the denominator by 4, the quotient is 3, and 4 the fractional unit becomes, which is 4 times as great as, because, if be divided into 4 equal parts each part will be. If we take this fractional unit 5 times, the result will be 4 times as great as; therefore, we have PROPOSITION II.-If the denominator of a fraction be divided by any multiplier, the value of the fraction will be increased as many times as there are units in that multiplier. by 3. OPERATION. 116. Let it be required to divide ANALYSIS.-In 191, there are 9 fractional units, each of which is, and these are to be divided by 3. But 9 things, divided by 3, gives 3 things of the same kind for a quotient; hence, the quotient is 3 elevenths, a number which is one-third of; hence, we have 11 PROPOSITION III.-If the numerator of a fraction be divided by any number, the fraction will be diminished as many times as there are units in the divisor. 115. What is proved in proposition II? 116. What is proved in proposition III? 117. Let it be required to divide by 3. ANALYSIS.--In, there are 9 fractional units, each of which is 59 OPERATION. Now, if we 11÷ 3 =11x8 = 8. multiply the denominator by 3, it becomes is 33, and the fractional unit becomes, which is one-third part of If, then, we take this fraciional unit 9 times, the result hence, we have divided the fraction just one-third part of by 3 therefore, we have, PROPOSITION IV.—If the denominator of a fraction be multiplied by any divisor, the fraction will be diminished as many times as there are units in that divisor. 118. Let it be required to multiply both terms of the fraction by 4. OPERATION. 3 X 4 5×4 ANALYSIS.-In, the fractional unit is, and it is taken 3 times. By multiplying the denominator by 4, the fractional unit becomes, the value of which is is one-fourth of. By multiplying the numerator by 4, we increase the number of fractional units taken, 4 times; that 117. If the denominator of a fraction be multiplied by any number, how will the value of the fraction be effected? 118. If both terms of a fraction be multiplied by any number, how will the value of the fraction be effected? is, we increase the number just as many times as we decrease the value ; hence, the value of the fraction is not changed: therefore, we have PROPOSITION V.-If both terms of a fraction be multiplied by the same number, the value of the fraction will not be changed. EXAMPLES. 1. Multiply both terms of the fraction by 4, by 6, and by 5. 8 2. Multiply both terms of by 5, by 8, by 9, and 11. 3. Multiply both terms of 16 by 7, by 8, and 9. 4. Multiply both terms 14 by 5, 8, 6, and 12. 5. Multiply both terms of 23 by 2, 3, 4, and 5. 119. Let it be required to divide the numerator and denominator of by 3. OPERATION. 6÷3 2 15÷35 ANALYSIS.--In, the fractional unit is, and it is taken 6 times. By dividing the denominator by 3, the fractional unit becomes, the value of which is 3 times as great as. By dividing the numerator by 3, we diminish the number of fractional units taken 3 times; that is, we diminish the number just as many times as we increase the value: hence, the value of the fraction is not changed; therefore, we have PROPOSITION VI.—If both terms of a fraction be divided by the same number, the value of the fraction will not be changed. EXAMPLES. 1. Divide both terms of by 2 and by 4. by 3. 3. Divide both terms of 34 by 2, 3, 4, 6, and 12. 24 36 4. Divide both terms of 4 by 2, 4, 8, and 16. 72 5. Divide both terms of 32 by 2, 3, 4, 6, and 12. 96 6. Divide both terms of 36 by 2, 3, 4, 6, and 36. 144 119. If both terms of a fraction be divided by any number, how will the value of the fraction be effected? REDUCTION OF FRACTIONS. 120. REDUCTION OF FRACTIONS is the operation of changing the fractional unit without altering the value of the fraction. A fraction is in its lowest terms, when the numerator and denominator have no common factor. CASE I. 121. To reduce a fraction to its lowest terms. 1ST. OPERATION. ANALYSIS. By inspection, it is seen that 5 is a common factor of the numerator and denominator. Dividing by it, we have . We then see that 7 is a common factor of 14 and 35: dividing by it, we have . Now, there is no factor common to 2 and 5: therefore, is in its lowest terms. 2D OPERATION. 2d. The greatest common divisor of 70 and 175 is 35. (Art. 93); if we divide both terms of the fraction by it, we obtain, f. . The value of the fraction is not changed in either operation, since the numerator and denominator are both divided by the same number (Art. 119): hence, the following 35). RULE.-Divide the numerator and denominator by their common factors, until they become prime with respect to each other. Or: 2d. Divide the numerator and denominator by their greatest common divisor. 120. What is reduction of fractions? When is a fraction in its lowest terms? 121. How do you reduce a fraction to its lowest terms? 11. Reduce 792 by 2d. meth. 17. Reduce 160. 1386 2340 122. To reduce an improper fraction to an equivalent whole or mixed number. 1. In 278 how many entire units? ANALYSIS. Since there are 5 fifths in 1 unit, there will be in 278 fifths as many units 1 as 5 is contained times in 278, viz., 55 and § times. Hence, the following OPERATION. 5)278 558. RULE.-Divide the numerator by the denominator, and the quo-. tient will be the equivalent whole or mixed number. EXAMPLES. Reduce the following fractions to whole, or mixed numbers 123. To reduce a mixed number to an equivaleut improper fraction. 1. Reduce 12 to its equivalent improper fraction. 122. What is an improper fraction? How do you reduce an inproper fraction to its equivalent whole, or mixed number? 123. What is a mixed number? How do you reduce a mixed number to an improper fraction? How do you reduce a whole number to a fraction having a given denominator? |