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54. Two kinds of Division. Since 2 x $4 = $8, therefore $8 - $4 = 2, and $8 = 2 = $4. That is,

If the dividend and the divisor are concrete, they must be like numbers, and the quotient is abstract.

For example, $8 = $4 = 2. This is sometimes called measuring because the $8 is measured by the $4.

If the dividend is concrete and the divisor abstract, the quotient is like the dividend.

For example, $8:2=$4. This is sometimes called partition, because the $8 is separated into 2 equal parts.

55. Check. If there is no remainder, the dividend is the product of the quotient and the divisor. To check the work in division, therefore, we multiply the quotient by the divisor; this product added to the remainder, if any, should equal the dividend.

For example, 45 – 5 = 9; then 5 x 9 = 45. Again 48 + 5 = 9 with remainder 3; then 5 x 9 = 45, and 45 + 3 = 48.

56. Short Division. When the divisor is so small that the work may be done mentally the process is called short division.

(1) Required to divide 342 by 3. 3)300 + 30 +12

If 342 is separated as shown, we can 100 + 10 + 4=114

divide each part by 3. This is what we

do mentally below. We write the numbers as shown; then 3 is contained in 3 once (that is, 100 times in 300); 3 is contained in 4 once with remainder 1 (that is, 10 times in 40 with

remainder 10); this 1 is 10 units, and with the 2 makes 12. 3)342 Then 3 is contained in 12 four times. Therefore the quotient 114

is 114. We think: "3 in 3, 1; in 4, 1; in 12, 4.” (2) Required to divide 42,364 by 7. 7)42364

Here 7 is not contained in 4, but 7 is contained in 42

six times. Then 7 is not contained in 3, so we write a 0 6052

beneath the 3. Then 7 is contained in 36 five times with remainder 1, and in 14 twice. Therefore the quotient is 6052.

(3) Required to divide 35,282 by 9. 9)35282

Here 9 is contained in 35 three times with remainder 3920

8; in 82 nine times with remainder 1; in 18 twice; in 2

no times with remainder 2. Therefore the quotient is 3920 with remainder 2, and this is usually written 3920.

(4) Required to divide $27.53 by 4. 4)$27.53

Dividing as before, we have $27 4= $6, with $3 $6.88 remaining ; $3.50 = 4 = $0.8 with $0.30 remaining ;

$0.33 · 4 = $0.08, with $0.01 remaining. Therefore the quotient is $6.884

EXERCISE 17

Find the quotient of: 1. 232 • 2.

19. 4213 4. 2. 345 : 3.

20. 2893 ; 5. 3. 416 - 4.

21. 6271 : 6. 4. 325 : 5.

22. 7356 : 7. 5. 738; 6.

23. 9123 +8. 6. 924 7.

24. 8077 9. 7. 592 : 8.

25. $47.24 : 2. 8. 837 : 9.

26. $33.15 - 3. 9. 3242 = 2.

27. $28.72_4. 10. 4122 · 3. 28. $32.75 +5. 11. 6004 : 4.

29. $21.30 : 6. 12. 3015 : 5. 30. $29.47 : 7. 13. 4032 : 6. 31. $65.68 + 8. 14. 3073: 7. 32. $83.61 : 9. 15. 7368 ; 8. 33. 81,004 - 4. 16. 8019: 9. 34. 29,005 : 5. 17. 5555 = 2. 35. 31,266 : 6. 18. 6014 - 3. 36. 40,040 - 7.

37. 70,328 8. 38. 32,013 - 9. 39. 31,201 • 2. 40. 34,078 + 3. 41. 26,231 • 4. 42. 70,023 + 5. 43. 80,335 : 6. 44. 29,321 - 7. 45. 42,345 : 8. 46. 76,209 = 9. 47. $35.21 : 2. 48. $72.43 3. 49. $61.23 - 4. 50. $72.36 +5. 51. $82.01 - 6. 52. $42.03 +7. 53. $65.00 8. 54. $93.65 : 9.

