11. How much is of 30? Since 1 third of 30 is 10, 2 thirds of 30 is 2 times 20. If a pound of sugar cost 15 cents, how much will 1 third of a pound cost? 2 thirds? 21. Paid $60 for a cow, and 2 twelfths as much for some hay; how much did the hay cost? 22. What part of 7 is 2? of 7, or 2 sevenths of 7. Therefore, 2 is 4 of 7. ·Since 1 is 1 seventh of 7, 2 is 2 times 1 seventh 27. What part of 16 yards is 5 yards? 11 yards? 28. What part of $20 is $7? $13? $17? 29. What part of 10 pounds is 5 pounds? 8 pounds? 30. What part of 27 tons is 3 tons? 9 tons? 31. What part of $32 is $12? $4? $16? 32. What part of 40 bushels is 20 bushels? Explain the solution of example 11. Of example 22. REDUCTION. 149. Reduction of Fractions is the process of changing their form, without changing their value. Case I. A Fraction to its Lowest Terms. 150. A fraction is in its Lowest or Smallest Terms, when the numerator and denominator have no common factor greater than 1. OPERATION. 4 2 X 2 2 10 5X2 5 Ans. Dividing both terms of a fraction by the same number, or canceling equal factors in both (Art. 135), will not change the value expressed. Now the only factor greater than 1, com mon to both terms of the given fraction, is 2, and canceling, we have . Therefore, in its lowest terms is . 10 Rule. Cancel in the numerator and denominator all the factors common to both. What is Reduction of fractions? When is a fraction in its low est terms? Why is not the value of a fraction changed by dividing both terms by the same number? What is the Rule? Case II. An Integer to a Fraction with a given Denomi nator. 151. 1. Reduce 8 to a fraction whose denominator is 7. Rule.-Multiply the whole number by the given denominator, and write the product over the denominator. 8. 9 is equal to what fraction having 15 for a denominator? Ans. 135. Ans. 1441. 9. Reduce 131 to a fraction whose denominator shall be 11. Case III. A Mixed Number to an Improper Fraction. 152. 1. Reduce 5 to thirds. 2 thirds, are 17 and 5 times 3 thirds plus, or 15 thirds plus thirds, or . Therefore, 5 is equal to . Explain the operation of reducing an integer to a fraction with a given denominator. Of reducing a mixed number to an improper fraction. Rule. - Multiply the integral part of the mixed number by the denominator of the fractional part; to the product add the numerator; and write the result over the denominator. 10. In 10 dollar, how many fourths of a dollar? 11. In 221 bushels, how many elevenths of a bushel? Case IV. An Improper Fraction to an Integer or Mixed Number. 153. 1. Reduce 17 to an integer or mixed number. OPERATION. 53, Ans. Since there are 3 thirds in 1 unit, there will be in 17 thirds as many units as 3 is contained times in 17, or 5. Therefore, y is equal to 5. Rule. - Divide the numerator by the denominator. Explain the operation of reducing an improper fraction to an in teger or mixed number. Recite the Rule. 10. In 1g of a dollar how many dollars? 11. What is the value of 8224 miles? Ans. $333. Ans. 514 miles. COMMON DENOMINATOR. 154. Fractions have a Common Denominator when they have the same number for a denominator. 1. Reduce and to fractions having a common denominator. 5 3 7 = OPERATION. 2×7 14 5×7 3 × 5 7 × 5 = 35 Ans. 15 35 Since multiplying both the numerator and denominator of a fraction by the same number does not change its value (Art. 148), we multiply both terms of the fraction by 7, the denomi nator of the second fraction, and have =; and both terms of the fraction by 5, the denominator of the first fraction, and have. Therefore, and are equal to 1 and 15. 2. Reduce,, and to fractions having a common denominator. When have fractions a Common Denominator? Explain the manner of finding the common denominator by the first operation. |