EXERCISES FOR THE SLATE. 105. When a number or thing is divided into equal parts, as halves, thirds, fourths, fifths, &c., these parts are called Fractions. Hence, A FRACTION denotes a part or parts of a number or thing. OBS. Fractions are used to express parts of a collection of things, as well as of a single thing; or parts of any number of units, as well as of one unit. Thus, we speak of 1 third of six oranges; 3 fifths of 75, &c. In this case the collec tion, or number to be divided into equal parts, is regarded as a whole. 106. Fractions are divided into two classes, Common and Decimal. (For the illustration of Decimal Fractions, see Section VIII.) 107. Common fractions are those which arise from dividing an integer into any number of equal parts. They are expressed by two numbers, one placed over the other, with a line between them. For example, one half is written thus, ; one third,; one fourth, ; nine tenths, 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 3, 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 numerator and denominator together, are called the terms of the fraction. OBS. 1. The term fraction, is derived from the Latin fractio, which signifios the act of breaking, a broken part or piece. Hence, Fractions are sometimes called broken numbers. 2. Common fractions are often called vulgar fractions. This term, however, is very properly falling into disuse. QUEST.-105. What are fractions? 106. Into how many classes are fractions divided? 107. What are Common Fractions? How are they expressed? What is the number below the line called? What does it show? What is the num ber above the line called? What does it show? What are the numerator and denominator, taken together, called? Obs. What is the meaning of the term Auction? What are common fractions sometimes called? 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 shows how many parts are expressed by the fraction. 108. Common Fractions are divided into proper, improper simple, compound, complex, and mixed numbers. A proper fraction is a fraction whose numerator is less than its denominator; as 1, 3, . An improper fraction is one whose numerator is equal to, or greater than its denominator; as, 3, 4. 9 A simple fraction is a fraction which has but one numerator and one denominator, and may be proper or improper; as, }, {. A compound fraction is a fraction of a fraction; as, of . A complex fraction is one which has a fraction in its numerator, or denominator, or in both; as, A mixed number is a whole number and together; as, 43, 2511. 4 2. a fraction written OBS. The original notion or definition of a fraction appears to have been, that It was a part of a unit. But it was seen, that those expressions whose numerator is equal to, or greater than their denominator, as 5 fifths, 9 fourths, &c., did not eone under this definition; therefore they were called improper fractio"s. Although it is not accurate to call 5 fifths, or 9 fourths a part of a unit, there is no inaccuracy in calling them fractions; for they donote parts of an integer. (Art. 105.) The impropriety, therefore, belongs not to this class of fractions, but to the definition which limits the meaning of the term fraction, to a part of a unit, and, consequently, is not sufficiently comprehensive to cover the whole ground, Read the following fractions, and name the kind of each: 6 9 2. 1. ; ; of 25; 17. 115. 5670. 6007 ; 15; 235; 4373; 5939; 7235; 8803. 13 of 28 of 7; 25 of 90% of 1000; 27 of 65. 8. 75; 299; 4115; 27315; 4273828; 706593; 88645. 731 88 33 6 87 100 825. 109. Fractions, it will be seen, both from the definition and the mode of expressing them, arise from division, and may be treated as expressions of unexecuted division, the numerator answering to the dividend, and the denominator to the divisor. (Arts. 67, 105.) QUEST.-Why is the lower number called the denominator? Why is the upper one called the numerator? 108. How are common fractions divided? What is a proper fraction? An improper fraction? A simple fraction? A compound fraction? A complex fraction? A nixed number? 109. From what do fractions arise? 110. The value of a fraction is the quotient of the numerator divided by the denominator. Thus the value of § is two; of is one; ofis one third; &c. Hence, 111. If the denominator remains the same, multiplying the numerator by any number, multiplies the value of the fraction by that number. For, the numerator and denominator answer to the dividend and divisor; therefore, multiplying the numer ator 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 =2. Multiplying the numerator by 3, the fraction becomes 13, whose value is 6, and is the same as 2 x 3, 18 112. 