EXERCISE 18

1. If 3 head of cattle cost $117, how much will 1 head cost ?

2. It is 108 feet around a square. What is the length of each side ?

3. There being 7 days in a week, how many weeks are there in 364 days ?

4. There being 3 feet in a yard, how many yards are there in 432 feet?

5. There being 4 quarts in a gallon, how many gallons are there in 236 quarts ?

6. It is 132 feet around a triangle whose sides are all equal. What is the length of each side ?

7. At $9 a dozen, how many dozen cups and saucers can be bought for $1116 ?

8. At $4 a dozen, how many dozen handkerchiefs can be bought for $1152 ?

9. A dealer pays $1224 for 9 wagons. What is the average price?

10. A wholesale grocer pays $1728 for a shipment of chocolate at $3

per
dozen cans. How

many

dozen cans does he buy?

11. A wholesale grocer pays $4320 for some cocoa at $6 per dozen cans. How

many dozen cans does he buy ? 12. A hotel pays $13.20 for 8 dozen packages of breakfast food. How much does it pay per dozen ?

13. If a grocer pays $12.15 for 9 dozen packages of breakfast cereal, how much does he pay per dozen ?

14. If a grocer pays $19.84 for 8 dozen tins of biscuit, how much does he pay per dozen ?

57. Dividing by 10 and its Powers. To divide by 10 is merely to find how many tens there are in the number, and this we can tell at a glance.

Thus in 1230 there are 123 tens and no units. In 2373 there are 237 tens and 3 units remainder; that is, 2373 + 10 = 2371 .

Therefore, to divide an integer by 10 cut off the right-hand figure of the dividend. The result is the quotient, and the right-hand figure expresses the remainder.

To divide an integer by any power of 10 cut off as many figures from the right of the dividend as there are zeros at the right of the divisor. The result is the quotient, and the right-hand figures express the remainder.

Thus 12,400 = 100 = 124 ; 43,203 = 100 = 432180 ; 624,000 = 1000 = 624 ; 736,117 - 1000 = 7361000.

58. Dividing by Multiples of Powers of 10. In this case we divide by the power of 10, as in $ 57, and then by the number indicating the multiple.

For example, to divide 42,800 by 200 we cut off two 290 )42800

zeros from divisor and dividend and then divide by 2. 214

So to divide 27,407 by 300 we cut off two zeros 398 )27407 from the divisor and two figures from the right of the 91187

dividend. We then divide by 3, and the quotient is

91, with remainder 1 in the hundreds' place, and to this we join the 07 cut off, obtaining the remainder 107.

EXERCISE 19

Find the quotient and the remainder (if any):

[blocks in formation]

Find the quotient and the remainder (if any): 13. 325 : 30. 21. 20,300 - 100. 29. 98,207 — 900. 14. 275 ; 40. 22. 27,400 = 200. 30. 126,000 - 3000. 15. 236 50. 23. 64,200 • 300. 31. 328,000 + 4000. 16. 327 ; 70. 24. 72,400 = 400. 32. 465,000 = 5000. 17. 542 : 70. 25. 31,501 - 500. 33. 564,700 - 6000. 18. 391 : 80. 26. 27,403 = 600. 34. 429,820 - 7000. 19. 278 ; 90. 27. 59,721 + 700. 35. 742,623 - 8000. 20. 777 ; 90. 28. 42,348 + 800. 36. 857,542 + 9000.

37. At $90 each, how many horses can I buy for $2160 ?

38. There being 60 minutes in an hour, how many hours are there in 1020 minutes ?

39. There being 60 seconds in a minute, how many minutes are there in. 1620 seconds ?

40. At $30 each, how many harnesses must a manufacturer sell to receive $1110 ?

41. A farmer pays $6600 for some land at $40 an acre. How many acres does he buy ?

42. A dealer sells some village lots for $10,200, at $600 a lot. How many lots does he sell ?

43. An investor has $87,750 with which to buy some, bonds at $500 each. What is the greatest number he can buy, and how much will he have left ?

44. An automobile dealer has $35,100 to invest in automobiles at $2000 each. What is the greatest number he can buy for this sum, and how much will he have left?

45. A ton is 2000 pounds. If a coal pocket contains 583,260 pounds of coal, how many tons does it contain, and how many pounds are left over ?

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