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A group of friends goes for dinner and gets bill of $2400\$$. $2$ of them say that they have forgotten their purse so all the others make an extra contribution of $100\$$ to pay up the bill.

Question : What is the number of persons in that group?

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up vote 5 down vote accepted

Have you tried to write the problem as an equation ? Let's call $n$ the number of people. We know that the total sum is $2400$. Each one should then pay $\frac{2400}{n}$. But $2*\frac{2400}{n}=\frac{4800}{n}$ haven't been paid by the ones who forgot their money. This money has been given by the $n-2$ others, with $100$ each.

So we have $\frac{4800}{n} = (n-2)*100$, so $100n^2-200n-4800 = 0$.

This can be simplified in $n^2-2n-48 = 0$, which is of the form $an^2+bn+c = 0$.

We calculate $\Delta = \sqrt{b^2-4ac} = \sqrt{196} = 14$

The two possible answers are then $\frac{-b-\sqrt{\Delta}}{2a}=-6$ and $\frac{-b+\sqrt{\Delta}}{2a}=8$.

I think you can now determine which of the two answers is admissible !

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Can you solve this further after this equation, What is exact answer? I don't know how to solve this. – Android developer Feb 4 '14 at 10:50

Say there are $n$ people. Then originally, they should each have to pay $\frac{2400}{n}$. Now, instead they are $2$ less to split the bill, so those who do pay have to pay $\frac{2400}{n - 2}$. But at the same time, they only have to pay $100$ more, that is $\frac{2400}{n} + 100$. I have now described the actual per person payment in two different ways. These two have to be equal. Which means you set them next to eachother and solve like an equation: $$ \frac{2400}{n - 2} = \frac{2400}{n} + 100 $$

If you don't know how to solve a second degree equation, then trial-and-error would probably be the best solution. If there were three people originally, the last person's bill would go up from $800$ to $2400$ when it turns out two of them cannot pay. That's too much. If there were $12$ people originally, the payment would go up from $200$ per person to $240$ per person, which is too little. Your answer is somewhere in between.

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How to estimate the result: there are $N$ friends and the bill is $B$ hence, originally, the individual share is $B/N$. When somebody declares they cannot pay, the share becomes $B/(N-1)$. This is roughly $B/N+B/N^2$ hence the effect of two people leaving is to raise the share by an extra contribution $E\approx2B/N^2$. Thus, $N\approx\sqrt{2B/E}$.

With $B=2400$ and $E=100$, $\sqrt{2B/E}=\sqrt{48}\approx7$ (which is only $1$ short of the true answer).

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