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Number crunching
Tons of examples
dtcoulson@gmail.com
2014
8.0
100
80
50
40
454
343
54
43




54
43
(actual answer = 0.796)
0.5% error
Convert to decimal:
.333...0
3
2
24
16
48
32
96
64

96
64
No approximation.
Answer is exact.
Convert to decimal:
.571428...0
7
5
70
50
35
25

35
25
No approximation.
Answer is exact.
Convert to decimal:
14-28-57-14-28-57....
.960
100
96
50
48
941
939
41
39




41
39
Convert to decimal:
(actual answer = 0.951...)
1% error
 
 
.333...0
3
1
133
131
39
13

39
13
Convert to decimal:
No approximation.
Answer is exact.
 
 
.333...0
3
1
173
171
51
17

51
17
Convert to decimal:
No approximation.
Answer is exact.
.120
100
12
50
6
25
3

25
3
Convert to decimal:
No approximation.
Answer is exact.
 
 
.1250
8
1
38
31
24
3

24
3
Convert to decimal:
No approximation.
Answer is exact.
 
 
.111...0
9
1
39
31
27
3

27
3
Convert to decimal:
No approximation.
Answer is exact.
.750
4
3
12
9
120
90

120
90
Convert to decimal:
No approximation.
Answer is exact.
.3750
8
3
16
6

16
6
Convert to decimal:
No approximation.
Answer is exact.
.050
100
5
99
5

99
5
Convert to decimal:
(0.01% error)
Small by about 1%
0.05050.0005.050 
.0750
10
1
4
3
40
3







40
3
Convert to decimal:
No approximation.
Answer is exact.
    ..33.310
3
1
10
6
2
6
20
620
Calculate or estimate:
No approximation.
Answer is exact.
    9010910
9
81
9
810
9.0
81
9.081
Calculate or estimate:
No approximation.
Answer is exact.
.63
10
36
5
18
15
54
150
540
75
270
75270
Calculate or estimate:
No approximation.
Answer is exact.
.777...1
9
7
1
9
16
27
48
270
480
135
240
135240
Calculate or estimate:
No approximation.
Answer is exact.
.030
100
3
50
5.1
505.1 
Calculate or estimate:
No approximation.
Answer is exact.
    3010
101
101
3
9
10
303
909
303
9090
3.30
909






3.30909
Calculate or estimate:
No approximation.
An...
5
100
500
101
500
01.1
5
01.15
Calculate or estimate:
0.01% error
Too big by about 1%
4.950.05-5 
10
42
420
2.4
42
2.442
Calculate or estimate:
No approximation.
Answer is exact.
4
9
36
90
360
90360
Calculate or estimate:
No approximation.
Answer is exact.
125
4
5
100
4
500






4500
Calculate or estimate:
No approximation.
Answer is exact.
  ..333.1333...333.11000
3
4
1000
3
4000






34000
Calculate or estimate:
No approximation.
Answer is exact.
$3.30333x11 
30$%11 of
Calculate or estimate:
No approximation.
Answer is exact.
$1080x
8
1

80$%5.12 of
Calculate or estimate:
No approximation.
Answer is exact.
12$%12 of
Calculate or estimate:
No approximation.
Answer is exact.
$1.4414412x12 
About one-tenth.
Close to $1.20
150$%15 of
Calculate or estimate:
No approximation.
Answer is exact.
  $22.502252520x1015x15 
About one-tenth.
Close...
40$%90 of
Calculate or estimate:
No approximation.
Answer is exact.
36$4-40 
One-tenth smaller
2$%47 of
Calculate or estimate:
No approximation.
Answer is exact.
$0.94492x47 
About half
2$%1 of
Calculate or estimate:
No approximation.
Answer is exact.
02.0$
2$%5.1 of
Calculate or estimate:
No approximation.
Answer is exact.
$0.03325.1 x
50$%51 of
Calculate or estimate:
No approximation.
Answer is exact.
$25.105125x15 
Expect an answer which is close to $25
200$%23 of
Calculate or estimate:
No approximation.
Answer is exact.
$46642x32 
Roughly a quarter of 200
45$%45 of
Calculate or estimate:
No approximation.
Answer is exact.
$20.25250x50454x54 
Roughly half of 50
35x35
Multiplication:
No approximation.
Answer is exact.
12252540x0353x53 
92x31
Multiplication:
No approximation.
Answer is exact.
8991-0392x13 22

