Sunday, October 31, 2010
Sunday, October 17, 2010
October 2010 Brain Teaser Solution
Q: What is the greatest possible product of two positive whole numbers whose sum is 100?
A: 2500 (50 x 50)
Let x = one number then 100 - x = other number
So we want to maximize the product of this algebra, x(100 - x) = 100x - x^2
see the graph below:
This result can also be achieved through Calculus. If we take the derivative of the algebra and set that equal to 0, then solve for x.
The derivative of 100x - x^2 is 100 - 2x, when 100 - 2x = 0 is solved x = 50.
When we substitute 50 into 100x - x^2, we get
100(50) - (50)^2
5000 - 2500 = 2500
So the maximum point is at x = 50 -- at the (x,y) point (50, 2500).
The list below shows:
Column 1 first number
Column 2 100 - first number
Column 3 product of Column 1 and Column 2
1 99 99
2 98 196
3 97 291
4 96 384
5 95 475
6 94 564
7 93 651
8 92 736
9 91 819
10 90 900
11 89 979
12 88 1056
13 87 1131
14 86 1204
15 85 1275
16 84 1344
17 83 1411
18 82 1476
19 81 1539
20 80 1600
21 79 1659
22 78 1716
23 77 1771
24 76 1824
25 75 1875
26 74 1924
27 73 1971
28 72 2016
29 71 2059
30 70 2100
31 69 2139
32 68 2176
33 67 2211
34 66 2244
35 65 2275
36 64 2304
37 63 2331
38 62 2356
39 61 2379
40 60 2400
41 59 2419
42 58 2436
43 57 2451
44 56 2464
45 55 2475
46 54 2484
47 53 2491
48 52 2496
49 51 2499
50 50 2500
51 49 2499
52 48 2496
53 47 2491
54 46 2484
55 45 2475
56 44 2464
57 43 2451
58 42 2436
59 41 2419
60 40 2400
61 39 2379
62 38 2356
63 37 2331
64 36 2304
65 35 2275
66 34 2244
67 33 2211
68 32 2176
69 31 2139
70 30 2100
71 29 2059
72 28 2016
73 27 1971
74 26 1924
75 25 1875
76 24 1824
77 23 1771
78 22 1716
79 21 1659
80 20 1600
81 19 1539
82 18 1476
83 17 1411
84 16 1344
85 15 1275
86 14 1204
87 13 1131
88 12 1056
89 11 979
90 10 900
91 9 819
92 8 736
93 7 651
94 6 564
95 5 475
96 4 384
97 3 291
98 2 196
99 1 99
A: 2500 (50 x 50)
Let x = one number then 100 - x = other number
So we want to maximize the product of this algebra, x(100 - x) = 100x - x^2
see the graph below:
This result can also be achieved through Calculus. If we take the derivative of the algebra and set that equal to 0, then solve for x.
The derivative of 100x - x^2 is 100 - 2x, when 100 - 2x = 0 is solved x = 50.
When we substitute 50 into 100x - x^2, we get
100(50) - (50)^2
5000 - 2500 = 2500
So the maximum point is at x = 50 -- at the (x,y) point (50, 2500).
The list below shows:
Column 1 first number
Column 2 100 - first number
Column 3 product of Column 1 and Column 2
1 99 99
2 98 196
3 97 291
4 96 384
5 95 475
6 94 564
7 93 651
8 92 736
9 91 819
10 90 900
11 89 979
12 88 1056
13 87 1131
14 86 1204
15 85 1275
16 84 1344
17 83 1411
18 82 1476
19 81 1539
20 80 1600
21 79 1659
22 78 1716
23 77 1771
24 76 1824
25 75 1875
26 74 1924
27 73 1971
28 72 2016
29 71 2059
30 70 2100
31 69 2139
32 68 2176
33 67 2211
34 66 2244
35 65 2275
36 64 2304
37 63 2331
38 62 2356
39 61 2379
40 60 2400
41 59 2419
42 58 2436
43 57 2451
44 56 2464
45 55 2475
46 54 2484
47 53 2491
48 52 2496
49 51 2499
50 50 2500
51 49 2499
52 48 2496
53 47 2491
54 46 2484
55 45 2475
56 44 2464
57 43 2451
58 42 2436
59 41 2419
60 40 2400
61 39 2379
62 38 2356
63 37 2331
64 36 2304
65 35 2275
66 34 2244
67 33 2211
68 32 2176
69 31 2139
70 30 2100
71 29 2059
72 28 2016
73 27 1971
74 26 1924
75 25 1875
76 24 1824
77 23 1771
78 22 1716
79 21 1659
80 20 1600
81 19 1539
82 18 1476
83 17 1411
84 16 1344
85 15 1275
86 14 1204
87 13 1131
88 12 1056
89 11 979
90 10 900
91 9 819
92 8 736
93 7 651
94 6 564
95 5 475
96 4 384
97 3 291
98 2 196
99 1 99
100 0 0
Subscribe to:
Posts (Atom)