dividend, and the denominator a divisor, and the value of the fraction is the quotient; thus, is the quotient of 1 (the numerator) divided by 2, (the denominator;) is the quotient arising from 1 divided by 4, and is 3 times as much, that is, 3 divided by 4; thus, one fourth part of 3 is the same as 3 fourths of 1. Hence, in all cases, a fraction is always expressed by the sign of division. 3 is the dividend, or numerator. expresses the quotient, of which is the divisor, or denominator. 3. If 4 oranges be equally divided among 6 boys, what part of an orange is each boy's share? A sixth part of 1 orange is ¿, and a sixth part of 4 oranges is 4 such pieces, = . Ans. of an orange. 4. If 3 apples be equally divided among 5 boys, what part of an apple is each boy's share? if 4 apples, what? if 2 apples, what? if 5 apples, what? 5. What is the quotient of 1 divided by 3? of 1 by 4? of 2 by 4? of 3 by 4? of 2 by 3? of 5 by 7? - of 6 by 8 ? of 2 by 14? of 3 oranges? of 3? 6. What part of an orange is a third part of 2 oranges? one fourth of 2 oranges? + of 3 oranges? of 4? of 2? of 2? of 5? A Proper Fraction. Since the denominator shows the number of parts necessary to make a whole thing, or 1, it is plain, that, when the numerator is less than the denominator, the fraction is less than a unit, or whole thing; it is then called a proper fraction. Thus,,, &c. are proper fractions. An Improper Fraction. When the numerator equals or ceeds the denominator, the fraction equals or exceeds unity, or 1, and is then called an improper fraction. Thus, &,,,, are improper fractions. é 10 A Mixed Number, as already shown, is one composed of a whole number and a fraction. Thus, 141, 137, &c. are mixed numbers. 7. A father bought 4 oranges, and cut each orange into 6 equal parts; he gave to Samuel 3 pieces, to James 5 pieces, to Mary 7 pieces, and to Nancy 9 pieces; what was each one's fraction? Was James's fraction proper, or improper? Why? Was Nancy's fraction proper, or improper? Why? To change an improper fraction to a whole or mixed number. ¶ 44. It is evident, that every improper fraction must contain one or more whole ones, or integers. 1. How many whole apples are there in 4 halves (1) of in 10? in 984? - in in § of a yard? in? in in? in 20? in 48? 3. How many bushels in 8 pecks? that is, in of a bushel? in 10? -in ? 100? in 31? in 13? in 24? in This finding how many integers, or whole things, are contained in any improper fraction, is called reducing an improper fraction to a whole or mixed number. 4. If I give 27 children of an orange each, how many oranges will it take? It will take 27; and it is evident, that OPERATION. 4)27 Ans. 62 oranges. dividing the numerator, 27, (= the sumber of parts contained in the fraction,) by the denominator, 4, (= the number of parts in 1 orange,) will give the number of whole oranges. Hence, To reduce an improper fraction to a whole or mixed number,-RULE: Divide the numerator by the denominator; the quotient will be the whole or mixed number. EXAMPLES FOR PRACTICE. 5. A man, spending of a dollar a day, in 83 days would spend 3 of a dollar; how many dollars would that be? Ans. $138. 6. In 1417 of an hour, how many whole hours? The 60th part of an hour is 1 minute: therefore the question is evidently the same as if it had been, In 1417 minutes, how many hours? Ans. 2387 hours. 7. In 8753 of a shilling, how many units or shillings? Ans. 730, shillings. 8. Reduce 1467 to a whole or mixed number. 706 9. Reduce 8, 100, 175, 1786, 3465, to whole or mir To reduce a whole or mixed number to an improper fraction. be ¶ 45. We have seen, that an improper fraction may changed to a whole or mixed number; and it is evident, that, by reversing the operation, a whole or mixed number may be changed to the form of an improper fraction. 1. In 2 whole apples, how many halves of an apple? Ans. 4 halves; that is, . In 3 apples, how many halves? in 4 apples? in 6 apples? in 10 apples? 170? in 492 ? in 24? in 60? in Reduce 23 yards to 2. Reduce 2 yards to thirds. Ans. §. thirds. Ans. §. Reduce 3 yards to thirds. 5 yards. 33 yards. yards. 31 yards. 5 yards. 63 252 bushels. of a dollar? 3. Reduce 2 bushels to fourths. 61 bushels. 4. In 16 dollars, how many + make 1 dollar: if, therefore, we, multiply 16 by 12, that is, multiply the whole number by the denominator, the product will be the number of 12ths in 16 dollars: 16 × 12 = 192, and this, increased by the numerator of the fraction, (5,) evidently gives the whole number of 12ths; that is, 12 of a dollar, Answer. OPERATION. 165 dollars. 12 192 5 12ths in 16 dollars, or the whole number. 197= 197, the answer. Hence, To reduce a mixed number to an improper fraction,— RULE: Multiply the whole number by the denominator of the fraction, to the product add the numerator, and write the result over the denominator. EXAMPLES FOR PRACTICE. 5. What is the improper fraction equivalent to 237 hours? 6. Reduce 730 shillings to 12ths. As of a shilling is equal to 1 penny, the question is evidently the same as, In 730 s. 3 d., how many pence? Ans. 163 of a shilling; that is, 8763 pence. 12 7. Reduce 118, 1748, 875, 46, and 728 to improper fractions. 8. In 15617 days, how many 24ths of a day? Ans. 7613761 hours. 9. In 3422 gallons, how many 4ths of a gallon? Ans. 1371 of a gallon = 1371 quarts. To reduce a fraction to its lowest or most simple terms. 146. The numerator and the denominator, taken together, are called the terms of the fraction. If of an apple be divided into 2 equal parts, it becomes The effect on the fraction is evidently the same as if we had multiplied both of its terms by 2. In either case, the parts are made 2 times as MANY as they were before; but they are only HALF AS LARGE; for it will take 2 times as many fourths to make a whole one as it will take halves; and hence it is that is the same in value or quantity as is 2 parts; and if each of these parts be again divided into 2 equal parts, that is, if both terms of the fraction be multiplied by 2, it becomes. Hence, ==, and the reverse of this is evidently true, that ==. It follows therefore, by multiplying or dividing both terms of the fraction by the same number, we charge its terms without altering its value. Thus, if we reverse the above operau.on, and divide both terms of the fraction by 2, we obtain its equal, ; dividing. again by 2, we obtain, which is the most simple form of th fraction, because the terms are the least possible by which the fraction can be expressed. The process of changing into its equalis called reducing the fraction to its lowest terms. It consists in dividing both terms of the fraction by any "umber which will divide them both without a remainder, and the quotient thence arising in the same manner, and so on, till it appears that no number greater than 1 will again divide them. A number, which will divide two or more numbers without a remainder, is called a common divisor, or common measure of those numbers. The greatest number that will do this is called the greatest common divisor. ! 1. What part of an acre are 128 rods? One rod is of an acre, and 128 rods are 18 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 again measure them. The operation may be presented thus: 128 32 4 4) 160 40 5 99 of an acre, Answer. 2. Reduce 58,297, 148, and 194 to their lowest terins. 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, 0, 195, and 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 128. 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. |