When the numbers are larger the process cannot be performed in the mind. For example. 5. Reduce and to a common denominator. When we multiply both terms of a Fraction by the same number, the value is not altered. (§ XLI.) Then we may multiply the 3, and the 24, in the last Fraction by 23, without altering the value. And, we may multiply the 1 and the 23 of the first Fraction by 24, without altering the value. Thus, 23x3=69; and 23×24=552. Therefore,—• 552. Therefore, same denominator. And 24X1-24; and 24×23= 24 552 552 In this process, we multiplied both terms of each Fraction, by the denominator of the other Fraction. 6. Reduce and to a com. denom. and 41: 1 and 1; 2 and 31; §§ and 4. It will be found most convenient, to reduce the Fractions first, to their lowest terms. 7. Reduce and to a common denominator. 126 11=2, and 3=7. A. 1 and 28. 27 and to a com. den. 18 and 3* ; 9. Reduce and § and ¦ to a com. den. A. 41, 44 and 4. Here we have thrée Fractions. If there were only the first two, viz. and, we should multiply the terms of by 6 and those of 5 by 4. But if we should multiply these terms, thus increased, by the other denominator, 2, it would not alter the value of the Fractions. And, if, then, we should multiply the terms of by 4 and 6, it would not alter its value. Therefore to reduce Fractions to a common denominator, MULTIPLY BOTH TERMS OF EACH FRACTION BY THE DENOMINATORS OF ALL THE OTHER FRACTIONS. NOTE. As all the denominators of the reduced Fractions are alike, when the denominator of one has been found it may be written down for all the others, without the trouble of Multiplication. EXAMPLES FOR PRACTICE. and 10. Reduce 2, 18, and 11 to a com. den., 37, 4, and ;,,, and 4; 3, 4, and 7: 151, 1, and 11. § XLV. A much smaller common denominator, may often be found, than the preceding rule would give. To determine what the smallest possible denominator would be, we must attend to the subject of multiples. 15 contains 3, an even number of times exactly. Therefore, 15 is called a multiple of 3. A NUMBER IS CALLED A MULTIPLE OF ANY NUMBER WHICH WILL DIVIDE IT WITHOUT A REMAINder. It will be seen that multiple and measure are merely relative terms. Any number is the multiple of its measure, and the measure of its multiple. 15 is a multiple of both 3 and 5. For 3 and 5 are measures of 15. Hence 15 is called a common multiple of 3 and 5. ANY MULTIPLE OF TWO OR MORE NUMBERS, IS CALLED THEIR COMMON MULTIPLE. If any two or more numbers be multiplied togther, the product is evidently a common multiple of all; for it may be divided by either of the factors which compose it. If two numbers be prime to each other, their product is likewise their least common multiple. Thus 3 and 5 are prime to each other, and their product, 15, is their least common multiple. For if you suppose that 3, repeated fewer times, will contain 5, you must suppose that 3 contains some factor, which, multiplied by 4 will produce 5. But, as 3 and 5 are prime to each other, they have no common factor. Therefore 5 times 3-15, is the least common multiple of 3 and 5. If there be more numbers than two, the same is true for similar reasons. Take a case, in which the numbers are not prime to each other. 15 and 10 have 5 for their greatest common measure. Divide one of them, as 15, by this common measure, and the quotient is 3. Multiply this into the other, and the product, 30, is the least common multiple of 15 and 10. For if you suppose that 10, repeated fewer times, as, for example 2 times, will contain 15, you must suppose that 10 contains some factor, which multiplied by 2, will produce 15. But this cannot be, for the quotient 3 was found by dividing 15 by the greatest common measure of the two numbers. Therefore, 3 times 10-30, is the least common multiple of 15 and 10. Hence, to find the least common multiple of two numbers, DIVIDE ONE OF THE NUMBERS BY THEIR GREATEST COMMON MEASURE, AND MULTIPLY THE QUOTIENT BY THE OTHER、 EXAMPLES FOR PRACTICE. 1. Find the least common multiple of 12 and 16. Their greatest com. meas. is 4. 16÷4-4. 4×12=48 Ans. 2. Find the least common multiple of 14 and 18. A. 126. Qf 9 and 12. A. 36. Of 24 and 30 A. 120. Of 25 and 45. Of 24 and 84. Of 36 and 48. Of 26 and 36. Of 32 and 56. Of 81 and 108. When there are more than two numbers, after finding the least common multiple of two of them, proceed with this multiple and the third number, to find another common multiple by the rule. After finding for three, proceed in the saine mammer with the fourth number, and so on. 3. Find the least common multiple of 15, 18 and 24. The least com. mult. of 15 and 18 is 90; of 90 and 24, 540 Ans. 4. Find the least com. mult. of 12, 16 and 30. A. 240. Of 9, 12, 16, 18 and 24. A. 144. Of 6, 10, 25 and 36. Of 224, 648, 936 and 872. Of 828, 333, 756 and 963. When there are many numbers and particularly if they are large, it is tedious to proceed as above. There is a shorter process. We have already seen, that when several numbers are prime to each other, their product is their least common multiple. If, then, we can contrive to find all the factors prime to each other, in the several numbers given, the product of these factors will be the least common multiple, not only of the factors, themselves, but also of the numbers from which we obtained them. These factors must be found by actual trial, thus. Place the numbers in a row and divide as many of them as possible by any number, which will divide them without a remainder, placing the quotients, with the numbers not divided, likewise in a row under the first. Divide as many numbers in this row as possible, in the same manner as before, and continue this process until no two numbers can be divided by any number greater than 1. The divisors used, and the last row of numbers, will be the factors sought. Of course, THE PRODUCT OF THESE FACTORS WILL BE THE LEAST COMMON MULTIPLE REQUIRED. 