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38 THE MARINE REVIEW ber of multiples. Thus, the multiple of 5 is 5 itself, 10, 15, 20, 25, etc. The multiple of 3 is 3, 6, 9, 12, 15, etc. A common multiple of two or more given numbers is a number exactly divisible by each of them; thus, 10 is a com- mon multiple of 2 and 5, also 20, 30, 40, etc., are common multiples of 2 and 5. The least common multiple of two or more given numbers is the least number exactly divisible by each of them. Two or more numbers can have but one least common multiple. Thus, 20 is the least common multiple of 2, 4 and 5. What is a multiple of 4, 6, 8? Answer 24, 18 and 32. What is a common multiple of 3, 4, and 6? Answer 12. What is the least common multiple of the denominators of 14, %, 44? Answer 8. - Of 2-3, 2-9 and 5-6? Answer 18. Reduce 2-3 and 1-9 to 18ths. Answer 2-3 = 12-18 and 1-9 = 2-18. : Name some fractions that can be changed to 16ths. Answer %, x% and IY, Name four fractions that can be changed to 24ths. Answer %, 5%, 1-6 and 3. A common denominator is a denominator common to two or more fractions. The least common denominator of two or more fractions is the least denominator to which they can all be reduced. Since all higher terms of a fraction are multiples of its ; corresponding lowest terms, hence the following: A common denominator of two or more fractions is a common multiple of their denominators. The least common denominator of two or more fractions is the least common multiple of their denominators. To re- duce two or more fractions to equivalent fractions having a common denominator, multiply the terms of each fraction by the denominators of all the other fractions. Reduce %, 2-3 and 3-5 to equivalent fractions having a common denominator. Multiply each denominator by the other two, thus, 2 X3 X 5 = 30, and the product 30, is a common denominator of the three. But since the value of the fraction is not to be changed each numerator must be multiplied by the same number as its denominator. Thus, % = 15-30, found thus: We multiply the denominator 2 by the product of the other two denominators (3 X 5 = 15) times the denominator 2 is 30 and multiplying the numerator 1 by the same number, which is 15, so as not to alter the value of our fraction we get 15-30. The next fraction 2-3 is found in the same manner, thus we multiply the denomina- tor 3 by the product of the other two which is 2 X 5 = Io X 3 = 30; and multiplying the numerator 2 by the same number, which is 10, we get 20-30. 3-5 would'be 2 X 3 = 6 X 5, the denominator equals 30 and multiplying the numerator 3 by the same number gives us 18-30. Reduce to fractions having a common denominator: 3-7 and 5-9. The common denominator of 3-7 and 5-9 is 63, and 3-7 = 27-63; found thus, multiplying the denominator 7 by the other denominator 9 we get 63, and multiplying the numerator 3 by the same number, so as not to change the value of our fraction (3 X 9 = 27) we get 27-63, and multi- plying the denominator 9 by the other denominator 7 we get 63 and multiplying the numerator 5 by the same number we get 35-63. To reduce fractions to their least common denominator. Rule----Find the least common multiple of the denomina- tors of the given fractions for their least common denomina- tor. Divide this common denominator by the denominator of each of the given fractions, and multiply its numerator by the quotient. The products are the new numerators. Mixed numbers must first be reduced to improper fractions. Change 2-3, 5-12 and 7-15 to equivalent fractions having the least common denominator. First find the least common multiple of the given denomina- tors, which is 60; found thus: 3|3 12 1 xX 1X 4 X 5=60 2-3 =40-60 | 5-12 = 25-60 | 7-15 = 28-60. To find the least common denominator of two or more fractions find the least common multiple of their denomina- tors; thus, first write down separately the denominators of each fraction in a line and divide by any prime number that will exactly divide two or more of the given numbers and write the quotients and undivided numbers in a line under- neath. In like manner divide the quotients and undivided numbers until they are no longer divisible. The product of the divisors and the final quotients and undivided numbers is the least common multiple. Reduce 2-3, 34, 7-12, 9-15 to similar fractions having their least common denominator: 91354 12. 10 a | 0 A 4 16 ot. 2 2 8 I I I 4 (3X 2K 2X 4=-- 4. Explanation--The least common denominator cannot al- ways be easily found by inspection. It may be found as above. Since the least common denominator must be the small- est number that will contain each of the denominators and no other factors. The prime factors are found as above. 3 is a prime factor of 3 and 12, and consequently a factor of the least common denominator. Dividing by 3, and writ- ing below the quotients and numbers of which 3 is not a factor, we have, I, 4, 4, 16. Dividing by 2, and again by 2, the factors of the denominator are found to be the divisors 3, 2, 2, and the factor 4 in the last row. Their product is 48, the least common denominator. The fractions thus be- come 32-48, 36-48, 28-48, 27-48. Note.--Fractions should first be reduced to their lowest terms. In finding the factors of the least common denomin- ator a number that is a factor of another number may be disregarded. Thus, since 3 and 4 are factors of 12, they 'might have -been: disregarded, and the factors of 12 and 16 only found. It is reported in San Francisco that the Oceanic Steam- ship Co. will be sold to Japan if congress defeats the ship- ping bill now before it. The statement of Mr. Eugene F. Loud before the committee on merchant marine and fisheries at Washington has been corroborated by Mr. Frederick F. Samuels, manager of the foreign shipping department of the Spreckels line. It is the contention - of the company that it is impossible to operate between San Francisco and Australia without government sup- port. Mr. Samuels states that the cost of operation is too high to compete with other lines. The Smooth-On Manufacturing Co., Jersey City, N. J., has just issued an illustrated catalog No. 5 concerning Smooth-On iron cements, Smooth-On sheet packing, Smooth- On corrugated steel gaskets, showing when, where and how to use them. The different Smooth-On cements are explained in the catalog and photographs are also submitted showing actual subjects upon which the composition has been used with great success. The catalog, while compact, is very thor- ough, and should be of great interest to users of machinery. George B. Hibbard, a well known marine lawyer, died in Buffalo last week at the age of eighty-two years. aati