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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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up vote 2 down vote accepted

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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Thanks for the answer , could you also tell me why my method of simply removing 15% from the Bill at the start is wrong ? – MistyD Sep 1 '12 at 2:22
@MistyD: Removing $15\%$ of $207$ from $207$ is wrong because the $15\%$ tip was on the bill(without tip) not on $207$(which includes tip) – Aang Sep 1 '12 at 2:37
Now I see its actually Bill + 15% of Bill = 207 this implies that Bill = 207 - 15% of Bill – MistyD Sep 1 '12 at 2:43
Consider this simpler example: A television store buys a television from the distributor for \$100 and marks it up 50% and offers it to you for \$150. You see the \$150 price and you want to know what the wholesale price was, before the 50% markup. Can you get it by subtracting 50% of \$150? No, that would give you \$75. – MJD Sep 1 '12 at 2:45
@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. – Ross Millikan Sep 1 '12 at 2:52

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