Q. When the numerator cannot be thus divided, how do you proceed? A. Multiply the denominator by the whole number, writing the result under the numerator. Exercises for the Slate. 1. If 8 yards of tape cost of a dollar, how much is it a yard How much is÷8.. Note. When a mixed number occurs, reduce it to an improper digestion, en divide as before. 7. Divide $63 among five men. A. 63=4+5=33=17, XLI. TO MULTIPLY ONE Frictive #7 (****** 1. A man, owning of a packed, the $ not like pod part of the whole packet did he ad? 3 X 5 = 15 4 X 8 = 32 Ans. from the following illustration 4 Once is, and 1 of 1 is evidbury & therand by winkly is done (¶XL) by santayang tan senyumasin, X by t making 32; that is, 11 Again, if of bed. How & will be $x much, that is, #2. Again, if of #2, the wil bly Ans, as before. consist in multiplying together the two numerators for a new numerator, and the two denominators for a new denominator. Should a whole number occur in any example, it may be reduced to an improper fraction, by placing the figure 1 under it; thus 7 becomes ; for, since the value of a fraction (¶ XXXIV.) is the numerator divided by the denominator, the value of is 7; for, 1 in 7, 7 times. From these illustrations we derive the following RULE. Q. How do you proceed to multiply one fraction by another? A. Multiply the numerators together for a new numerator; and the denominators together for a new denominator. Note.-If the fraction be a mixed number, reduce it to an improper fraction: then proceed as before. Q. What are such fractions as these sometimes called? A. Compound Fractions. Q. What does the word of denote ? A. Their continual multiplication into each other. Exercises for the Slate. 1. A man, having of a factory, sold part of the whole did he sell? How much is =T90=25, Ans. = 2. At of a dollar a yard, what will § of a yard of cloth. 6. Multiply 383 by 3. 315 A. . = = 500 A. 1883-933-1333. Nete. If the denominator of any fraction be equal to the numerator of any other fraction, they may both be dropped on the principle explained in ¶ XXXVII.; thus 2 of 2 of 1⁄2 may be shortened, by dropping the numerator 3, and denominator 3; the remaining terms, being multiplied together, will produce the fraction required in lower terms; thus, of of of 5 = 2 of &=10=12, Ans. The answers to the following examples express the fraction in its lowest terms. XLII. TO FIND THE LEAST COMMON MULTIPLE OF TWO OR MORE NUMBERS. Q. 12 is a number produced by multiplying 2 (a factor) by some other factor; thus, 2×6=12; what, then, may the 12 be called? A. The multiple of 2. Q. 12 is also produced by multiplying not only 2, but 3 and 6, likewise, each by some other number; thus, 2×6=12; 3 × 4 = 12; 6×2=12; when, then, a number is a multiple of several factors or numbers, what is it called? A. The common multiple of these factors. Q. As the common multiple is a product consisting of two or more factors, it follows that it may be divided by each of these factors without a remainder; how, then, may it be determined, whether one number is a common multiple of two or more numbers, or not? A. It is a common multiple of these numbers, when it can be divided by each without a remainder. Q. What is the common multiple of 2, 3, and 4, then? A. 24. Q. Why? A. Because 24 can be divided by 2, 3, and 4, without a remainder. Q. We can divide 12, also, by 2, 3, and 4, without a remainder; what, then, is the least number, that can be divided by 2 or more numbers, called? A. The least common multiple of these numbers. Q. It sometimes happens, that one number will divide several other numbers, without a remainder; as, for instance, 3 will divide 12, 18, and 24, without a remainder; when, then, several numbers can be thus divided by one number, what is the number called? A. The common divisor of these numbers. Q. 12, 18, and 24, may be divided, also, each by 6, even; what then, is the greatest number called, which will divide 2 or more numbers without a remainder? A. The greatest common divisor.* * In ¶ XXXVII., in reducing fractions to their lowest terms, we were sometimes obliged, in order to do it, to perform several operations in dividing, but, had we only known the greatest common divisor of both terms of the fraction, we might have reduced them by simply dividing once; hence it may sometimes be convenient to have a rule To find the greatest common divisor of two or more numbers. 1. What is the greatest common divisor of 72 and 84? OPERATION. 72) 84 (1 72 12) 72 (6 72 A. 12, common divisor. In this example, 72 is contained in 84, 1 timo, and 12 remaining; 72, then, is not a factor of 84 Again, if 12 be a factor of 72, it must also be a factor of 84; for 72+12=84. By dividing 72 by 12, we do find it to be a factor of 72, (for 72 +12=6 with no remainder); therefore, 12 is a common factor or divisor of 72 and 84; and, as the greatest common divisor of two or more numbers never exceeds their difference, so 12, the difference between 84 and 72, must be the greatest common divisor. Hence, the following RULE:-Divide the greater number by the less, and, of there be no remainder, the less number itself is the common divisor; but, if there be a remainder, divide the divisor by the remainder, always dividing the last divisor by the last remainder, till nothing remain: the last divisor is the divisor sought. Note. If there be more numbers than two, of which the greatest common divisor is to be found, find the common divisor of two of them first, and then of that common divisor and one of the other numbers, and so on. 2. Find the greatest common divisor of 144 and 132. 3. Find the greatest common divisor of 168 4. Find the greatest common divisor of 24, 48, and 96. A. 12. A. 24. Let us apply this rule to reducing fractions to their lowest terms. 5. Reduce to its lowest terms. 12), Ans. In this example, by using the common divi sor, 12, found in the answer to sum No. 2, we have a number that will reduce the fraction to its lowest terms, by simply dividing both terms but once. After the same manner perform the following examples: Find the common divisor of 750 and 1000; also reduce 7050 to its lowest term. A. 250, and 3. 1. What is the least common multiple of 6 and 8? OPERATION. 2)6.8 In this example, it will be perceived that the divisor, 2, is a factor, both of and 8, and that dividing 6 by 2 gives its 3 4 other factor, 3 (for 6÷2=3); likewise dividing the 8 by 2 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, 4 × 3 × 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 be contained in any one number, without a remainder, it is evident, that the divisor is not a factor of that number; consequently, it may be omitted, and reserved to be divided by the next divisor. 2. What is the common multiple of 6, 3, and 4 ? OPERATION. 3) 6 2) 2 3 4 1 1 4 2 Ans. 3 X 2 X 2 = 12. In dividing 6, 3 and 4 by 3, I find that 3 is not contained in 4 even; therefore, I write the 4. down with the quotients, after which I divide by 2, as before. Then, the divisors and quotients multiplied together, thus, 2 × 2 X 3=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? A. Z. Should it be preferred to reduce fiactions to their lowest terms by ¶ XXXVII., 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. |