44. 8 is 1 seventh of what number? 1 sixth? 1 tenth? 1 ninth 1 twelfth ? 45. 4 is two thirds of what number? Solution. First find 1 third. Now if 4 is 2 thirds, 1 third is 1 half of 4, which is 2; and 3 thirds is 3 times 2, or 6. Ans. 6. 46. 9 is 3 fourths of what number? 105. When a number or thing is divided into equal parts, these parts are called FRACTIONS. The number thus divided is called an Integer. 106. Fractions are divided into two classes, Common and Decimal. (For the illustration of Decimal Fractions, see Section VIII.) 107. Common Fractions are expressed by two numbers, one placed over the other, with a line between them. One half is written thus ; one third, ; one fourth, 4; nine tenths, thirteen forty fifths, 13, 13 &c. ; The number below the line is called the denominator, and shows into how many parts the number or thing is divided. The number above the line is called the numerator, and shows how many parts are expressed by the fraction. Thus in the fraction, the denominator 3, shows that the number is divided into three equal parts; the numerator 2, shows that two of those parts are expressed by the fraction. The denominator and numerator are called the terms of the fraction. QUEST.-105. What are fractions? What is an integer? 106. Of how many kinds are fractions? 107. How are common fractions expressed What is the number below the line called? What does it show? What is the number above the line called? What does it show? What are the denominator and numerator, taken together, called? OBS. 1. The word fraction is of Latin origin, and signifies broken, or separated into parts. Hence fractions are sometimes called broken numbers. 2. Common fractions are often called vulgar fractions. This term, however, is very properly falling into disuse. 3. The number below the line is called the denominator, because it gives the name or denomination to the fraction; as halves, thirds, fifths, &c. The number above the line is called the numerator, because it numbers the parts, or designates how many are expressed by the fraction. 108. When the numerator is less than the denominator, the fraction is called a proper fraction. Thus, 1⁄2, 1, , &c., are proper fractions. When the numerator is equal to, or greater than the denominator, it is called an improper fraction. Thus, 3, 4, 1, 12, &c., are improper fractions. A compound fraction is a fraction of a fraction; as of of. A mixed number is a whole number and a fraction expressed together; as 4, 2511. A complex fraction is one which has a fraction in its 24 2 numerator or denominator, or in both; as 5' 5'8 109. Fractions, it will be seen, both from the definition and the mode of expressing them, arise from division, and may be regarded as expressions of uncxecuted division, the numerator answering to the dividend, and the denominator to the divisor. (Arts. 67, 105.) Hence, 110. The value of a fraction is the quotient of the numerator divided by the denominator. Thus the value of is two; of 4 is one; of is one third, &c. Hence, 111. If the denominator remains the same, multiply QUEST.-Obs. What is the meaning of the word fraction? What are common fractions sometimes called? Why is the lower number called the denominator? Why is the upper one called the numerator? 108. What is a proper fraction? What, an improper fraction? What, a compound fraction? What, a mixed number? What, a complex fraction? 109. From what do fractions arise? 110. What is the value of a fraction? ing the numerator by any number, multiplies the value of the fraction by that number. Since the numerator and denominator answer to the dividend and divisor, multiplying the numerator is the same as multiplying the dividend. Now multiplying the dividend, we have seen, multiplies the quotient, (Art. 83,) which is the same as the value of the fraction. (Art. 110.) Thus the value of Multiplying the numerator by 3, the fraction becomes 18, whose value is 6, and is the same as 2 x 3. 112. Dividing the numerator by any number, divides the value of the fraction by that number. For, dividing the dividend, divides the quotient. (Art. 84.) Thus, =2. Now dividing the numerator by 2, the fraction becomes 3, whose value is 1, and is the same as 2÷2. Hence, OBS. With a given denominator, the greater the numerator, the greater will be the value of the fraction. 113. If the numerator remains the same, multiplying the denominator by any number, divides the value of the fraction by that number. For, multiplying the divisor is dividing the quotient. (Art. 85.) Thus, 244. Now multiplying the denominator by 2, the fraction becomes 21, whose value is 2, and is the same as 4÷2. 