# Question on some arithmetic calculations

When $6272$ is multiplied by $0.94$ the answer is $5895.68$. When it is divided by $1.06$ the answer is $\approx 5916.9811$. Why is it so?

Just as a little background, I am using the default Microsoft calculator for this calculation. I haven't pulled out Mathematica yet. Could this be because of some weird variable conversion (int to double)? Any help would be much appreciated.

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I vote to close this as it currently stands. In its current form, it is a pointless question. Why should anyone be surprised by these two results? – Steve D Jun 6 '12 at 5:39
@SteveD Maybe since $\frac1{1+0.06}$ is approximately $1-0.06=0.94$, so the results are expected to be closed to each other. Or perhaps OP considered them both as calculations of subtracting 6% from the original number. Just my guesses - I would not mind clarification from OP for his motivation to ask this. – Martin Sleziak Jun 6 '12 at 5:52
Obviously I'm not the most advanced mathematician in the world. I'm actually still in the middle school math curriculum and I needed to solve a simple equation. I was perplexed when I ended up with two different solutions for x = (0.94)6272 and 1.06x = 6272 because we were taught that the two equations were equivalent in elementary school. My apologies for the basic question. – Confused Jun 6 '12 at 6:00
@Confused: I think you didn't explain well where the confusion came from. It's a much better question now that you said it. – Gigili Jun 6 '12 at 6:09
Yes, all that was needed was an explanation of why you were confused. – Steve D Jun 6 '12 at 6:15

You may be doing rounding too much and too early. You may figure, correctly, that multiplying by $0.94$ is equivalent to dividing by $\frac{1}{0.94}$.

Calculate $\frac{1}{0.94}$. It is approximately $1.0638298$. You may have thrown away the part after the $1.06$, figuring it wouldn't make much difference. Well, percentagewise it is not much difference, but in dollars the difference is significant.

There is no mysterious fault in the Windows calculator, it is doing the calculation correctly, to good accuracy. To get good performance out of a calculator, let it do its thing, only round off at the end.

Remark: In general, if $a\ne 0$, multiplying by $a$ is equivalent to dividing by $\frac{1}{a}$.

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Thank you, in school we were taught that 1.06x = 6272 is the same as x = (0.94)6272. Well it seems like I still have much to learn! – Confused Jun 6 '12 at 5:52
@Confused: That is very roughly true. The error gets much bigger if we use that kind of approximation with $1.4$ and $0.6$ instead of $1.06$ and $0.94$. – André Nicolas Jun 6 '12 at 5:57

Because $$6272(0.94) = 6272 \left(\dfrac{94}{100}\right)$$ while $$\dfrac{6272}{1.06} =\dfrac{6272}{\dfrac{106}{100}} = 6272 \left(\dfrac{100}{106}\right).$$

As you can see, $$\dfrac{94}{100} \neq \dfrac{100}{106},$$ because if the were then $$106 \cdot 94 = 100 \cdot 100,$$ which is not the case.

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 Nice clean explanation. +1 ! – Inquest Jun 6 '12 at 12:35

If you take $5196$ and multiply it by $1.06 \times 0.94$ you are, in fact multiplying by a product of the form:

$$(1-x)(1+x)=1-x^2$$

In this case we have $x=0.06$. This is a useful thing to know if you are working with figures like this all the time (e.g. in finance, if prices go up 6% and then go down 6%, they end up a little lower).

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