82x32
Multiplication:
No approximation.
Answer is exact.
8962-0382x23 22

52x35
Multiplication:
No approximation.
Answer is exact.
8755-0352x53 22

52x36
Multiplication:
No approximation.
Answer is exact.
900100x9
4
100
x6352x63 
521x888
Multiplication:
No approximation.
Answer is exact.
  111,0001000x1111000
8
1
x888521x888 
12x11
Multiplication:
No approximation.
Answer is exact.
3121}12{212x11 
114x11
Multiplication:
No approximation.
Answer is exact.
52141}11}{14{4114x11 
.51x42
Multiplication:
No approximation.
Answer is exact.
631224.51x42 
51x4.2
Multiplication:
No approximation.
Answer is exact.
6312245.1x4251x4.2 
99x011
Multiplication:
No approximation.
Answer is exact.
999999x011 
08$of
5
4
Multiplication:
No approximation.
Answer is exact.
$6408$x
10
8
08$x
5
4

05$of
25
4
Multiplication:
No approximation.
Answer is exact.
$8
25
50
x405$x
25
4

03$of
15
4
Multiplication:
No approximation.
Answer is exact.
$8
15
30
x403$x
15
4

01$of
90
45
Multiplication:
No approximation.
Answer is exact.
$510$ofhalf01$of
90
45

41$of
21
2
Multiplication:
No approximation (except
for the dollars and cents
rounding)
.331$
3
4
3
2
x2
21
14
x241x
21
2
...
001$of
99
19
Multiplication:
0.01% error
19.19$0.1991
100
011
x19
99
001
x19001x
99
19

001$of
69
34
Multiplication:
0.5% error
$49001x
100
49
001x
50
5.24
001x
69
34

52$of
35
18
Multiplication:
22.12$10
7
9
14
180
140
1800
4
100
x
35
18







exact
34
Square roots:
..833.5
6
1
6
36
1
16
36
2
16
36
34
634 






0.05% error
30
Square roots:
5.5
12
6
6
36
3
16
36
6
16
36
30
630 






0.5% error
26
Square roots:
1.5
50
5
5
50
1
15
25
1
15
25
26
526 






0.02% error
18
Square roots:
52.4
16
1
14
16
2
14
16
18
418 






0.02% error
160
Square roots:
1316010x61601 
3% error
640
Square roots:
  3.25
10
13
24
80
13
3816.381064640 






0.007% error
[ENOUGH!]
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Number crunching tons of examples

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Number crunching tons of examples

