(c) The Numerator shows how many of those parts are taken or expressed by the fraction. (d) The numerator and denominator are called the terms of the fraction. 3 Write the following fractions: three fourths, two thirds, seven eighths, nine tenths, seven elevenths, eight fifteenths. Read the following fractions: 7, 1, 8, 9, &, PT, 18, 15. 134. A SIMPLE FRACTION has but one numerator and one denominator; as 3, 4, 13. 6 135. A COMPOUND FRACTION is a fraction of a fraction; as .of, of 1. 136. A PROPER FRACTION is one whose numerator is less than the denominator; as §, 3, 3. 137. An IMPROPER FRACTION is one whose numerator equals or exceeds the denominator; as 1, 7, 2, 4. 138. A MIXED NUMBER is a whole number and a fraction united; as, 7, 51, 274. 139. The terms of a fraction sustain to each other the relation of dividend and divisor, the numerator answering to the dividend and the denominator to the divisor. That is, a fraction may be regarded as an expression of division. Hence, The VALUE of a fraction is the QUOTIENT of the numerator divided by the denominator, as = 93 = 3. It follows from this that the GENERAL PRINCIPLES of DIVISION (Arts. 85, 86, and 87) apply to all fractions. 134. What is a Simple Fraction? 135. Compound? 136. Proper? 137. Improper? 138. What is a mixed number? 139. What relation do the terms of a fraction sustain to each other? Which term answers to the dividend? Which to the divisor? How may a fraction be regarded? To what is the value of a fraction equivalent? What principles before stated apply to fractions? 1. Multiplying the numerator, if the denominator remains unaltered, multiplies the value of the fraction by the same number, as 2 x 2 = 1. × 2. Dividing the numerator, if the denominator remains unaltered, divides the value of the fraction by the same number, as 2 ÷ 2 = 4. In the above cases it will be seen that the size of the parts, (fourths,) remains the same, but the number of the parts is increased or diminished. 3. Multiplying the denominator, if the numerator remains unaltered, divides the value of the fraction by the same number, as 2 × 2 = 4. Dividing the denominator, if the numerator remains unaltered, multiplies the value of the fraction by the same number, as ÷ 2 In the last two cases it will be seen that the number of parts (numerators) remains the same, but the size of the parts (denominators) is increased or diminished. 5. If the numerator and denominator are both multiplied or divided by the same number the value of the fraction is not 2 X 2 4 22 4 X 2 altered, as -- or = 1 Hence, the following general law in regard to Fractions may be stated, That any change in the NUMERATOR causes a LIKE change in the value of the fraction; and any change in the DENOMINATOR causes an OPPOSITE change in the value of the fraction. Upon these principles all the following operations upon fractions depend. 139. Give the 1st principle and illustrate it. The 2d principle. The 3d principle. The 4th principle. The 5th principle. What general law is given ? CASE 1. 140. To reduce a mixed number to an improper fraction. Ex. 1. In 73 how many fifths? OPERATION. 73 5 38 Ars. 5 In a unit there are five fifths; and in seven units there are seven times five fifths, or 35 fifths, which with the 3 fifths in the example 38 fifths38. RULE. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and under the sum write the denominator. 9. Reduce 85 to an improper fraction. J In 19 how many fourteenths? 14 Reduce 4913 to an improper fraction. NOTE. To reduce a whole number to a fraction having any given denominator, multiply the whole number by the proposed denominator, and under the product write the denominator. CASE 2. 141. To reduce an improper fraction to a whole or mixed number. Ex. 1. How many units in 17? 4=17÷÷4=4} In one unit there are four fourths, and in seventeen fourths there are as many units as four is contained times in seventeen. 140. Explain the operation in Case 1. Rule for reducing a mixed number to an improper fraction? 141. Rule for reducing an improper fraction to a whole or mixed number? RULE. Divide the numerator by the denominator; if there is any remainder, place it over the divisor, and annex the fraction so formed to the quotient. 2. Reduce 19 to a mixed number. 3. Reduce 29 to a mixed number. 5 Ans, 2. Ans. 5. Ans. 3. Ans. 98. Ans. 1012. Ans. 7. NOTE. The denominator of a fraction being a divisor, it follows that whenever the denominator exactly measures the numerator, the quotient will be a whole number. (See Exs. 9 and 10.) CASE 3. 142. To reduce a fraction to its lowest terms. Ex. 1. Reduce 1st OPERATION 2334=18 313= 3 Ans 2d OPERATION. to its lowest terms. RULE 1. Divide each term by any factor common to them, then divide these quotients by any factor common to THEM, and so proceed till the quotients are mutually prime. (Art. 139, 5th.) Find the greatest common divisor, (Art. 125,) Ans. and by it divide both terms of the fraction. RULE 2. Divide each term by their greatest common divisor. 2. Reduce to its lowest terms. 3. Reduce 56 to its lowest terms. Ans. 3. Ans. . 142. Rule for reducing a fraction to its lowest terms? Second rule for reducing a fraction to its lowest terms? 143. To multiply a fraction by a whole number. Ex. 1. Multiply 1st OPERATION. 1 × 4 = 28 == 2d OPERATION. by 4. It is just as evident that 4 times 7 eighths (3) are 28 eighths (28) as it is that 4 times 7 boys are 28 boys. If we divide the denominator by 4 we obtain the same result as before. In the first operation we increase the number of the parts four-fold, and in the second, we increase the size or value of the parts four-fold while the number of parts remains the Hence the following same. RULE 1 Multiply the numerator by the whole number. Or, Ans. 345 864 143. First rule for multiplying a fraction by a whole number? Second rule? 6. Multiply by 3. |