ART. 128. To find the least common multiple. Ex. 1. What is the least common multiple of 6, 9, 12 ? FIRST OPERATION. 6 = 2 × 3 9 = 12= 2 × 2 × 3 2 × 2 × 3 × 3 = 36 Ans. 36. we Resolving the numbers into their prime factors, thus, 6=2 × 3, and 93 X 3, and 12: 2× 2 × 3, find their different prime factors to be 2 and 3. The greatest number of times the 2 occurs as a factor in any of the numbers is twice, as 2 X 2 in 12; and the greatest number of times the 3 occurs in any of the numbers is also twice, as 3X3 in 9. Hence 2 X2 X3 X3 must be all the prime factors that are necessary in composing 6, 9, and 12; and, consequently, the product of these factors must be the least number that can be exactly divided by 6, 9, and 12. Therefore 2 × 2 × 3 × 3 = 36 is the least common multiple required. SECOND OPERATION. 316 9 12 22 3 4 1 3 2 3 × 2 × 3 × 2 = 36 Another method, and one usually preferred, is as by second operation. Having arranged the numbers on a horizontal line, we divide by 3, a prime number that will divide all of them without a remainder, and write the quotients in a line below. We next divide by 2, a prime number that will divide without a remainder most of them, writing down the quotients and undivided numbers as before. Then, since these numbers are prime to each other, we multiply together the divisors and the numbers on the lower line, which are all the prime factors of 6, 9, and 12, and thus obtain 36 for the least common multiple. RULE 1. Resolve the given numbers into their prime factors. The product of these factors, taking each factor the greatest number of times it occurs in any of the numbers, will be the least common multiple. Or, RULE 2.-Having arranged the numbers on a horizontal line, divide by such a prime number as will divide most of them without a remainder, and write the quotients and undivided numbers in a line beneath. So continue to divide until no prime number greater than 1 will divide two or more of them. The product of the divisors and the numbers of the line below will be the least common multiple. NOTE 1.-When numbers are prime to each other, their product is their least common multiple. NOTE 2. When one or more of the given numbers are factors of any one of the other numbers the factor or factors may be cancelled. QUESTION.-Art. 128. What are the rules for finding the least common multiple? EXAMPLES FOR PRACTICE. 2. What is the least common multiple of 7, 14, 21, and 15? Ans. 210. OPERATION. 77 14 21 15 2 3 15 7X2X15=210 Since 7 is a factor of 14, another of the numbers, we cancel it; and since 3 is a factor of 15, we also cancel that (Note 2) : thus the work is rendered shorter. 3. What is the least common multiple of 3, 4, 5, 6, 7, and 8? Ans. 840. 4. What is the least number that 10, 12, 16, 20, and 24, will divide without a remainder? Ans. 240. 5. What is the least common multiple of 9, 8, 12, 18, 24, 36, and 72? Ans. 72. 6. Five men start from the same place to go round a certain island. The first can go round it in 10 days; the second, in 12 days; the third, in 16 days; the fourth, in 18 days; the fifth, in 20 days. In what time will they all meet at the place from which they started? Ans. 720 days. XVIII. FRACTIONS. ART. 129. A FRACTION is an expression denoting one or more equal parts of a unit. The term fraction is derived from the Latin word frango, which signifies to break; from the idea that a number or thing is broken or separated into parts. Fractions are of two kinds, Common and Decimal. COMMON FRACTIONS. ART. 130. A COMMON FRACTION is expressed by two numbers one above the other, with a line between them. The number below the line is called the denominator; and the number above, the numerator. QUESTIONS. Art. 129. What is a fraction? From what is the term derived, and what does it signify? How many kinds of fractions, and what are they called? -Art. 130. How is a common fraction expressed? What is the number below the line called? The number above the line? The denominator shows into how many parts the whole number is divided, and gives a name to the fraction. The numerator shows how many of these parts are taken, or expressed by the fraction. A proper fraction is one whose numerator is less than the denominator; as, 4. An improper fraction is one whose numerator is equal to, or greater than, the denominator; as,, . NOTE.A fraction, strictly speaking, is less than a unit; hence, if the numerator is equal to, or greater than, the denominator, it expresses a unit or more than a unit, and is therefore called an improper fraction. 