sor and one of the other numbers, and so on to the last number. Then will the greatest common divisor last found be the answer. 4. Find the greatest common divisor of the terms of the fraction , and, by it, reduce the fraction to its lowest terms. Note. Let these examples be wrought by both methods; by several divisors, and also by finding the greatest common divisor. 6. Reduce 384 to its lowest terms. 1152 Ans.. Ans. . Ans. 11 29 . Ans.. ¶ 48. 1. If 2 yards of cloth cost of a dollar, what does 1 yard cost? how much is 2 divided by 2? 2. If a cow consume of a bushel of meal in 3 days, how much is that per day?÷3 = how much? 3. If a boy divide of an orange among 2 boys, how much will he give each one?÷2 how much? 4. A boy bought 5 cakes for 19 of a dollar; what did 1 cake = how much? 12÷5 cost? 5. If 2 bushels of apples cost of a dollar, what is that per bushel? 1 bushel is the half of 2 bushels; the half of is. Ans. # dollar. 6. If 3 horses consume of a ton of hay in a month, what will 1 horse consume in the same time? are 12 parts; if 3 horses consume 12 such parts in a month, as many times as 3 are contained in 12, so many parts 1 horse will consume. 7. If 2 of a barrel of flour be divided equally among 5 families, how much will each family receive? is 25 parts; 5 into 25 goes 5 times. Ans. of a barrel. The process in the foregoing examples is evidently dividing a fraction by a whole number; and consists, as may be seen, in dividing the numerator, (when it can be done without a remainder,) and under the quotient writing the denominator. But it not unfrequently happens that the numerator will not contain the whole number without a remainder. 8. A man divided of a dollar equally among 2 persons; what part of a dollar did he give to each? of a dollar divided into two equal parts will be 4ths. Ans. He gave of a dollar to each. 9. A mother divided a pie among 4 children; what part of the pie did she give to each? 4 how much? 10. A boy divided of an orange equally among 3 of his companions; what was each one's share? 3 how much? 11. A man divided of an apple equally between 2 children; what part did he give to each? divided by 2 = what part of a whole one? is 3 parts; if each of these parts be divided into 2 equal parts, they will make 6 parts. He may now give 3 parts to one, and 3 to the other; but 4ths divided into 2 equal parts, become 8ths. The parts are now twice so many, but they are only half so large; consequently is only half so much as 1. Ans. of an apple. In these last examples, the fraction has been divided by multiplying the denominator, without changing the numerator. The reason is obvious; for, by multiplying the denominator by any number, the parts are made so many times smaller, since it will take so many more of them to make a whole one; and if no more of these smaller parts be taken than were before taken of the larger, that is, if the numerator be not changed, the value of the fraction is evidently made so many times less. ¶ 49. Hence, we have two ways to divide a fraction by a whole number: I. Divide the numerator by the whole number, (if it will contain it without a remainder,) and under the quotient write the denominator. Otherwise, II. Multiply the denominator by the whole number, and over the product write the numerator. EXAMPLES FOR PRACTICE. 1. If 7 pounds of coffee cost of a dollar, what is that per pound? how much? Ans. of a dollar. 21+7= 23 2. If of an acre produce 24 bushels, what part of an acre will produce 1 bushel? 1924-how many? 3. If 12 skeins of silk cost of a dollar, what is that a skein how much? }} ÷ 12 = Note. When the divisor is a composite number, the intelligent pupil will perceive that he can first divide by one component part, and the quotient thence arising by the other; thus he may frequently shorten the operation. In the last example, 168 X 2, and Ans. 1. <÷8= and ÷ 2 = 78° 5. Divide by 12. Divide by 21. Divide 28 by 24. ΤΟ 40 6. If 6 bushels of wheat cost $43, what is it per bushel? Note. The mixed number may evidently be reduced to an improper fraction, and divided as before. Ans. 39= 18 of a dollar, expressing the fraction in its lowest terms. (46.) 7. Divide $411 by 9. 8. Divide 126 by 5. 9. Divide 14 by 8. 10. Divide 184 by 7. Quot. 7 of a dollar. Ans. 26. 265 Note. When the mixed number is large, it will be most convenient, first, to divide the whole number, and then reduce the remainder to an improper fraction; and, after dividing, annex the quotient of the fraction to the quotient of the whole number; thus, in the last example, dividing 184 by 7, as in whole numbers, we obtain 26 integers, with 2 remainder, which, divided by 7, gives, and 26 = +8=26.5, Ans. 11. Divide 27864 by 6. Ans. 4643. 12. How many times is 24 contained in 764611? Ans. 318347. 13. How many times is 3 contained in 462}? To multiply a fraction by a whole number. 50. 