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 aividing the numerator by 2, the fraction becomes }, whose value is 1, and is the same as 2÷2. Hence, OBS. With a given denominator, the greater the numerator, the greater win 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 divides the quotient. (Art. 85.) Thus, 24=4. Now multiplying the dʊnominator by 2, the fraction becomes 24, whose value is 2, ¿nd 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, 2=4. Now dividing the denominator by 2, the fraction becomes 2, whose value is 8, and is the same as 4 × 2. Hence, 24 OBS. With a given numerator, the greater the denominator, the less will be the value of the fraction. QUEST.-110. What is the value of a fraction? 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? Why? Obs. With & given denominator, what is the effect of increasing the numerator? 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.) And, 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. (Arts. 88, 109.) 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 ; whose value is 3; that is, 12=2=1=3. 117. Since the value of a fraction is the quotient of the nu merator divided by the denominator, it follows that If the numerator and denominator are equal, the value is a unit or one. Thus, =1, 7=1, &c. If the numerator is greater than the denominator, the value is greater than one. Thus, 2, 5=13. If the numerator is less than the denominator, the value is less than one. Thus, =1 third of 1, 3=4 fifths of 1. OBS. The best method to estimate the comparative magnitude or value of fractions, is to compare them with a unit or one, and not with each other. Thus, pupils are often puzzled to tell which is the greater, 7 eighths or 8 sevenths, when they attempt to compare the given fractions with each other; but by comparing each with a unit, the difficulty vanishes at once. 118. It will be seen from the preceding exercises that fractions may be added, subtracted, multiplied, and divided, as well as whole numbers. 16 OBS. In order to perform these operations, it is often necessary to make cer tain changes in the terms of the fractions, while the value remains the same. Thus, the terms of the fraction may be changed into, &, Ī, 4, 8, &c., without altering its value; for in each case the value is 2. Hence, For any given fraction, we may substitute any other fraction of equal value. QUEST. 115. What may be done to the denominator to produce the same effect on the value of the fraction, as multiplying the numerator by any given number? What, to produce the same effect, as dividing the numerator by any 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 numer When the ator and denominator are equal, what is the value of the fraction? numerator is the larger, what? When smaller, what? REDUCTION OF FRACTIONS. 119. Reduction of Fractions is the process of changing their terms into others, without altering the value of the fractions. CASE I.-Reducing fractions to their lowest terms. 119.a. A fraction is said to be reduced to its lowest terms, when its numerator and denominator are expressed in the smallest numbers possible. Ex. 1. Reduce to its lowest terms. Suggestion.-Dividing both terms First Operation. of the fraction by 2, it becomes . 2)=3: then 3)}=1⁄2 Ans. Then, dividing both by 3, we ob tain, whose terms are the lowest to which the given fraction can be reduced. Or, divide both terms by their greatest Second Operation. common divisor, which is 6, and the given 6). Ans. fraction will be reduced to its lowest terms by a single division. (Art. 96.) 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; then divide this result as before, and so on till no number greater than 1 will exactly divide them; the last two quotients will be the lowest terms to which the given fraction can be reduced. Or, divide both the numerator and denominator by their greatest common divisor; and the quotients will be the lowest terms of the given fraction. (Art. 96.) OBS. 1. The value of a fraction is not altered by reducing it to its lowest terms; for the numerator and denominator are divided by the same number. (Art. 116. 2. When the terms of the fraction are small, the former method will generally be found to be the shorter and more convenient; but when the terms are large, it is often difficult to determine whether the fraction is in its simplest form, without finding their greatest common divisor. 2. Reduce to its lowest terms. 16 Ans.. QUEST.119. What is reduction of fractions? 119.a. What is meant by lowest terms of a fraction? 120. How is a fraction reduced to its lowest terms? Obs. Is the value of a fraction altered by reducing it to its lowest terms? Why not? |