1 XL. To divide a Fraction by a Whole Number. 1. If 3 apples cost of a cent, what will 1 apple cost? How much is ÷3? 2. If a horse eat or of a bushel of meal in 2 days, how much will he eat in one day? How much is ÷2? 3. A rich man divided § of a barrel of flour among 6 poor men; how much did each receive? How much is §-6? 4. If 3 yards of calico cost of a dollar, how much is it a yard? How much is ÷3? 5. If 3 yards of cloth cost of a dollar, how much is it a yard? The foregoing examples have been performed by simply dividing their numerators, and retaining the same denominator, for the following reason, that the numerator tells how many parts any thing is divided into; as, are 4 parts, and, to divide 4 parts by 2, we have only to say, 2 in 4, 2 times, as in whole numbers. But it will often happen, that the numerator cannot be exactly divided by the whole number, as in the following examples. 6. William divided of an orange among his 2 little brothers; what was each brother's part? We have seen, ¶ XXX",’II., that the value of the fraction is not altered by multiplying both of its terms by the same number; hence, 2. Now, & are 6 parts, and William can give 3 parts to each of his two brothers; for 2 in 6, 3 times. A. of an orange apiece. Q. In this last example, if (in ) we multiply the denominator 4 by 2, (the whole number,) we have, the same result as before; why is this? A. Multiplying the denominator makes the parts so many times smaller; and, if the numerator remain the same, no more are taken than before; consequently, the value is lessened so many times. From these illustrations we derive the following RULE. 1. When the numerator can be divided by the whole number without a remainder, how do you proceed? A. Divide the nu merator by the whole number, writing the denominator under the quotient. II. 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? 5. Divide by 8. (Divide the numerator.) A. zo. Note. When a mixed number occurs, reduce it to an improper fraction, then divide as before. ¶ XLI. To multiply one Fraction by another. 1. A man, owning of a packet, sells of his part; what part of the whole packet did he sell? How much is † of § ? 3 x 5 = 15 X 32 Ans. The reason of this operation will 4x 8 appear from Once is, and of is evidently divided by 4, which is done, ¶ XL., by multiplying the denominator 8 by the 4, making 32; that is, of 1=32. Again, if much, that is, 2. of be, then of will be 5 times as Again, if of be, then will be 3 times 25 Ans., as before. The above process, by close inspection, will be found to consist in multiplying together the two numerators for a new numerator, and the two denominators for a new de nominator. 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 or denote? A. Their continual multiplication into each other. Exercises for the State. 1. A man, having of a factory, sold of his part; what part of the whole did he sell? How much is of 3x50-180-25, Ans. 2. At of a dollar a yard, what will of a yard of cloth cost? How much is 23 of §? A. 4o. 3. Multiply of by . 4. Multiply of by . 8 X 7 X 7 45 3 X 5 X 3 A. 6. Multiply 6 by 7. A. 1888–933=1138. Note. 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 T XXXVII.; thus of of 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: 2 of off of Z ==x, 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 seve ral factors or numbers, what is it called? A. The commoi multiple of these factors. As the common multiple is a product consisting of tw or more factors, it follows that it may be divided by each o these factors without a remainder; how, then, may it be de termined, whether one number is a common multiple of two o more numbers, or not? A. It is a common multiple of thes 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, with out a remainder. Q. We can divide 12, also, by 2, 3, and 4, without a remair der; what, then, is the least number, that can be divided by or more numbers, calied? A. The least common multiple o these numbers. Q. It sometimes happens, that one number will divide sev ral other numbers, without a remainder; as, for instance, 3 wi divide 12, 18, and 24, without a remainder; when, then, sev ral numbers can be thus divided by one number, what is th 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 coinmen 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 S4? OPERATION. 72 )84 (1 72 12) 72 (6 A. 12, common divisor. In this example, 72 is contained in 84 1 time, 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, if 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 lest 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. A. 12. A. 84. A. 24. 5. Reduce 132 124 to its lowest terms. 12)|1=1, 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. 6 Find the common divisor of 750 and 1000; also reduce to its low est terms. A. 250, and 2. 7. Reduce to its lowest terms. 8. Reduce 660 to its lowest terms. A. 3. Should it be preferred to reduce fractions 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 |