1. If1 yard of cloth cost of a dollar, what will 2 yards cost? X2: how much? 2. If a cow consume 4 of a bushel of meal in 1 day, how much will she consume in 3 days? X 3 = how much? 3. A boy bought 5 cakes, at of a dollar each; what did he give for the whole? X5 = how much. 6. If a man spend of a dollar per day, how much will he spend in 7 days? is 3 parts. If he spend 3 such parts in I day, he will evidently spend 7 times 3, that is, 21=2§ in 7 days. Hence, we perceive, a fraction is multiplied by multiplying the numerator, without changing the denominator. But it has been made evident, (¶ 49,) that multiplying the denomi nator produces the same effect on the value of the fraction as dividing the numerator: hence, also, dividing the denominator will produce the same effect on the value of the fraction as multiplying the numerator. In all cases, therefore, where one of the terms of the fraction is to be multiplied, the same result will be effected by dividing the other; and where one term is to be divided, the same result may be effected by multiplying the other. This principle, borne distinctly in mind, will frequently enable the pupil to shorten the operations of fractions. Thus, in the following example: At of a dollar for 1 pound of sugar, what will 11 pounds cost? 66 Multiplying the numerator by 11, we obtain for the product §§= of a dollar for the answer. ¶ 51. But, by applying the above principle, and dividing the denominator, instead of multiplying the numerator, we at once come to the answer, 5, in its lowest terms. Hence, there are two ways to multiply a fraction by a whole number. I. Divide the denominator by the whole number, (when it can be done without a remainder,) and over the quotient write the numerator. Otherwise, II. Multiply the numerator by the whole number, and under the product write the denominator. If then it be an improper fraction, it may be reduced to a whole or mixed number. EXAMPLES FOR PRACTICE. 1. If1 man consume 5 of a barrel of flour in a month, how much will 18 men consume in the same time? men? 2. What is the product of multiplied by 40? how much? Note. When the multiplier is a composite number, the pupil will recollect, (¶ 11,) that he may first multiply by one component part, and that product by the other. Thus, in the last example, the multiplier 60 is equal to 12 X 5; therefore, 13 X 12-13, X5=65=55 Ans. 4. Multiply 5 by 7. = and 13 Ans. 404. Note. It is evident that the mixed number may be reduced to an improper fraction, and multiplied, as in the preceding examples; but the operation will usually be shorter to multiply the fraction and whole number separately, and add the results together. Thus, in the last example, 7 times 5 are 35; and 7 times are 21=54, which added to 35, make 404, Ans. Or, we may multiply the fraction first, and, writing down the fraction, reserve the integers to be carried to the product of the whole number. 5. What will 913 tons of hay come to at $17 per ton? Ans. $164. *6. If a man travel 26 miles in 1 hour, how far will he travel in 5 in 12 hours? - in 3 days, hours? in 8 hours? supposing he travel 12 hours each day? Ans. to the last, 77% miles. Note. The fraction is here reduced to its lowest terms; the same will be done in all following examples. To multiply a whole number by a fraction. T52. 1. If 36 dollars be paid for a piece of cloth, what costs of it? 36 X = how much? of the quantity will cost of the price; a time 36 dollars, that is, of 36 dollars, implies that 36 be first divided into 4 equal parts, and then that 1 of these parts be taken 3 times; 4 into 36 goes 9 times, and 3 times 9 is 27. Ans. 27 dollars. From the above example, it plainly appears that the object in multiplying by a fraction, whatever may be the multiplicand, is, to take out of the multiplicand a part, denoted by the multiplying fraction; and that this operation is composed of two others, namely, a division by the denominator of the multiplying fraction, and a multiplication of the quotient by the numerator. It is matter of indifference, as it respects the result, which of these operations precedes the other, for 363427, the same as 36 4 X 327. Hence, To multiply by a fraction, whether the multiplicand be a whole number or a fraction, RULE. Divide the multiplicand by the denominator of the multiplying fraction, and multiply the quotient by the numerator; or, (which will often be found more convenient in practice,) first multiply by the numerator, and divide the product by the denominator. Multiplication, therefore, when applied to fractions, does not always imply augmentation or increase, as in whole numbers; for, when the multiplier is less than unity, it will always require the product to be less than the multiplicand, to which it would be only equal if the multiplier were 1. We have seen, (T 10,) that when two numbers are multiplied together, either of them may be made the multiplier, without affecting the result. In the last example, therefore, instead of multiplying 16 by, we may multiply by 16, (T 50,) and the result will be the same. EXAMPLES FOR PRACTICE. 2. What will 40 bushels of corn come to at of a dollar per bushel? 40 X = how much? 3. What will 24 yards of cloth cost at g of a dollar per yard? 24 how much? 4. How much is of 90? of 369? 5. Multiply 45 by Multiply 20 by . 1 of 45? |