Common Fractions. As explained in Art. 23, Note 13, an expression of division in which the dividend is written above the divisor is called a Fraction. It is evident that an application of the principles of division will enable one to multiply or to divide a fraction or to change its form without affecting its value. When the dividend and the divisor of a fraction have no common factor, the fraction is said to be in its Lowest Terms. 46. To Reduce a Fraction to its Lowest Terms. There are 128 cubic feet in a cord. What part of a cord, then, is one cubic foot? What part of a cord are 2 cubic feet? 9 cubic feet? 27 cubic feet? 64 cubic feet? 84 cubic feet? 116 cubic feet? A wood-chopper cuts 84 cubic feet of wood on Monday and 116 cubic feet on Tuesday. What part of a cord does he cut on Monday? What part of a cord on Tuesday? We wish to reduce the expressions 18 84 and 84 128 118 = 32 116 to the simplest equivalent expressions. Explain the reduction of the first expression. Give a rule for reducing a fraction to its lowest terms. Ex. 1. We are to reduce 1 to its lowest terms. 256 314 Dividing both dividend and divisor by the same number does not change the value of an expression of division. We therefore divide both terms of the fraction 144 by 8, and both terms of, the fraction thus obtained, by 2. We thus obtain as the simplest expression for the given fraction. Ex. 2. Explain the solution of Ex. 2. 47. To Add or to Subtract Fractions having a Common Denominator. Suppose that the chopper had cut one cord on Monday, and two cords on Tuesday. How would we find the amount cut by him on both days? What is the common unit to be added? If instead, as previously stated, he cuts?! of a cord on Monday, and of a cord on Tuesday, what is the common unit to be added? How many of these units does he cut On Monday? On Tuesday? On both days? Write, then, an expression indicating the amount he cuts on both days. Give a rule for adding fractions having a common denominator. Write an expression indicating the difference between the amount he cuts on Monday and the amount he cuts on Tuesday. Give, then, a rule for subtracting fractions having a common denominator. These numbers have a common unit,. Therefore, we proceed with our addition as though we were adding tens or units; thus, 7, 12, 21 eighteenths. Our answer, therefore, is, or 15, or 11. Ex. 2. Explain the solution of Ex. 2. NOTE. A fraction in which the indicated operation can be performed is called an Improper Fraction. An improper fraction evidently can be changed to a mixed number by performing the indicated operation. 48. To Add or to Subtract Fractions not having a Common Denominator. Suppose that the chopper had cut two cords on Monday and 256 cubic feet on Tuesday. Could we have directly united these two amounts into one equivalent amount? Why not? How can we change the form of one of these amounts so as to render it capable of being united with the other? Suppose that he had cut of a cord on Monday, and of a cord on Tuesday. Could we have directly united these two amounts into one equivalent amount? Why not? What, then, must we do with fractions that have not a common denominator to render them capable of addition? Give Prin. 5 of Division. By what must both terms of the fraction be multiplied to change it to an equivalent fraction with the divisor 64? 21, then, equals how many sixty-fourths? Give a rule for changing a fraction to an equivalent fraction with higher terms. Suppose that the second fraction had been Could we have conveniently changed tion with the divisor 80? instead of. to an equivalent frac What is the least common multiple of 32 and 80 ? 3 and § can, then, be changed to equivalent fractions having what common denominator? equals how many one hundred sixtieths? equals how many one hundred sixtieths? Give a rule for reducing fractions to equivalent fractions having a common denominator. Give a rule for adding fractions not having a common denominator. EXPLANATIONS. Ex. 1. We are to add 15, 2, and 3. The first step is to change these fractions to equivalent fractions having a common denominator. The least common multiple of the denominators 15, 20, and 25 is 300. To change the denominator 15 to 300 we must multiply it by 20. But if we multiply the divisor 15 by 20 we must also multiply the dividend 7 by 20. The product of 7 by 20 is 140; therefore, 15 3009 108 140 120 Proceeding in the same way with the remaining fractions, we find that equals 128, and that equals 388. Adding 38, 338, and 108 we find that the sum is 365, or 13. Therefore, 173 +2% + 2=143. 3009 Ex. 2. Explain the solution of Ex. 2. Ex. 62. Read carefully the Notes at the end of this Article, and then find the value of the following expressions: NOTE 2. Observe that the following are the principal steps of the solution. We find the least common multiple of the denominators to be 96 X3X7, or 288 X 7, or 2016. To find by what 42 must be multiplied to produce the least common multiple, we imagine 42 to be written under the expression 96 X3X7. We next mentally cancel 3 X 7 from the dividend and 21 from the divisor of this new expression. We next divide 96, the remaining factor of the dividend, by 2, the remaining factor of the divisor. The quotient thus obtained is 48. We next multiply 33 by 6 and by 8, the factors of 48. The product thus obtained is 1584. Our first numerator, therefore, is 1584. Explain the finding of the second numerator; of the third numerator. Explain the remaining steps of the solution. 49. To Add or to Subtract Mixed Numbers. A decimal, in the sense that it represents one or more equal parts of a unit, is evidently a fraction. In distinction, fractions that are expressed by writing the dividend above and the divisor below a horizontal line are called Common Fractions. The term Mixed Number is used in the same sense with common fractions as with decimals. We wish to add the mixed numbers 21, 43, and 5¦. 2 + 3 + 1 = ? 2+4+5=? 11+1=? 2+4+5, then, = ? Give, then a rule for adding mixed numbers. We wish to subtract 23 from 53. Change and to equivalent fractions having a common denominator. What are these fractions? Can we subtract from ? Suppose that we add 1 to the minuend. How many integers must we add to the subtrahend to counterbalance this addition? Give, then, a rule for subtracting mixed numbers. |