- #1

pob

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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