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Dividing any number by a fraction by inverting the divisor and multiplying.

1. 11= how many ? 4. 1+ = how many ? 2. 1+ 4 = how many ? 5. 1+ & = how many ? 3. 1+1 = how many ? 6. 1+1= how many ? Observe that each fraction is contained in 1 as many times as the numerator of the fraction is contained in the denominator.

7. 3+ 1= how many ? 9. 3+ 1 = how many ? 8. 4+ 1 = how many ? 10. 4+ 8 = how many ? Since 1 is contained in 1, 4 or 4 times, 4 is contained in 3, 3 x 4 = 1, or 12 times. A short method of dividing any number by a fraction is to multiply the given number by the number of times the fraction is contained in 1.

Since 1+1=3, then 1= must equal 1 of 3 or 3. Divide the denominator of 3 by the numerator. Show that 1- ģ=1x . Observe, then, that the number of times any fraction is contained in 1 equals the fraction inverted ; thus, is contained in 1, times. 11. Divide 8 by 4.

4 is contained in 1, 4 times. In 8 it is contained 8 x 4 times. 8x = 8 x 4

- 16.

2

Show 12. State the two methods of dividing fractions. that improper fractions and whole and mixed numbers are divided on the same principles as proper fractions.

Find quotients :
13. 4 }
16. 11 +

19. 10:
14. 11 + 3
17. 8 ; 1

20. 11+1 18.7

21. 12:56

15. 5

=

Written Work 1. Divide f by : 2

Since 1 + $ = , + = f of 4 2

5 5

Х 3

Canceling, the result is z. 5 3 4

6 2

1 divided by 8 equals $; 5$, or 2. Divide 54 by 8.

, divided by 8, equals ze times }, 54 +8= 4 x 5 = 48

or 3. Divide 3. by 45.

1 divided by 43, or 4, equals it; 23 3 23

33, or 2*, divided by 4s, equals seus 35 - 43 х 6 14 28

times it or it.
2
Find the quotients :
4. 13:5 10. 11 - 5 16. 11 + 11
5. 25; } 11. 11 = 1 17. 1495 + 1
6. 121 - 11 12. 15 = 1 18. 36 = 36
7. 256 = 1

13. 16 = 1 19. +
8. 42:
14. 18 +

20. 14 = 5
9. 55: A1

15. 21 - 1 21. *7 = 33 Change both numbers to fractions ; then invert the divisor and multiply.

Notes. — 1. When possible, cancellation should be used.

2. In solving problems that contain fractions, sometimes the method of changing to like fractional units is the shorter. 22. 2) = 21 28.

184 = 7 34, 6716 + 7 23. 12= 163 29. 193 + 1,3 35. 8841 163 24. 8} = 48 30. 160 ;

36. 30*6 = 54 25. 5} = 41 31. 163 = 6 37. 95+ 73 26. 7: 61 32. 77 = 21 38. 178} + 27. 84 : 41 33. 103 - 10%

39. 200

• 67

21

9

5

40. When common laborers earn $1.75 in a day of 91 hours, how much do they earn per hour ?

41. The product of two numbers is 213. One of the numbers is 4%, what is the other?

42. The material for a dress, containing 111 yards of cloth, cost $18. What was the cost per yard ?

43. I invested $447 in books at $3} per volume. How many volumes did I buy?

44. Duffy ran 100 yards in 9 seconds. What was his rate per second ?

45. Dan Patch paced a mile in 1161 seconds. How many yards did he cover in a second ?

46. At $ apiece, how many tablets can be bought for $.

47. By the Pennsylvania Railroad the distance from Pittsburg to Cleveland is 150 miles. A "fast" train runs the distance in 31 hours. What is the rate of the train per hour?

48. When cement is worth $37 per barrel, how many barrels can be bought for $1027 ?

49. At $14 each, how many baseballs can be purchased for $221?

50. There are 4124 grains of silver in a silver dollar. How many dollars can be made from 9900 grains ?

51. A franc is worth 19% cents. How many francs are equal to 38% cents ?

52. How many barrels, each holding 24 bushels, can be filled from 899] bushels of apples ?

53. Into how many pieces, each 11 yards long, can a piece of ribbon 12 yards long be cut ?

54. How many cans, each holding 27 gallons, can be filled from a barrel of oil containing 50 gallons ?

SHORT METHODS

Learn the following parts of $1:

64 cents = 16 of $.1. 84 cents = 1 of $1. 124 cents = of $1. 163 cents = į of $1. 20 cents =

of $1. 25 cents =

4 of $1. 33} cents = í of $1.

37] cents = g of $1.
50 cents = of $1.
621 cents = of $1.
66 cents = of $1.
75 cents å of $1.
83 cents = of $1.
87} cents = f of $1.

644 = $ to Since at $1 a 1. Find the cost of 32 pounds pound, 32 lb. would cost $ 32, at of rice @ 61¢ a pound.

$ to a pound, they will cost to of $32, or $2.

Find, by a short method, the cost of the following articles : 2. 24 ft. hose @ 834. 8. 48 neckties @ 371¢. 3. 40 rd. fence @ 12%. 9. 30 shirts @ 50%. 4. 36 yd. challis @ 163€. 10. 72 gal. sirup @ 121¢. 5. 35 gal. molasses @ 20%. 11. 150 yd. oilcloth @ 663$. 6. 44 qt. strawberries @ 254. 12. 36 doz. collars @ 75%. 7. 21 yd. flannel @ 331¢. 13. 64 bu. wheat @ 8734.

14. At 614 per gal., how many gallons of oil can be bought for $3?

1
16

Since 64¢ = $ 16, as many gallons 15. $ 3

3 x =48,
16
1

can be bought as $1 is contained the number of gallons.

times in $3, which is 48 times.

Hence 48 gallons can be bought. 16. At 121¢ per yd., how many yards of ribbon can be bought for 25$? for 50¢ ? for $1.50 ? for $3? for $5?

17. At 81$ per lb., how many pounds of raisins can be bought for $1? for $5? for $2.50 ?

18. At 25 cents each, how many fishing lines can be bought for $.50 ? for $1 ? $ 1.50 ?

19. At 373¢ a gallon, how many gallons of molasses can be bought for $2? for $4?

20. At 33}$ per pair, how many pairs of stockings can be bought for $1? for $3? for $5? for $15?

21. At 25$ per meal, how many persons can be fed for $2? for $5? for $9? for $20 ?

22. At 164¢ per gallon, how many gallons of gasoline can be bought for $.50 ? for $2? for $6 ? for $9?

how often

23. 25 cents is contained how often in $1 ? in $5?

24. At 25 cents a pound, how many pounds of butter can be bought for $5 ?

25. At 624 cents a yard, how many yards of serge can be bought for $1.25 ?

26. At 33} cents a pound, how many pounds of butter can be bought for $4? for $6? for $10?

Divide the following at sight : 27. $8 by 25 cents.

32. $32 by 20 cents. 28. $11 by 33} cents. 33. $40 by 64 cents. 29. $18 by 50 cents. 34. $50 by 122 cents. 30. $27 by 163 cents. 35. $72 by 25 cents. 31. $12 by 8} cents. 36. $90 by 33} cents.

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