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I am having trouble judging where I am going wrong with the following question

The price of lunch for 15 people was 207.00 including a 15% gratuity for service.What was the average price per person EXCLUDING the gratuity ? a)11.73 b)12.00 c)13.80 d)14.00 (Ans=12.00)

Here is how I am solving it

Without tip = 207-\frac{15}{207} = 175.93

Per Person = \frac{175.95}{15} = 11.73$

Why is my answer wrong ?

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Let $x$ be the total bill, without tip. Then the bill with tip is $x$ plus $15\%$ of $x$. So it is $x(1.15)$.

Thus $x(1.15)=207$. Now you can find $x$, and divide by $15$.

The above is probably the most natural way to solve the problem. One could alternately say that the amount paid per person is $\frac{207}{15}=13.80$. If $y$ is the bill per person before tip, then $y(1.15)=13.80$.

Remark: To see that the procedure you used is not correct, suppose that the bill before tip is $100$. Then the tip will be $15$, so the bill after tip is $115$. What happens if you remove $15\%$ of this? You end up below $100$.

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    @MistyD: This is about the fact that if you mark up by $x$ you multiply by $1+x$. If you mark down by $x$, you multiply by $1-x$. If you do one and then the other, you multiply by $(1+x)(1-x)=1-x^2\lt 1.$ If $x \ll 1, x^2 \lll 1$ and you might be able to ignore it. $15\%^2=2.25\%$ which may not be small.2012-09-01