THE MEANING OF DIVISION
PROBLEMS
11. a) What number times 100 will be 287? Answer using the division
1. a) sign ÷ .
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Do the problem yourself first!
287 ÷ 100 = 2.87
11. b) Read your answer to part a).
287 divided by 100 equals (or is) 2.87.
11. c) What is 287 called? The dividend.
11. d) What is 100 called? The divisor.
11. e) What is 2.87 called? The quotient.
11. f) Prove your answer to part a). 2.87 × 100 = 287
12. Practice the following.
| 3 × 6 = 18. | 4 × 8 = 32. | 5 × 9 = 45. | 6 × 9 = 54. | 7 × 6 = 42. | |||||
| 8 × 6 = 48. | 9 × 3 = 27. | 3 × 8 = 24. | 4 × 6 = 24. | 5 × 6 = 30. | |||||
| 6 × 5 = 30. | 7 × 4 = 28. | 8 × 9 = 72. | 9 × 6 = 54. | 3 × 9 = 27. | |||||
| 4 × 9 = 36. | 5 × 7 = 35. | 6 × 7 = 42. | 7 × 3 = 21. | 8 × 5 = 40. | |||||
| 3 × 4 = 12. | 4 × 4 = 16. | 6 × 3 = 18. | 4 × 5 = 20 | 9 × 6 = 54. | |||||
13. a) Let this straight line represent 24, and illustrate 24 ÷ 4.
13. b) Prove 24 ÷ 4 = 6. 6 × 4 = 24
13. c) Illustrate 24 ÷ 6.
13. d) Prove 24 ÷ 6 = 4. 4 × 6 = 24
14. a) How many times could you subtract 8 from 40? 5
13. b) Write that problem using the division sign ÷ .
40 ÷ 8 = 5
14. c) Write that problem using the division bar.
| 40 8 |
= 5 |
14 d) In that form, what is 40 called? The dividend.
14. e) What is 8 called? The divisor.
14. f) What is 5 called? The quotient.
15. a) From a bottle that contains 48 oz, how many times could you fill
14. a) a 6 oz glass? 8 times. 48 ÷ 6 = 8.
15. b) From a bottle that contains 2 quarts, how many times could you
14. b) fill an 8 oz glass? (1 quart = 32 oz)
8 times. 64 oz ÷ 8 oz = 8
To divide, the units must be of the same kind.
16. A farmer has a field that is 100 yards long, and he wants to put a
16. fence post every 4 feet. How many fence posts must he put?
How many times is 4 feet contained in 100 yards? To answer, we have to change yards to feet because, again, the units must be the same.
Since 1 yard = 3 feet, then 100 yards = 300 feet. How many 4's are there in 300?
Now, 100 is made up of twenty-five 4's: 100 = 25 × 4. Therefore, 300 = (3 × 25) × 4 = 75 × 4. The farmer must put 75 fence posts.
(Alternatively, to divide a number by 4 is equivalent to taking its fourth part. And to take the fourth part, we can take half of half. Lesson 15, Question 8. Half of 300 is 150. Half of 150 is 75.)
But what about the fence post at the beginning of the field -- at the "zero" point? In reality, then, he would put 76 fence posts!
17. If you divide a number by 8, what are the possible remainders?
1, 2, 3, 4, 5, 6, 7.
18. Divide each number by 8 mentally. Write the whole number
16. quotient and the remainder.
17 18 2 R 2. 29 3 R 5. 39 4 R 7. 54 6 R 6. 76 9 R 4.
19. Divide each number by 7 mentally. Write the whole number
19. quotient and the remainder.
19. 17 2 R 3. 25 3 R 4. 31 4 R 3. 41 5 R 6. 61 8 R 5.
10. Divide each number by 9 mentally. Write the whole number
18. quotient and the remainder.
19. 17 1 R 8. 25 2 R 7. 31 3 R 4. 41 4 R 5. 62 6 R 8.
11. Prove: 157 ÷ 14 = 11 R 3
11 × 14 = 140 + 14 = 154, plus 3 is 157.
12. How will 436 change if you multiply it by 10 and then divide by 10?
It will not change!
