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, T 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?÷7= how much? Ans. 2 of a dollar. 2. If 28 of an acre produce 24 bushels, what part of an acre will produce 1 bushel? ÷ 24 = how much? 3. If 12 skeins of silk cost of a dollar, what is that a skein?÷12 = how much? 4. Divide by 16. 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, 16 = 8 × 2, and ÷8= 1, and ÷ 2 Ans. T = 5. Divide by 12. Divide by 21. Divide by 24. 6. If 6 bushels of wheat cost $ 47, what is it per bushel? Note. The mixed number may evidently be reduced to an improper fraction, and divided as before. Ans. 1 of a dollar, expressing the fraction in its lowest terms. (Π 46.) 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 K the whole number; thus, in the last example, dividing 1841 by 7, as in whole numbers, we obtain 26 integers, with 22 = remainder, which, divided by 7, gives, and 26+4 26, Ans. 11. Divide 27861 by 6. 12. How many times is 24 contained in 764611? Ans. 4648. Ans. 31847 13. How many times is 3 contained in 4624? To multiply a fraction by a whole number. Ans. 1544. 50. 1. If 1 yard of cloth cost of a dollar, what will 2 yards cost? X 2 = how much? 2. If a cow consume + of a bushel of meal in 1 day, how much will she consume in 3 days? 1×3= how much? 3. A boy bought 5 cakes, at of a dollar each; what did he give for the whole? × 5 = how much? 4. How much is 2 times ? - 3 times ? times ?? by 2.- by 7. 2 5. Multiply by 3. 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 1 day, he will evidently spend 7 times 3, that is, = 28 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 denominator 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? Multiplying the numerator by 11, we obtain for the product of a Jollar 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, &, 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. If 1 man consume of a barrel of flour in a month, how much will 18 men consume in the same time? men? 6 9 men? Ans. to the last, 14 barrels. 2. What is the product of 1 multiplied by 40? 12 X 40 how much? Ans. 233. 3. Multiply by 12.- by 18. - by 21. by by 48. 36. by 60. 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 × 5; therefore, T × 12 = 1, and 1 × 5 ==512, Ans. 4. Multiply 5 by 7. Ans. 401. 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 24=51, which, added to 35, make 401, 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 943 tons of hay come to at $17 per ton? Ans. $1642. 6. If a man travel 24 miles in 1 hour, how far will he travel in 5 hours? - in 8 hours? - in 12 hours ? in 3 days, supposing he travel 12 hours each day? Ans. to the last, 773 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. 1 52. 1. If 36 dollars be paid for a piece of cloth, what costs of it? 36 × = 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 mutiplicand, is, to take out of the multiplicand a part, denoted by the multiplying fraction; and that this operation is composed of wo others, namely, a division by the denominator of the 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 36×3÷4=27, the same as 364×3 = 27. 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, (IT 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 × = how much? 3. What will 24 yards of cloth cost at & of a dollar per yard? 24 × 8 = how much? 4. How much is + of 90? - of 369? 5. Multiply 45 by To. Multiply 20 by . To multiply one fraction by another. 53. 1. A man, owning of a ticket, sold share; what part of the whole ticket did he sell? how much? To of 45? of his ofiz We have just seen, (IT 52,) that, to multiply by a fraction, is to divide the multiplicand by the denominator, and to multiply the quotient by the numerator. divided by 3, the denominator of the multiplying fraction, (149,) is, which, multiplied by 2, the numerator, (1151,) is 15, Ans The process, if carefully considered, will be found to consist in multiplying together the two numerators for a new numerator, and the two denominators for a new denominuior. EXAMPLES FOR PRACTICE. 2. A man, having of a dollar, gave of it for a dinner; what did the dinner cost him? 3. Multiply by . Multiply to by. 4. How much is of of of ? Ans. & doilar. Product, Note. Fractions like the above, connected by the word of, are sometimes called compound fractions. The word ofF implies their continual multiplication into each other. Ans. 188=20 When there are several fractions to be multiplied continually together, as the several numerators are factors of the new numerator, and the several denominators are factors of the new denominator, the operation may be shortened by dropping those factors which are the same in both terms, on the principle explained in 146. Thus, in the last exampie, t, 3, 4, 4, we find a 4 and a 3 both among the numerators and among the denominators; therefore we drop them, multiplying together only the remaining numerators, 2 × 7 = 14, for a new numerator, and the remaining denominators, 5 × 8 = 40, for a new denominator, making = 40, Ans, as before. K* 7 |