8. Explain why changing the fractions in example 6 to twenty-fourths does not change the value of the fractions. Draw figures to illustrate. 9. Explain why changing the fractions in example 7 to sixths does not change the value of the fractions. Draw figures to illustrate. Changing fractions to higher terms. When several fractions are equal in value, the one having the largest numerator and denominator is said to have the highest terms, and the one having the smallest numerator and denominator is said to have the lowest terms. 1. In j= = * = it, which fraction has the highest terms ? the lowest terms ? 2. Explain why the terms in & are larger than in ž? 3. In changing a fraction to higher terms, do we multiply or divide both terms of the fraction ? 4. Explain the meaning of equivalent fractions. 5. Change 1, s, į, 6, 7, 11, to 48ths. 6. In changing 1 to 8ths, into how many parts is the fractional unit divided? What is the size of each part? What are the number of parts? Written Work 1. Change & to 12ths. 3 x 3= 9 Since 12 = 4= 3, we must multiply both terms of the fraction by 3 to obtain a fraction of the 4 x3 =12 same value with 12 for the denominator. To change a fraction to higher terms divide the required denominator by the denominator of the fraction, and multiply both terms by the quotient. 2. Change to 40ths 1, 4, Bo, f, f. : 8. $= 9. *= 144: 10. = 15 14. i to 96ths 15. i to 121sts 13. á to 60ths 16. 16 to 128ths Changing a fraction to its lowest terms. 1. In the square 1&=\; &=; 18 = $ 2. = ; 16 = $. 3. Draw figures to show that 1= }; 14 = }; f= . 4. Which do you prefer, 18 of a | = 16/16 dollar or i of a dollar? Explain why. 5. By what number do we divide both terms of 26 to change the fraction to f? 6. When is changed to $, is it in its lowest terms? Express it in its lowest terms. 7. Change to lowest terms 4, g, 12, 13, 15, 16, 17, 1o, 11. Observe that in changing these fractions to the lowest terms, both terms of the fraction are divided by their greatest common divisor. 8. Explain by drawing figures why dividing both terms of a fraction by the same number does not change its value. Written Work 2 = 1. Change to its lowest terms. Since the value of a fraction is not changed 36 +2=183=6 by dividing both terms by the same number, we 42 21 : 3=7 divide both numerator and denominator first Or by 2; then we divide the terms of the resultg.c.d. = 6 ing fraction di by 3. Since 6 and 7 have no 36 +6=6 common divisor except 1, $ is in the lowest 42 +6=7 terms. Or we may, in one step, divide both terms of the fraction by their g. c. d., 6. To change a fraction to its lowest terms cancel all common factors from both terms or divide both terms by their greatest common divisor. Change to lowest terms : 14. 18 160 3. 6. 18 9. 141 12. ja 15. 11% 4. 49 7. && 13. 56 144 16. 378 2. 14 5. 8. 108 84 11. 100 48 180 10. 132 192 Changing to similar fractions. 2. To what common denominator may you change and to make them similar? {= 0; $= to: Change to similar fractions: 3. 4, 7. 11,3 11. 2; } 15. &, g 4.1, 12 8. } 12. , 16. 11, 5. , 9. 4, } 13. & 17. 11, q 유 6. 1o, 10. 14. s, 1 18. 15 % Note. — 4 and 1 may be changed to 12ths or 24ths, and in either case have common denominators, but when changed to 12ths, they are expressed in their least common denominator (1. c.d.). 69 Changing fractions to their least common denominator. 1. What is the least common multiple of 2, 3, and 4 ? 2. What is the l.c. m. of the denominators of 1, , and ? 3. Change the fractions i, j, and į to equivalent fractions having the l. c. m. of the denominators for a like denominator. How, then, do you change fractions to their least common denominator ? Written Work 1. Change 1, y, and to similar fractions having the least common denominator. 2)4 10 2 5 In finding the least common multiple we may re1. c. d. = ject 5, since 10 is a multiple of 5. 2 x 2 x 5 = 20 The least common multiple of the denominators 3 x 5 15 4 and 10 is 2 x 2 x 5, or 20, which is the least common denominator of the given fractions. 4 x 5 20 To change to 20ths divide the required denomi2x4 8 nator by the given denominator and multiply both 5x4 20 terms by the quotient. Proceed in the same way 7 x 2 14 with the other fractions. 10 x 220 Change to similiar fractions having the l. c. d. : 2. }, } 11. , , 20. lk, 24, 3 3. 1, 12. 4, 5, } 21. 29, , 4. , 13. 3, 4, Bo 22. 1, $, 26 5. , 14. , , Í 23. 15, , 6.6 15. 4, 1, 24. 12, 1; } 7. o 16. 16 25. 20, , 8. s, iz 17. , La 26. $,$, } 9. , , 1 18. 6, 4, 13 27. 8, , 1 10. á, , iš 19. 11, 1 28. 6, 4, 1 ADDITION OF FRACTIONS 1. Observe the different kinds of units in each of the following numbers": 5, 10 ft., 4 bu., 7, 9 ft., 8 oz., $6, 12 oz., 10, 6 bu. 2. Add the numbers that have units of the same kind. 3. Why can you not add 10 feet and 4 bushels ? 4. What change must you make in 3 bushels, 3 pints, and 3 quarts before you can add them ? 5. Can you add f and without change ? Can you add them after changing both to 20ths ? 6. What change must be made in fractions whose denominators are not alike before they can be added ? Written Work Adding fractions. 1. Add f and 5. The l.c.d. is 36; {=}} and = 18; 3% + 38 = . 36 = 1.c.d. $7, or 138. Observe the three steps in adding fractions : f=17 1. If necessary, make the fractions similar, that is, change them to a common denominator. 2. Write the sum of the numerators over * + 5 = 4] or 131 the common denominator. 3. Change the sum to its simplest form. 14. , }, 15. , 16. , , 17. 9, 1, 18. ], to s 13. 4, , ) 19. 11, 4, 1 |