5. For example: find the least common multiple of 72, 64, 21, 18, and 98. 2 72; 64; 21; 18; 98 3 12: 32; ; 3; 49 7 4; 32; 7 ; 1; 49 4 4; 32; ; 1; 7 1; 8; 1; 1; 7 I observe that all, but 21, may be divided by 2. Therefore, I divide by 2 and bring down 21. Then, that all the next row, except 32 and 49, may be divided by 3. Therefore I divide by 3, and bring down 32 and 49. In this way, I proceed, dividing, each time as many as possible. I obtain then the factors 2, 3, 3, 7, 4, 1, 8, 1, 1, 7. Therefore 2×3×3×7X 4X8X7-28,224 is the least common multiple of the given numbers. In multiplying, I neglect the factors, 1, 1, 1, since they will not alter the product. The intelligent pupil will at once reply that these factors are not all prime to each other. But, it should be recollected, that, by this process, the factors of some of the numbers become resolved into their component parts. If these component parts be re-multiplied, the compound factors, thus obtained, will be all prime to each other. Thus, the two factors, 3 and 3, were derived, successively from 18. Therefore the compound factor of 18, is 3X3= 9. In like manner 2, 4 and 8 were derived successively from 64, and the compound factor is 2X4X8-64. So 7 and 7 were derived from 98, and the compound factor is 49. It will now be seen, that these compound factors, 64, 9 and 49, are all prime to each other; and that they contain all the factors, which are component parts of the given numbers, and no more. By the process, it will be seen, that when any number is a factor of several numbers, it is excluded from them all, by Division, and is then made once a factor in the multiple obtained. If all the given numbers of which it is a factor be not divided by it, then we shall employ it more than once as a factor in the result, and thus make the multiple obtained too large. From this consideration, the pupil will see the necessity of the following caution in employing the above mode of operation. In making each division, choose divisors which will divide AS MANY NUMBERS AS POSSIBLE. This will be best understood by illustration. 6. Find the least com. mult. of 6, 9, and 24. 6 and 24 can be divided by 9. But all the numbers can be divided by 3. If, then we first divide by 6, we shall leave the 9 undivided by 3, and so obtain too great a multiplier. First divide by 6. | 1, 9, 4, We can divide no further. Therefore we should expect 6X9 X4 216 to be the least com. mult., which the other operation shows to be incorrect. First divide by 3. 2|2, 3, 8, | 1, 3, 4, Hence, 3×2×3×4=72 is the least com. mult. None of our common Arithmetics provide against this source of error, and hence the learner is often perplexed by finding his result wrong, when he has strictly followed the directions of his rule. By this mode, perform the following. 7. Find the least com. mult. of 32, 72 and 120. 30,48 and 56. A. 1,680. Of 250, 180 and 540. 375, 125, 320 and 45. Of 872, 16, 98 and 75. 1,131, 340 and 460. A. 1,440. Of A. 13,500. Of Of 196, 762, In finding a common denominator for several Fractions, we mul. tiply all the denominators together. The common denominator, therefore, is a common multiple of the given denominators. Of course, the least com. den. must be the least com. mult. 8. Reduce and to the least com. den. 24 is the least com. mult. of 5 and 8. We must therefore bring the Fractions to 24ths. 24 24ths make a whole one. Then of 1 is of 24 24ths; and of 1 is 3 of 24 24ths. A. & and. Hence, for the numerators of the reduced Fractions, multiply the com. denom. by each of the given Fractions. ( XXXIII.) 9. Reduce and to the least com. den. A. 21 224 A shorter mode of finding numerators, may often be employed. Thus, 10. Reduce 3112, 27, and to the least com. den. A. and Process for the least com. mult. Then 3×4X9-108 is the least com. mult. 3 4, 9, and 4 (the first two of these factors) multiplied, make 12, the first given denominator. The remaining factor, being 9, shows that the com. den. is 9 times the given den. 12. Of course the corresponding numerator, in the answer, ought likewise, to be 9 times the given numerator 5. Hence, 9X5-45 the first numerator. It will be seen that this is obtained by excluding from the fac. tors of the com. den. just enough to produce 12, the given den., and multiplying the numerator 5, by the remaining factor 9. The process is similar for the other numerator. Thus, 3x9-27 the sec. ond den. Therefore, exclude 3 and 9, and multiply 4 the second numerator by 4, the remaining factor. Then-4x4-16 the second required numerator. When seeking the least com. den. we must of course take care first to bring the given Fractions to their lowest terms. 13. Reduce 17, 18, 77, and A. 14. Reduce, 4, 4, 4, 44 den. 40 135 4o to the least com. den. 135 2169 %, 17, 18, and 4. and 32 to the least com. Sometimes the denominator of one of the given Fractions may be a measure of that of another. In this case, by multiplying both terms of the former, it may be brought to the same denominator with the latter: or, if the numerator of the latter admits of being divided, we may bring this one to the same denominator with the former. Common denominators, and sometimes a least common denominator may often be readily found in this way. Sometimes both Division and Multiplication may be used, but we leave the pupil, in this respect, to exercise his own ingenuity. 15. Take, for example the Fractions and Multiplying the former by 2 gives us and, having a com.. den. Dividing the latter by 2 gives and, having a least com. den. to a com. den. A. and 1. to a com. den. A. 1 and. § and 3. 16. Reduce and 17. Reduce and To a least con. den. A. 18. Reduce and and and A. 15. and and and 4. 19. Reduce and and 20. Reduce and 3 and to a least com. den. A. and 3. and to a least com. den. |