1. What is the least common multiple of 6 and 8? OPERATION. In this example, it will be perceived 2) 6.8 that the divisor, 2, is a factor, both of 6 and 8, and that dividing 6 by 2 gives its 3 4 other factor, 3 (for 6+2=3); likewise dividing the 8 by? gives its other factor, 4 (for 8+2=4); consequently, if the divisors and quotients be multiplied together, their product must contain all the factors of the numbers 6 and 8; hence this product is the common multiple of 6 and 8, and, as there is no other number greater than 1, that will divide 6 and 8, 4X3 X 2= 24 will be the least common multiple of 6 and 8. Note.-When there are several numbers to be divided, should the divisor not he contained in any one number, without a remainder, it is evident, that the divisor is not a factor ní that number ; consequently, it may be omittad, and reserved to be divided by the next divisor. 2. What is the common multiple of 6, 3, and 4 ? OPERATION. In dividing 6, 3 and 4 by 3, I find that 3 is not contained in 3) 6.3.4 4 even; therefore, I write the 4 2 ) 2.1.4 down with the quotients, after which I divide by 2, as before. 1 1.2 Then, the divisors and quotients multiplied together, thus, 2 X 2 Ans. 3 X 2 X 2=12. X3=12, Ans. From these illustrations we derive the following RULE. Q. How do you proceed first to find the least common multiple of two or more numbers ? A. Divide by any number that will divide two or more of the given numbers without a remainder, and set the quotients, together with the undivided numbers, in a line beneath. Q. How do you proceed with this result ? . 7. Reduce 38 to its lowest terms. A. 8. Reduce % to its lowest terms. A. 38 Should it be preferred to reduce fractions to their lowest terms by 1 XXX VII., the following rules may be found serviceable : Any number ending with an even number or cipher is divisible by 2 Any number ending with 5 or 0 is divisible by 5; also if it end in 0, it is divisible by 10. A. Continue dividing, as before, till there is no number greater than 1 that will divide two or more numbers without a remainder; then multiplying the divisors and numbers in the last line together, will give the least common multiple required. More Exercises for the Slate. 3. Find the least common multiple of 4 and 16. A. 16. 4. Find the least common multiple of 10 and 15. A. 30. 5. Find the least common multiple of 30, 35 and 6. A. 210. 6. Find the least common multiple of 27 and 51. A. 459. 7. Find the least common multiple of 3, 12 and 8. A. 24, 8. Find the least common multiple of 4, 12, and 20. A. 60. 9. Find the least common multiple of 2,7,14 and 49. A. 98. 1 XLIII. TO REDUCE FRACTIONS OF DIFFERENT DE NOMINATORS TO A COMMON DENOMINATOR. Q. When fractions have their denominators alike, they may be added, subtracted, &c. as easily as whole numbers'; for example, } and are &; but in the course of calculations by numbers, we shall meet with fractions whose denominators are unlike; as, for instance, we cannot add, as above, and together: what, then, may be considered the object of reducing fractions of different denominators to a common denominator ? A. To prepare fractions for the operations of addition, subtraction, &c. of fractions. Q. What do you mean by a common denominator ? In performing this example, we Denom. 3 X 6=18, com. denom. take , and multi ply both its terms Numer. 5 X3=15, new numer. by the denominator Denom. 6 X 3= 18, com. denom. of ; also, we mul tiply both the terms of by 3, the denominator of; and, as both the terms of each faction are multiplied by the same number, consequently the value of the fractions is not altered ; 11 XXXVII. From these illustrations we derive the following. RULE. A. By all the other denominators. A. By the same numbers (denominators) that Note.--As, by multiplying in this manner, the same denominators are continually multiplied into each other, the process may be shortened; for, having found one denominator, it may be written under each new numerator. This, however, the intelligent pupil' will soon discover of himself; and, perhaps, it is best he should. 63 More Exercises for the Slate. 2. Reduce and } to a common denominator. A., S. 3. Reduce and to a common denominator. A. 1, 19. 