- #1
pob
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Hi,
I have $150. This value is multiplied by 2 or more factors. Each factor is less than or equal to 1 and greater than or equal to 0. Each factor has a maximum of 9 decimal places. The sum of the factors equals 1. For example, the following set of factors meets these 3 conditions:
0.500033333
0.499966667
So does this set of factors:
0.099703630
0.095107000
0.035644140
0.264757680
0.050352750
0.144806740
0.145405230
0.110790870
0.053431960
Suppose I perform the following operations:
So far, I've worked the problem backwards by supposing I had 2 factors that resulted in the following unrounded products:
75.005
74.995
And the following rounded products sum to $150.01,
75.01
75.00
However, I find it impossible to produce the products 75.005 and 74.995 using factors limited to 9 decimal places. The closest I get are the following factors:
0.500033333
0.499966667
Which, when multiplied by $150, yield the following products. These rounded products also sum to $150.
75.00499995
74.99500005
Given this example, I believe I've shown that, given 2 factors, the sum of the rounded products is always equal to $150. However, I'm trying to generalize this to any number of factors.
Thank you,
pob
I have $150. This value is multiplied by 2 or more factors. Each factor is less than or equal to 1 and greater than or equal to 0. Each factor has a maximum of 9 decimal places. The sum of the factors equals 1. For example, the following set of factors meets these 3 conditions:
0.500033333
0.499966667
So does this set of factors:
0.099703630
0.095107000
0.035644140
0.264757680
0.050352750
0.144806740
0.145405230
0.110790870
0.053431960
Suppose I perform the following operations:
- Multiply the $150 by each of the factors and store each product, storing up to 13 decimal places
- Round each product from #1 to 2 decimal places
- Sum the rounded products
So far, I've worked the problem backwards by supposing I had 2 factors that resulted in the following unrounded products:
75.005
74.995
And the following rounded products sum to $150.01,
75.01
75.00
However, I find it impossible to produce the products 75.005 and 74.995 using factors limited to 9 decimal places. The closest I get are the following factors:
0.500033333
0.499966667
Which, when multiplied by $150, yield the following products. These rounded products also sum to $150.
75.00499995
74.99500005
Given this example, I believe I've shown that, given 2 factors, the sum of the rounded products is always equal to $150. However, I'm trying to generalize this to any number of factors.
Thank you,
pob