196. To reduce an improper fraction to a whole, or mixed number. Divide the numerator by the denominator, and the quotient will be the whole, or mixed number required. 20. Reduce 34 to a whole or mixed number. Ans. 64. Reduce the following fractions to whole or mixed numbers: 31. Reduce the mixed number 273 to an improper fraction, Analysis.-In 1 there are 5 fifths, and in 27 there are 27 times as many. Now 5X27=135, and 2 fifths make 137 fifths. Hence, Operation. 273 5 그림도 Ans 197. To reduce a mixed number to an improper fraction. Multiply the whole number by the denominator of the fraction, and to the product add the given numerator. The sum placed over the given denominator, will form the improper fraction required. OBS. 1. Any whole number may be expressed in the form of a fraction without altering its value, by making 1 the denominator. 2. A whole number may also be reduced to a fraction of any denominator, by multiplying the given number by the proposed denominator; the product will be the numerator of the fraction required. QUEST.-196. How is an improper fraction reduced to a whole or mixed number? 197. How reduce a mixed number to an improper fraction ? Obs. How express a whole number in the form of a fraction? How reduce it to a fraction of a given denominator? Thus, 25 may be expressed by 25, 100, or 400, &c., for 2526–100=400, &c. So 12-12=24=36=40, for the quotient of each of these numerators divided by its denominator, is 12. 32. Reduce 14 to an improper fraction. Ans. 22. Reduce the following numbers to improper fractions : 33. Reduce 172. 34. Reduce 25%. 35. Reduce 489. 36. Reduce 70%. 37. Reduce 1152. 38. Reduce 85645. 39. Reduce 13041. 40. Reduce 4725. 41. Reduce 445 to tenths. 42. Reduce 672 to eighths. 43. Reduce 3830 to one hundred and fifteenths. 44. Reduce 5743 to six hundred and twenty-fifths. CASE IV. 45. Reduce of to a simple fraction. Analysis.— ofis 2 times as much as 1 third of 7. Now of 7 is 7 or 4; for, multiplying the denominator divides the 8X3' value of the fraction. (Art. 188.) And 2 thirds is 2 times, or 7X2 which is equal to 14, or . (Art. 195.) The answer is . 24 OBS. This operation consists in simply multiplying the two numerators together and the two denominators. Hence, 198. To reduce compound fractions to simple ones. Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator. OBS. 1. That a compound fraction may be expressed by a simple one, is evident from the fact that a part of a part, must be equal to some part of the whole. 2. The reason of the rule may be seen from the analysis of the preceding example. 46. Reduce of 4 of % of % to a simple fraction. 47. Reduce of 1 of 12 of 1 to a simple fraction. 16 7 12 48. Reduce of 1 of 2 of 4 of to a simple fraction. QUEST.-198. How are compound fractions reduced to simple ones? 3, and 4, which are common to both; for, this is dividing the terms of the new fraction by the same number, (Art. 148,) and therefore does not alter its value. (Art. 191.) Multiplying the remaining factors together, we have, which is the answer required. Hence, 199. To reduce compound fractions to simple ones by CANCELATION. Cancel all the factors which are common to the numerators and denominators; then multiply the remaining terms together as before. (Art. 198.) OBS. 1. The reason of this rule depends upon the fact that the numerator and denominator of the new fraction are, in effect, divided by the same numbers; for, canceling a factor of a number divides the number by that factor. (Art. 148.) Consequently the value of the fraction is not altered. (Art. 191.) 2. This method not only shortens the operation of multiplying, but at the same time reduces the answer to its lowest terms. A little practice will give the student great facility in its application. 51. Reduce of 15 of Operation. 3 of 15 $ 3 $ 24 7 7 24 Ans. to a simple fraction. First we cancel the 3 and 8 in the numerator, then the 24 in the denominator, which is equal to the factors 3 'into 8. Finally, we cancel the 5 in the denomina tor and the factor 5 in the numerator 15, placing the other factor 3 above. We have 3 left in the numerator, and 7 in the denominator. Ans. 4. 52. Reduce 3 of 4 of 3 of 1% to a simple fraction. 53. Reduce % of 4 of 4 of 4 of to a simple fraction. QUEST.-199. How by cancelation? How does it appear that this method will give the true answer? Obs. What advantages does this method possess? |