4. Reduced and li to a common denominator. A. , 44 5. Reduce , and to a common denominator. A. 126126, 6. Reduce , and to a common denominator. A. 45, tis. Compound fractions must be reduced to simple fractions before finding the common denominator; also the fractional parts of mixed numbers may first be reduced to a common denominator, and then annexed to the whole numbers. 7. Reduce of į and to a common denominator. A. 14, 49 8. Reduce 14 and to a common denominator. A. 1414, 24 9. Reduce 103 and 1 of á to a common denominator. A. 1036, 48 10. Reduce 811 and 147 to a common denominator. A. 8147, 1444 Notwithstanding the preceding rule finds a common denomi nator, it does not always find the least common denominator. But, since the common denominator is the product of all the given denominators into each other, it is plain, that this product (T XLII.) is a common multiple of all these several denominators; consequently, the least common multiple found by 11 XLII. will be the least common denominator. b be . . 11. What is the least common denominator of $, and f? OPERATION. Now, as the denominator of 3 ) 3 6 . 2 each fraction is 6ths, it is evident that the numerator must be pro2 ) 1 2 2 portionably increased; that is, we must find how many 6ths 1 each fraction is; and, to do this, 1 1 Ans. 2 X 3 :6 we can take , , and of the 6ths, thus: of 6=4, the new numerator, written over the 6,=*. of 6=5, the new numerator, written over the 6,=* 1 of 6=3, the new numerator, written over the 6,=.. Ans. $, &, Hence, to find the least common denominator of several fractions, find the least common multiple of the denominators, for the common denominator, which, multiplied by each fraction, will give the new numerator for said fraction. 12. Reduce 1 and to the least common denominator. A. B. 13. Reduce and Ty to the least common denominator. A. to 14. Reduce 14 and 13 to the least common denominator. 1. 1412, 13% Fractions may be reduced to a common, and even to the least common denominator, by a method much shorter than either of the preceding, by multiplying both the terms of a fraction by any number that will make its denominator like the other denominators, for a common denominator; or by dividing both the terms of a fraction by any numbers that will make the denominators alike, for a common denominator. This method oftentimes will be found a very convenient one in practice. 15. Reduce 1 and to a common, and to a least common denominator. #X2=f; then f and = common denominator, A 2)=; then fand 1 = least common denominator, A. In this example, both the terms of one fraction are multiplied, and both the terms of the other divided, by the same number; consequently, (1 XXXVII.) the value is not altered. 16. Reduce 12 and to the least common denominator. A. 1, it • 17. Reducer and to the least common denominator. A. 4,1 18. Reduce any and to the least common denominator. 4. ਹੰਸ • 19. Reduce and to the least common denominator. A. 26, zor ADDITION OF FRACTIONS. 1 XLIV. 1. A father gave money to his sons as follows; to William $ of a dollar, to Thomas S, and to Rufus ; how much is the amount of the whole? How much are $, , and $, added together? 2. A mother divides a pie into 6 equal pieces, or parts, and gives to her son, and is to her daughter; how much did she give away in all? How much are å and added together? 3. How much are f + g + f? 4. How much are in tirtit? 5. How much are g + + is? 6. How much are otto? When fractions like the above have a common denominator expressing parts of a whole of the same size, or value, it is plain, that their numerators, being like parts of the same whole, may be added as in whole numbers; but sometimes we shall meet with fractions, whose denominators are unlike, as, for example, to add } and together. These we cannot add as they stand; but, by reducing their denominators to a common denominator, by T XLIII., they make g and s, which, added together as before, make , Ans. 1. Bought 3 loads of hay, the first weighing 191 cwt., the second 20 cwt., and the third 22 cwt.; what was the weight of the whole ? }, }, }, reduced to a common denominator, are equal to 3, 11 * |