13. Divide.
| a) | 40 5 |
= 8 | b) | 35 7 |
= 5 | c) | 28 4 |
= 7 | d) | 63 9 |
= 7 | |||
| e) | 54 6 |
= 9 | f) | 24 4 |
= 6 | g) | 72 9 |
= 8 | h) | 27 3 |
= 9 | |||
| i) | 35 7 |
= 5 | j) | 45 9 |
= 5 | k) | 32 8 |
= 4 | l) | 48 6 |
= 8 | |||
| m) | 240 6 |
= 40 | n) | 420 6 |
= 70 | o) | 720 8 |
= 90 | p) | 120 4 |
= 30 | |||
| q) | 320 4 |
= 80 | r) | 450 5 |
= 90 | s) | 480 8 |
= 60 | t) | 400 5 |
= 80 | |||
| u) | 540 9 |
= 60 | v) | 300 6 |
= 50 | w) | 630 7 |
= 90 | x) | 360 9 |
= 40 | |||
| y) | 2100 3 |
= 700 | z) | 6400 8 |
= 800 | a') | 4200 7 |
= 600 | b') | 1000 5 |
= 200 | |||
| c') | 2000 5 |
= 400 | d') | 3000 5 |
= 600 | e') | 2000 4 |
= 500 | f') | 4000 5 |
= 800 | |||
| 14. | a) | 245 10 |
= 24.5 | b) | 245 100 |
= 2.45 | c) | 245 1000 |
= .245 | ||
| d) | 9 10 |
= .9 | e) | 9 100 |
= .09 | f) | 9 1000 |
= .009 | |||
15. The divisor composed of factors.
| a) | 12 4 |
= 3 | b) | 12 4 × 10 |
= .3 | c) | 12 40 |
= .3 |
| d) | 12 30 |
= .4 | e) | 42 60 |
= .7 | f) | 54 90 |
= .6 |
| g) | 54 900 |
= .06 | h) | 64 800 |
= .08 | i) | 56 700 |
= .08 |
16. Say that the quotient of two numbers is 10, and
16. a) you multiply the dividend by 5. What will be the new quotient?
50
16. b) you multiply the divisor by 5. What will be the new quotient?
2
16. c) you multiply the dividend and the divisor by 5. What will be the
16. c) new quotient?
10
17. What property of division allows us to write the following:
| 4.5 .005 |
= | 4500 5 |
If we multiply the dividend and divisor by the same number
-- 1000, in this case -- the quotient will not change.
18. To divide, the divisor must be a whole number.
| a) | 12 .3 |
= | 120 3 |
= 40 | b) | 24 .4 |
= | 240 4 |
= 60 | ||
| c) | 45 .5 |
= | 450 5 |
= 90 | d) | 42 .7 |
= | 420 7 |
= 60 | ||
| e) | 12 .03 |
= | 1200 3 |
= 400 | f) | 24 .04 |
= | 2400 4 |
= 600 | ||
| g) | 45 .005 |
= | 45,000 5 |
= 9,000 | h) | 32 .008 |
= | 32,000 8 |
= 4,000 |
| i) | 1.4 .07 |
= | 140 7 |
= 20 | j) | 3.5 .005 |
= | 3500 5 |
= 700 | |
| k) | .27 .009 |
= | 270 9 |
= 30 | l) | 3.6 .006 |
= | 3600 6 |
= 600 | |
19. The order of multiplying and dividing does not matter.
| a) | 9 × 12 3 |
= | 9 × 4 = 36 | b) | 8 × 12 4 |
= | 2 × 12 = 24 | |
| c) | 6 × 22 2 |
= | 6 × 11 = 66 | d | 25 × 25 5 |
= | 5× 25 = 125 | |
20. Show three ways of doing this problem:
| 1) | 9 × 12 3 |
= | 9 × 4 = 36 | 2) | 9 × 12 3 |
= | 3 × 12 = 36 |
| 3) | 9 × 12 3 |
= | 108 3 |
= | 90 + 18 3 |
= | 30 + 6 = 36 |
21. Find a multiple of the divisor in the dividend.
| 81 3 |
= | 9 × 9 3 |
= | 3 × 9 = 27. |
| 144 6 |
= | 12 × 12 6 |
= | 2 × 12 = 24 |
| 500 125 |
= | 2 × 250 125 |
= | 2 × 2 = 4. |
| 1000 125 |
= | 4 × 250 125 |
= | 4 × 2 = 8 |
22. A stack of CD's is 8 inches high. Each CD is .25 inches thick. How
18. many CD's are there?
| 8 .25 |
= | 800 25 |
= | 8 × 100 25 |
= | 8 × 4 = 32 |
23. Write the first five multiples of
23. a) 12. 12, 24, 36, 48, 60.
23. b) 18. 18, 36, 54, 72, 90.
23. c) 75. 75, 150, 225, 300, 375.
23. d) If you divide $6,000 equally among 12 people, how much will
21. d) each one get? $500
23. e) How many 18's are there in 540? 30
23. f) How many times could you subtract 75 from 3,000? 40
24. Inexact division. Write each quotient as a mixed number.
| a) | 7 5 |
= | 1 | 2 5 |
b) | 26 3 |
= | 8 | 2 3 |
c) | 23 9 |
= | 2 | 5 9 |
d) | 31 8 |
= | 3 | 7 8 |
|||
| e) | 46 7 |
= | 6 | 4 7 |
f) | 26 3 |
= | 8 | 2 3 |
g) | 65 12 |
= | 5 | 5 12 |
h) | 53 6 |
= | 8 | 5 6 |
|||
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Return to the Section 1 of the Lesson
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