  1. 1. Number crunching Tons of examples dtcoulson@gmail.com 2014
  2. 2. 8.0 100 80 50 40 454 343 54 43     54 43 (actual answer = 0.796) 0.5% error Convert to decimal:
  3. 3. .333...0 3 2 24 16 48 32 96 64  96 64 No approximation. Answer is exact. Convert to decimal:
  4. 4. .571428...0 7 5 70 50 35 25  35 25 No approximation. Answer is exact. Convert to decimal: 14-28-57-14-28-57....
  5. 5. .960 100 96 50 48 941 939 41 39     41 39 Convert to decimal: (actual answer = 0.951...) 1% error
  6. 6.     .333...0 3 1 133 131 39 13  39 13 Convert to decimal: No approximation. Answer is exact.
  7. 7.     .333...0 3 1 173 171 51 17  51 17 Convert to decimal: No approximation. Answer is exact.
  8. 8. .120 100 12 50 6 25 3  25 3 Convert to decimal: No approximation. Answer is exact.
  9. 9.     .1250 8 1 38 31 24 3  24 3 Convert to decimal: No approximation. Answer is exact.
  10. 10.     .111...0 9 1 39 31 27 3  27 3 Convert to decimal: No approximation. Answer is exact.
  11. 11. .750 4 3 12 9 120 90  120 90 Convert to decimal: No approximation. Answer is exact.
  12. 12. .3750 8 3 16 6  16 6 Convert to decimal: No approximation. Answer is exact.
  13. 13. .050 100 5 99 5  99 5 Convert to decimal: (0.01% error) Small by about 1% 0.05050.0005.050 
  14. 14. .0750 10 1 4 3 40 3        40 3 Convert to decimal: No approximation. Answer is exact.
  15. 15.     ..33.310 3 1 10 6 2 6 20 620 Calculate or estimate: No approximation. Answer is exact.
  16. 16.     9010910 9 81 9 810 9.0 81 9.081 Calculate or estimate: No approximation. Answer is exact.
  17. 17. .63 10 36 5 18 15 54 150 540 75 270 75270 Calculate or estimate: No approximation. Answer is exact.
  18. 18. .777...1 9 7 1 9 16 27 48 270 480 135 240 135240 Calculate or estimate: No approximation. Answer is exact.
  19. 19. .030 100 3 50 5.1 505.1  Calculate or estimate: No approximation. Answer is exact.
  20. 20.     3010 101 101 3 9 10 303 909 303 9090 3.30 909       3.30909 Calculate or estimate: No approximation. Answer is exact.
  21. 21. 5 100 500 101 500 01.1 5 01.15 Calculate or estimate: 0.01% error Too big by about 1% 4.950.05-5 
  22. 22. 10 42 420 2.4 42 2.442 Calculate or estimate: No approximation. Answer is exact.
  23. 23. 4 9 36 90 360 90360 Calculate or estimate: No approximation. Answer is exact.
  24. 24. 125 4 5 100 4 500       4500 Calculate or estimate: No approximation. Answer is exact.
  25. 25.   ..333.1333...333.11000 3 4 1000 3 4000       34000 Calculate or estimate: No approximation. Answer is exact.
  26. 26. $3.30333x11  30$%11 of Calculate or estimate: No approximation. Answer is exact.
  27. 27. $1080x 8 1  80$%5.12 of Calculate or estimate: No approximation. Answer is exact.
  28. 28. 12$%12 of Calculate or estimate: No approximation. Answer is exact. $1.4414412x12  About one-tenth. Close to $1.20
  29. 29. 150$%15 of Calculate or estimate: No approximation. Answer is exact.   $22.502252520x1015x15  About one-tenth. Close to $15
  30. 30. 40$%90 of Calculate or estimate: No approximation. Answer is exact. 36$4-40  One-tenth smaller
  31. 31. 2$%47 of Calculate or estimate: No approximation. Answer is exact. $0.94492x47  About half
  32. 32. 2$%1 of Calculate or estimate: No approximation. Answer is exact. 02.0$
  33. 33. 2$%5.1 of Calculate or estimate: No approximation. Answer is exact. $0.03325.1 x
  34. 34. 50$%51 of Calculate or estimate: No approximation. Answer is exact. $25.105125x15  Expect an answer which is close to $25
  35. 35. 200$%23 of Calculate or estimate: No approximation. Answer is exact. $46642x32  Roughly a quarter of 200
  36. 36. 45$%45 of Calculate or estimate: No approximation. Answer is exact. $20.25250x50454x54  Roughly half of 50
  37. 37. 35x35 Multiplication: No approximation. Answer is exact. 12252540x0353x53 
  38. 38. 92x31 Multiplication: No approximation. Answer is exact. 8991-0392x13 22 
  39. 39. 82x32 Multiplication: No approximation. Answer is exact. 8962-0382x23 22 
  40. 40. 52x35 Multiplication: No approximation. Answer is exact. 8755-0352x53 22 
  41. 41. 52x36 Multiplication: No approximation. Answer is exact. 900100x9 4 100 x6352x63 
  42. 42. 521x888 Multiplication: No approximation. Answer is exact.   111,0001000x1111000 8 1 x888521x888 
  43. 43. 12x11 Multiplication: No approximation. Answer is exact. 3121}12{212x11 
  44. 44. 114x11 Multiplication: No approximation. Answer is exact. 52141}11}{14{4114x11 
  45. 45. .51x42 Multiplication: No approximation. Answer is exact. 631224.51x42 
  46. 46. 51x4.2 Multiplication: No approximation. Answer is exact. 6312245.1x4251x4.2 
  47. 47. 99x011 Multiplication: No approximation. Answer is exact. 999999x011 
  48. 48. 08$of 5 4 Multiplication: No approximation. Answer is exact. $6408$x 10 8 08$x 5 4 
  49. 49. 05$of 25 4 Multiplication: No approximation. Answer is exact. $8 25 50 x405$x 25 4 
  50. 50. 03$of 15 4 Multiplication: No approximation. Answer is exact. $8 15 30 x403$x 15 4 
  51. 51. 01$of 90 45 Multiplication: No approximation. Answer is exact. $510$ofhalf01$of 90 45 
  52. 52. 41$of 21 2 Multiplication: No approximation (except for the dollars and cents rounding) .331$ 3 4 3 2 x2 21 14 x241x 21 2 
  53. 53. 001$of 99 19 Multiplication: 0.01% error 19.19$0.1991 100 011 x19 99 001 x19001x 99 19 
  54. 54. 001$of 69 34 Multiplication: 0.5% error $49001x 100 49 001x 50 5.24 001x 69 34 
  55. 55. 52$of 35 18 Multiplication: 22.12$10 7 9 14 180 140 1800 4 100 x 35 18        exact
  56. 56. 34 Square roots: ..833.5 6 1 6 36 1 16 36 2 16 36 34 634        0.05% error
  57. 57. 30 Square roots: 5.5 12 6 6 36 3 16 36 6 16 36 30 630        0.5% error
  58. 58. 26 Square roots: 1.5 50 5 5 50 1 15 25 1 15 25 26 526        0.02% error
  59. 59. 18 Square roots: 52.4 16 1 14 16 2 14 16 18 418        0.02% error
  60. 60. 160 Square roots: 1316010x61601  3% error
  61. 61. 640 Square roots:   3.25 10 13 24 80 13 3816.381064640        0.007% error
  62. 62. [ENOUGH!]

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