How to approximate the division by a number like prime number?

Question

I was solving some mathematical questions and have come across the situation, where I need to divide 3900/139. Here is my question,

a. Can I assume 139 to 140 for the ease of division?

If so, how will I know what percentage of error I am introducing? How can I ensure that I am adding very less value to a number and the results will not be tremendously affected?

Answer 1

N/139 = real answer

N/140 = your answer

your answer/real answer = (N/140)/(N/139) = 139/140.

Your answer will be 1/140 too small.

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in general if you replace p with (p + n) your result will by factor of n/(p+n)

Replace 487 with 500 and your be off by a factor of 13/500.

Basically your answer will be off be the same proportion as your rounding was off.

Answer 2

In general, if y = c/x (c is a constant) and you make a small change in x, say by h (> 0), then the following happens:

1. The exact relative error is (-h/(c+h))
2. This is approximately equal to (-h/c) [If you take the log and then the differential, that will be evident].