114. Dividing the denominator by any number, multiplies the value of the fraction by that number. For, dividing the divisor, multiplies the quotient. (Art. 86.) Thus, 244. Now dividing the denominator by 2, the fraction becomes 24, whose value is 8, and is the same as 4×2. Hence, OBS. With a given numerator, the greater the denominator, the less will be the value of the fraction. QUEST.-111. What is the effect of multiplying the numerator, while the denominator remains the same? Explain the reason. 112. What is the effect of dividing the numerator? Obs. With a given denominator, what is the effect of increasing the numerator? Explain the reason. 113. What is the effect of multiplying the denominator? Why? 114. What is the effect of dividing the denominator? Why? Obs. With a given numerator, what is the effect of increasing the denominator? 115. It is evident from the preceding articles, that, multiplying the numerator by any number, has the same effect on the value of the fraction, as dividing the denominator by that number. (Arts. 111, 114.) Dividing the numerator has the same effect, as multiplying the denominator. (Arts. 112, 113.) 116. If the numerator and denominator are both multiplied or both divided by the same number, the value of the fraction will not be altered. (Art. 88.) 8 Thus, 12=3. Now if the numerator and denominator are both multiplied by 2, the fraction becomes 24; whose value is 3. If both terms are divided by 2, the fraction becomes 9, whose value is 3; that is, 12-24=3. 117. Since the value of a fraction is the quotient of the numerator divided by the denominator, it follows, that If the numerator and denominator are equal, the value is a unit or one. Thus,1,1, &c. If the numerator is greater than the denominator, the value is greater than one. Thus, 1=2, §=1}, &c. If the numerator is less than the denominator, the value is less than one. Thus, 1 third of one, 4 fifths of one, &c. 118. It will be seen from the preceding exercises, that fractions may be added, subtracted, multiplied, and divided, as well as whole numbers. OBS. 1. In order to perform these operations, it is often necessary to make certain changes in the terms of the fractions. QUEST.-115. What may be done to the denominator to produce the same effect on the value, as multiplying the numerator by a given number? What, to produce the same effect as dividing the numerator by a given number? 116. What is the effect if the numerator and denominator are both multiplied or both divided by the same number? 117. When the numerator and denominator are equal, what is the value of the fraction? When the numerator is the larger, what? When smaller, what? 2. It is evident that any changes may be made in the terms of a fraction, which do not alter the quotient; for, if the quotient is not altered, the value remains the same. Thus the terms of the fraction may be changed into 2, 8, 16, &c., without altering its value; for in each case the quotient of the numerator divided by the denominator is 2. Hence, for any given fraction, we may substitute any other fraction, which will give the same quotient. REDUCTION OF FRACTIONS. 119. The process of changing the terms of a fraction into others, without altering its value, is called reduction of fractions. CASE I. EXERCISES FOR THE SLATE. Ex. 1. Reduce to its lowest terms. First Operation. 2) Dividing both terms of again, 3). Ans. the fraction by 2, it becomes again, dividing both by 3, we obtain, whose terms are the lowest to which the given fraction can be reduced. Second Operation. 6)2=1. Ans. If we divide both terms by 6, their greatest common divisor, (Art. 96.) the given fraction will be reduced to its lowest terms by a single division. Hence, 120. To reduce a fraction to its lowest terms. Divide the numerator and denominator by any number which will divide them both without a remainder; and thus continue the operation, till there is no number greater than 1 that will divide them exactly. Or, divide both the numerator and denominator by their greatest common divisor: the two quotients thus arising will be the least terms to which the given fraction can be reduced. (Art. 96.) OBS. 1. A fraction is reduced to its lowest terms, when its numerator and denominator are expressed in the smallest numbers possible. QUEST-Obs. What changes may be made in the terms of a fraction? 119. What is meant by reduction of fractions? 120. How is a fraction reduced to its lowest terms? Obs. What is meant by lowest terms? Is the value of a fraction altered by reducing it to its lowest terms? |