The denominator shows into how many parts the whole number is divided, and gives a name to the fraction. The numerator shows how many of these parts are taken, or expressed by the fraction. A proper fraction is one whose numerator is less than the denominator; as, §. An improper fraction is one whose numerator is equal to, or greater than, the denominator; as, g, g. 8 NOTE.A fraction, strictly speaking, is less than a unit; hence, if the numerator is equal to, or greater than, the denominator, it expresses a unit or more than a unit, and is therefore called an improper fraction. A mixed number is a whole number with a fraction; as, 71, 5g. A simple or single fraction has but one numerator and one denominator, and may be either proper or improper; as, 7, §. A compound fraction is a fraction of a fraction, connected by the word of; as, 7 of 1⁄2 of §. A complex fraction is a fraction having a fraction or a mixed number for its numerator or denominator, or both; as 27 81 71 ART. 131. The terms of a fraction are its numerator and denominator. The unit of a fraction is the unit or whole thing from which its fractional parts are obtained. A whole number may be expressed fractionally, by writing 1 for the denominator. Thus, 5 may be written, and read 5 ones; and 9 may be written, and read 9 ones. ART. 132. Fractions originate from division; the numerator answers to the dividend, and the denominator to the divisor. Thus, when we divide 479956 by 6 (Art. 49, Ex. 12), we had a remainder of 4, which could not be divided by 6, and therefore we wrote it over the divisor, with a line between them. This expression for an unexecuted division is a fraction; the number above the line being the numerator, and the one below the denominator. QUESTIONS.-What does the denominator of a fraction show? What does the numerator show? What is a proper fraction? What is an improper fraction? What is a mixed number? What is a simple fraction? What is a compound fraction? What is a complex fraction?-Art. 131. What are the terms of a fraction? What is the unit of a fraction? How may a whole number be expressed fractionally? From what do fractions originate? ART. 133. From what has preceded, we perceive that the value of a fraction is the quotient arising from the division of the numerator by the denominator, or from the expression of this division. Thus, the quotient of or 6÷2 is 3; and the quotient of or 34 is REDUCTION OF COMMON FRACTIONS. ART. 134. Reduction of Fractions is the process of changing their form of expression without altering their value. A fraction is in its lowest terms, when its terms are prime to each other. (Art. 112.) ART. 135. To reduce a fraction to its lowest terms. Ex. 1. Reduce OPERATION. 2)=3 We divide the terms of the fraction by 2, a factor common to them both, and obtain 3. We divide, again, both terms of 3 by 3, a factor common to them, and obtain. Now, as 1 and 3 are numbers prime to each other, the fraction is in its lowest terms. The same result would have been produced, if we had divided the terms by 6, the greatest common divisor. Since by dividing the numerator and denominator of a fraction by the same number, we cancel equal factors in both (Art. 115), and diminish them in the same proportion, their relation to each other is not changed, and the value of the fraction remains the same. Therefore, Dividing the numerator and denominator of a fraction by the same number does not alter the value of the fraction. RULE. Divide the numerator and denominator by any number greater than 1, that will divide them both without a remainder, and thus proceed until they are prime to each other. Or, Divide both the numerator and denominator by their greatest common divisor. What is reduc QUESTIONS. What is the value of a fraction?- Art. 134. tion of fractions? When is a fraction in its lowest terms?— Art. 135. Why does dividing both terms of a fraction by the same number not alter the value? Has the same value as? Why? Repeat the rule. 9. Reduce to its lowest terms. 10. What is the lowest expression of 48? 8116 Ans. 7891 Ans. 173 13. ART. 136. To reduce a mixed number to an improper fraction. Ex. 1. In 7% how many fifths? OPERATION. 79 5 35 fifths. 3 38 = 38 Ans. 38. Since there are 5 fifths in 1 whole one, there will be 5 times as many fifths as whole ones; therefore, in 7 there are 35 fifths, and the 3 fifths being added make 38 fifths, which are expressed thus, 38. RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place the sum over the given denominator. NOTE. To reduce a whole number to a fraction of the same value, having a given denominator, we multiply the whole number by the given denominator, and make the product the numerator; thus, 5, reduced to a fraction, having 3 for a denominator, becomes 15. EXAMPLES FOR PRACTICE. 2. In 8 dollars how many sevenths? 8. In 161 how many one hundred and Ans. 169. seventeenths? Ans. 18848. 9. Change 43114 to an improper fraction. 10. What improper fraction will express 27? 11. Change 111 to an improper fraction. 12. Change 125 to an improper fraction. 13. Change 25 to an improper fraction, having nominator. 14. Reduce 75 to ninths. 15. Change 343 to the form of a fraction. 16. Reduce 84 to fifteenths. 117 13 Ans. 5142. Ans. 675. Ans. 343. Ans. 1260. QUESTIONS. Art. 136. What is the rule for reducing a mixed number to an improper fraction? Give the reason. How do you reduce a whole num ber to a fraction of the same value, having a given denominator? ART. 137. To reduce improper fractions to whole or mixed numbers. Ex. 1. How many dollars in 2 dollars? OPERATION. 16) 37 (2 32 5 RULE. Ans. $21. This question may be analyzed by saying, As 16 sixteenths make one dollar, there will be as many dollars in 37 sixteenths as 37 contains 16, which is 25 times = $216. Divide the numerator by the denominator, and the quotient will be the whole or mixed number. 876 EXAMPLES FOR PRACTICE. 2. Reduce 96 to a whole number. 10. Reduce 1848 to a mixed number. 459 ART. 138. To reduce a compound fraction to a simple fraction. Ex. 1. Reduce of to a simple fraction. OPERATION. X= Ans.. To show the reason of the operation, this question may be analyzed by saying, that, if of an apple be divided into 5 equal parts, of an apple; and, if of be, it is will be 7 times as much. 7 times is be 5, of will be 4 times as much. 4 one of these parts is evident that of 55; and, if of times are. Or, by multiplying the denominator of by 5, the denominator of, it is evident we obtain of 71 , since the parts into which the number or thing is divided are 5 times as many, and consequently only as large as before. Again, since QUESTIONS. Art. 137. What is the rule for reducing improper fractions to whole or mixed numbers? Give a reason for the rule. Art. 138. How do you reduce a compound fraction to a simple one? Give the reason for the operation. of 7=55, of 7 will be 4 times as much; and 4 times 5=28. This process will be seen to be precisely like the operation. Ex. 2. Reduce of of of § of to a simple fraction. Ans. OPERATION BY CANCELLATION. 2 RULE. 3 2 11 Since some of the numerators and denominators to be multiplied together are alike, we may cancel these common factors, according to the principles of cancellation. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. NOTE 1. - All whole and mixed numbers in the compound fraction must be reduced to improper fractions, before multiplying the numerators and denominators together. NOTE 2.-When there are factors common to both numerator and denominator, they may be cancelled in the operation. Ans.. of 7 to a simple fraction. of 14 of 1 of 59. 8. Reduce of of of of to a simple fraction. § I Ans.. 9. Reduce of of 7 of 1 of 44 to a simple fraction. 10 Ans. 26 Ans. 38. 35 to a simple fraction. QUESTIONS.-When there "are common factors in the numerator and denominator, how may the operation be shortened? What is the rule? What must be done with all whole and mixed numbers in the compound fraction? How may the operation be shortened by cancelling? |