1. What part of an acre are 128 rods? One rod is of an acre, and 128 rods are 188 of an acre. Let us reduce this fraction to its lowest terms. We find, by trial, that 4 will exactly measure both 128 and 160, and, dividing, we change the fraction to its equal 23. Again, we find that 8 is a divisor common to both terms, and, dividing, we reduce the fraction to its equal , which is now in its lowest terms, for no greater number than 1 will agair measure them. The operation may be presented thus: 8/32 4) 1280 4 40 5 99 of an acre, Answer. 2. Reduce #88, 7, 148, and 16 to their lowest terms. Ans.,,, and . Note. If any number ends with a cipher, it is evidently divisible by 10. If the two right hand figures are divisible by 4, the whole number is also. If it ends with an even number, it is divisible by 2; if with a 5 or 0, it is divisible by 5. 3. Reduce 488, 4, 195, and 2 to their lowest terms. ¶ 47. Any fraction may evidently be reduced to its lowest terms by a single division, if we use the greatest common divisor of the two terms. The greatest common measure of any two numbers may be found by a sort of trial easily made. Let the numbers be the two terms of the fraction 8. The common divisor cannot exceed the less number, for it must measure it. We will try, therefore, if the less number, 128, which measures itself, will also divide or measure 160. 128) 160(1 128 32) 128 (4 128 128 in 160 goes 1 time, and 32 remain; 128, therefore, is not a divisor of 160. We will now try whether this remainder be not the divisor sought; for if 32 be a divisor of 128, the former divisor, it must also be a divisor of 160, which consists of 128 +32. 32 in 128 goes 4 times, without any remainder. Consequently, 32 is a divisor of 128 and 160. And it is evidently the greatest common divisor of these numbers; for it must be contained at least once more in 160 than in 128, and no number greater than their difference, that is, greater than 32, can do it. Hence the rule for finding the greatest common divisor o two numbers :-Divide the greater number by the less, and that divisor by the remainder, and so on, always dividing the last divisor by the last remainder, till nothing remain. The last divisor will be the greatest common divisor required. Note. It is evident, that, when we would find the greatest common divisor of more than two numbers, we may first find the greatest common divisor of two numbers, and then of that common divisor 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. T 48. 1. If 2 yards of cloth cost of a dollar, what does 1 yard cost? how much is 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 cake cost? ÷ 5 = how much? 2 of a dollar; what did 1 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 horse consume in the same time? 13 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. Ans. of a ton. 7. If of a barrel of flour be divided equally among families, how much will each family receive? 5 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 2 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 2. 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? ÷ 7 = how much? Ans. of a dollar. 2. If 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, 168 x 2, and ÷ 8, and ÷2 = 18. Ans. 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. (T 46.) 7. Divide $4+1 by 9. 8. Divide 12 by 5. 9. Divide 14 by 8. 10. Divide 1841 by 7. Quot. of a dollar. Quot. 48=24. Ans. 26 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 21 remainder, which, divided by 7, gives, and 26+ 26, Ans. 11. Divide 27861 by 6. Ans. 4643. 12. How many times is 24 contained in 7646↓↓ ? Ans. 31834. 13. How many times is 3 contained in 4624? To multiply a fraction by a whole number. ¶ 50. 1. If 1 yard of cloth cost 2 yards cost? X2 how much? Ans. 154. of a dollar, what will 2. If a cow consume of a bushel of meal in 1 day, how much will she consume in 3 days? × 3 = how much? 3. A boy bought 5 cakes, at of a dollar each; what did he give for the whole? X 5 = how much? 4. How much is 2 times ? times? 5. Multiply by 3.- g by 2. 3 times ? f by 7. 2 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, 22 in 7 days. Hence, we perceive, a fraction is multiplied by multiplying the numerator, without changing the denominator. But it has been made evident, (T 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. |