Ex. 3.-Multiply together 21, 1, 4, and 11. Reduce the mixed numbers to improper fractions. Throw the whole number into a fractional form, by giving it 1 for its denominator. Then proceed as in Example 2. Cancel 17 in the first numerator and second denominator. Cancel 4 in the third numerator and first denominator. Cancel 9 in the second numerator and third denominator. Multiply the remaining factors. Reduce the improper fraction obtained to a mixed number. 165. RULE.-1. Cancel factors common to any nume rator and denominator. Then multiply the numerators together for a new numerator, and the denominators for a new denominator. 2. Whole numbers must first be reduced to a fractional form, and mixed numbers to improper fractions. Reduce the result, when necessary, to a whole or mixed number. Go through Example 2. Explain Example 3.-165. Recite the rule for multiply. ing a fraction by a fraction, or reducing a compound fraction to a simple one. 10. Reduce of § of of to its simplest form. 11. Find the product of ‡ of 11 and 3 of 14. Ans. 143. Ans. 44 147 Add the products. 15. Multiply 6 by 211 by 8%. Add the products. 16. Find the difference between 31× 8 and 55 × 2. Ans. 11. 19. How much more is 6 times than 18 times? 20. Multiply 3+3+37 by 1+1. 21. Reduce of 18 of 30 of 5 of 11. Ans. Ans.. Ans. 28. Ans. 20%. 22. Reduce of of of 1 of 31. 23. Multiply×3×× by 7. Division of Fractions, Reduction of Complex Fractions. Three fourths divided by fire sixths. Ans. 48. Same ans. for both. 星÷8 Or, 166. A fraction divided by a fraction may be expressed in two ways with the sign of division, or in the form of a complex fraction. Whichever the division is exway pressed, the operation is the same. Hence, to reduce a complex fraction to a simple one, take the denominator as a divisor, and proceed as in division of fractions. 167. CASE I.-To divide a fraction by a whole number. We found in § 137 that dividing the numerator or multiplying the denominator by any number divides the fraction by that number. Hence the rule: RULE.-Divide the numerator of the fraction by the whole number when it can be done without a remainder; when not, multiply its denominator. 166. In what two ways may a fraction divided by a fraction be expressed?-167. What is the first case of division of fractions? Recite the rule for dividing a fraction by a whole number. Dividing the numerator diminishes the number of parts as many times as there are units in the divisor. Multiplying the denominator diminishes the size of the parts as many times as there are units in the multiplier. EXAMPLE 1.-Divide 36 by 6. 36 is exactly divisible by 6. Divide it. Ans. 168. CASE II.-To divide a mixed by a whole number. RULE.-1. Divide the integral and the fractional part separately, and combine the quotients. What is the effect of dividing the numerator? Of multiplying the denominator? Solve the examples.-168. What is the second case of division of fractions? Explain the given examples. Recite the rule for dividing a mixed by a whole number. 2. If, on dividing the integral part, there is a remainder, prefix it to the fractional part, reduce to an improper` fraction, divide as in Case I., and combine this quotient with that obtained by dividing the integral part. 2671 19 36 24 17. Reduce 4; 25; 10; 1444, 28878; 451; 169. CASE III.-To divide a fraction, whole, or mixed number, by a fraction or mixed number. Ex. 1.-How many times is contained in ? is contained in 1, 7 times. In 3 it is contained of 7 times, or 2 times. But is twice as great as, and hence is contained only half as many times. of 1 = 2 Ans. Now, what have we done to the dividend, to produce the quotient ? We have multiplied it by the divisor inverted. Hence the rule: RULE.-1. Multiply the dividend by the divisor inverted. 2. Whole and mixed numbers must first be reduced to improper fractions. 169. What is the third case of division of fractions? How many times is contained in? What have we done to the dividend, to produce the quotient? Recite the rule for dividing one fraction by another 31 EXAMPLE 2.-Reduce to its simplest form. 211 Reduce the numerator to an improper fraction: Reduce the denominator to an improper fraction: Multiply the dividend by the divisor inverted, cancelling common factors. Reduce the result to a mixed number. EXAMPLE 3.-Divide 4 by 4. 13 11 X = = 13 Ans The denominators, being the same, are cancelled when the divisor is inverted, and we have only to divide 4, the numerator of the dividend, by 2, the numerator of the divisor. Hence, When the fractions have a common denominator, reject it, and divide the numerator of the dividend by that of the divisor. EXAMPLES FOR PRACTICE. Find the value of the following: 7/2 =2 Ans. 18. How many times can a pitcher holding 17 quarts be filled from a pail containing 5 quarts? Solve and explain the given examples. When may cancellation be brought to bear? When the fractions have a common denominator, what is the best mode of proceeding? |