# Whats the difference between multiplying by 0.999 and dividing by 1.001?

1/1.001 - 1*0.999 is non zero. But whats the difference between multiplying by 0.999 and dividing by 1.001?

If say I was doing some foreign exchange at the airport and they wanted a 0.5% transaction fee, and I wanted to exchange \$100 USD for AUD at a rate of 2. Would I divide the result by 1.005 or multiply it by 0.995?

I feel like if they "take 0.5% commission" it would mean:

``````AUD/2=x
AUD=(USD)*2
AUD= (x+0.005x)*2
where 1.005x= 1USD
``````

# we take away the commision and so:

``````AUD= x*2
AUD=(USD*2)/1.005
``````

However, multiplying by 0.995 also makes sense, when they say, we take 0.5% of what you get as comission, leaving you with 0.995 of what you had originally.

Does one interpretation actually make more sense than another when talking about taking comissions or is one chosen over the other by convention? would you always rather they divide by (1+commission) rather than multiply by (1-commission)?

• I don't understand the equations you've written there. What is `AUD/2=x` supposed to mean? Are you assigning a value to `x` on the right? If so, then what is `AUD = USD*2` supposed to mean, where you now appear to be assigning a value to the left? – BrenBarn Jul 16 '16 at 5:33
• What do you mean by "what's the difference"? As you said right up front, these are different operations and produce different results. You use the one appropriate for the task you are trying to perform. Can you rephrase the question so it asks what you actually want an answer for? – keshlam Jul 16 '16 at 15:22
• The difference is about 0.0001%, which is pretty incredibly small. – David Schwartz Jul 19 '16 at 9:16

I would think that the transaction fee would be either US\$0.50 or AU\$1.00, depending on which currency they want.

You give them your US\$100, they take out US\$0.50, and hand you back 99.50 times 2 = AU\$199.00.

Or, you give them your US\$100, they exchange it for 100.00 times 2 = AU\$200.00, take out AU\$1.00, and hand you back the other AU\$199.00.

Either way, it's a 0.5% transaction fee. What you get back is worth 0.995 of whatever you put in.

Now, if you have the amount after the fee has been taken out, and want the original amount you put in, then you'd divide by 0.995 to get the original amount you put in.

But, you're correct in observing that 0.995X is not the same as X/1.005. I think 0.995X is the way to go.

1/1.001 - 1*0.999 is non zero. But whats the difference between multiplying by 0.999 and dividing by 1.001?

.999 is 999/1000. 1.001 is 1001/1000. To "undo" multiplying by 999/1000, you would need to multiply by 1000/999, which is 1.001001... In this case the difference is tiny, but for larger percentages it can be significant.

I agree with mbhunter that in a currency transaction both ways of thinking about it are equivalent. In a single currency transaction, you would never be doing two "reversed" operations as you are describing here. You would either do USD*0.5*.995 or USD*.995*.5, which are the same thing. If you then rebought the original currency you would do AUD*2*.995 or AUD*.995*2, which are again the same.

Right now I don't understand from your question how the number 1.005 figures into such a transaction at all, any more than, say, the number 1.2 figures into calculation of a 20% discount on a pair of pants.