# Dealing with very large/small numbers

Anyone got any strategies for dealing with estimating with large or small numbers.
Examples I’d like to solve would be:
37,000 out of 80,000,000 as a percentage
0.003% expressed as 1 in ?

Am only after estimates but am interested in how others go about these type of calculations.

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Drop the zeros you don’t need if you just want an estimate.

The above simplifies to 37 / 80,000 to begin with. Use 40 instead of 37, so now you got 4 in 8,000 which is 1 in 2,000; or 5 in 10,000 which is a percent of a percent, so 0.05%

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If you’re familiar with scientific notation, that can be useful for dealing with very large or small numbers.

For the first one, it’s not necessary because most of the zeros drop out straight away.

37,000 / 80,000,000
= 37 / 80,000
= (37 / 800) %
~ (40 / 800) %
~ (1 / 20) %
~ 0.05 %

For the second one, I would use scientific notation:

1 / 0.003%
= 1 / (3 x 10^-3 %)
= 1 / (3 x 10^-5)
= 10^5 / 3
= 100000 / 3
~ 33333

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Great answers already by Bjoern and Simon^

I’ll add a little trick to make these easier when working mentally. I’d first do the main work of the calculation (e.g. 37 ÷ 8) and afterwards put the decimal in the right place.

So 37 ÷ 8 = 4 rem 5 = 4 5/8 = 4.625

Then I’d say okay fine, 37 ÷ 80 must be 46.25%, same as 37,000 ÷ 80,000. Then we need to divide by 1000 and the final answer will be 0.04625%

Alternatively you could estimate like Bjoern did, and then calculate the 37 ÷ 8 at the end to figure out the exact digits for the “about 0.05%” …since they wre “4625” the answer must be 0.04625%

The main takeaway here is that you can do two types of low-intensity calculation (the rough estimate fiddling with the zeros, and the exact division disregarding the zeros) and easily put them together into the correct answer.

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Thanks so much Bjoern,Simon and Daniel. All great contributions that will help me with something that has bothered me for some time.
I have to admit the concept of a percent of a percent is a little hard to visualise but it’s becoming more apparent as I’m working with it. The scientific notation will be a big help when dealing with extremely large numbers, there may be no other way to perform calculations mentally without it.
Daniel, your tip is so straightforward yet so clever, I wasn’t expecting that level of accuracy and hence I asked only about estimates so appreciate your contribution.

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