58. A mixed number is a whole number with a fraction; as, 71, A simple or single fraction has but one numerator and one denominator, and may be either proper or improper; as,, . A compound fraction is a fraction of a fraction, connected by the word of; as, 7 of of 8. A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as 278171 ART. 131. The terms of a fraction are its numerator and denominator. The unit of a fraction is the unit or whole thing from which its fractional parts are obtained. A whole number may be expressed fractionally, by writing 1 for the denominator. Thus, 5 may be written, and read 5 ones; and 9 may be written, and read 9 ones. ART. 132. Fractions originate from division; the numerator answers to the dividend, and the denominator to the divisor. Thus, when we divide 479956 by 6 (Art. 49, Ex. 12), we had a remainder of 4, which could not be divided by 6, and therefore we wrote it over the divisor, with a line between them. This expression for an unexecuted division is a fraction; the number above the line being the numerator, and the one below the denominator. QUESTIONS.-What does the denominator of a fraction show? What does the numerator show? What is a proper fraction? What is an improper fraction? What is a mixed number? What is a simple fraction? What is a compound fraction? What is a complex fraction?-Art. 131. What are the terms of a fraction? What is the unit of a fraction? How may a whole number be expressed fractionally? From what do fractions originate? ART. 133. From what has preceded, we perceive that the value of a fraction is the quotient arising from the division of the numerator by the denominator, or from the expression of this division. Thus, the quotient of or 62 is 3; and the quotient of or 3÷4 is . REDUCTION OF COMMON FRACTIONS. ART. 134. Reduction of Fractions is the process of changing their form of expression without altering their value. A fraction is in its lowest terms, when its terms are prime to each other. (Art. 112.) ART. 135. To reduce a fraction to its lowest terms. Ex. 1. Reduce OPERATION. 2)=3 3)= We divide the terms of the fraction by 2, a factor common to them both, and obtain 3. We divide, again, both terms of 3 by 3, a factor common to them, and obtain. Now, as 1 and 3 are numbers prime to each other, the fraction is in its lowest terms. The same result would have been produced, if we had divided the terms by 6, the greatest common divisor. Since by dividing the numerator and denominator of a fraction by the same number, we cancel equal factors in both (Art. 115), and diminish them in the same proportion, their relation to each other is not changed, and the value of the fraction remains the same. Therefore, Dividing the numerator and denominator of a fraction by the same number does not alter the value of the fraction. RULE. -Divide the numerator and denominator by any number greater than 1, that will divide them both without a remainder, and thus proceed until they are prime to each other. Or, Divide both the numerator and denominator by their greatest common divisor. 56 Ans. Ans. Ans. Ans. Ans. . Ans. 18. QUESTIONS. What is the value of a fraction?- Art. 134. What is reduction of fractions? When is a fraction in its lowest terms? - Art. 135. Why does dividing both terms of a fraction by the same number not alter the value? Has the same value as 1? Why? Repeat the rule. 789 9. Reduce to its lowest terms. 10. What is the lowest expression of 18? ART. 136. To reduce a mixed number to an improper fraction. Ans. 38. Ex. 1. In 7% how many fifths? OPERATION. 7号 35 fifths. 3 38= RULE. -38 Since there are 5 fifths in 1 whole one, there will be 5 times as many fifths as whole ones; therefore, in 7 there are 35 fifths, and the 3 fifths being added make 38 fifths, which are expressed thus, 38. Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place the sum over the given denominator. NOTE.-To reduce a whole number to a fraction of the same value, having a given denominator, we multiply the whole number by the given denominator, and make the product the numerator; thus, 5, reduced to a fraction, having 3 for a denominator, becomes 15. Ans. 169. seventeenths? Ans. 18848. Ans. 5142. 9. Change 43114 to an improper fraction. 10. What improper fraction will express 27? 11. Change 111 to an improper fraction. 12. Change 125 to an improper fraction. 13. Change 25 to an improper fraction, having nominator. 14. Reduce 75 to ninths. 15. Change 343 to the form of a fraction. 16. Reduce 84 to fifteenths. QUESTIONS. 117 13 Ans. 360. Art. 136. What is the rule for reducing a mixed number to an improper fraction? Give the reason. How do you reduce a whole num ber to a fraction of the same value, having a given denominator? |
