# 551 used items, it takes 6 used to create one new… how many new items can you make?

I had this question recently in an online exam that really confused me (it's not hard).

A business decided to recycle all their old paper cups and produce new ones from the recycled cardboard. For every new cup it would require $6$ used cups. They have $551$ used cups, how many new cups can be created?

I thought easy, divide $551$ by $6$ thus allowing the business to create $91$ new cups.

The answer was (according to the creator) $110$. Which is $551$ divided by $5$. I still can't wrap my head around the logic, I have written the question as I was given it, could anyone please help me shed some light on this?

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The freshly made cups can be recycled in turn? – Jyrki Lahtonen Mar 18 '13 at 16:26
Logically I would think so, though the question is exactly as it was worded. – David Barker Mar 18 '13 at 16:27

at first you make $$\frac{551}{6} =91+ \frac{5}{6}$$ and having 5 rest. now we 96 paper cups that are $$\frac{96}{6}=16$$ 16, now we make $$\frac{16}{6}=2+\frac{4}{6}$$ $$\frac{6}{6}=1$$ so we have $$91+16+2+1=110$$ all together.

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Thank you for the explanation, that makes perfect sense – David Barker Mar 18 '13 at 16:33
@david you are welcome, i am glad i could help – Dominic Michaelis Mar 18 '13 at 16:41

They can create $91$ new cups with $5$ used cups left over. After they use the $91$ cups, they’ll have $96$ used cups, from which they can create $\frac{96}6=16$ new cups with nothing over. After these are used, they’ll have $16$ used cups, from which they can create $2$ new cups with $4$ used cups left over. And finally, when the $2$ new cups have been used, they can make one more new cup from the $6$ used cups that they will then have. Along the way they’ve created a total of

$$91+16+2+1=110$$

new cups. (But I think that the question is badly worded, perhaps deliberately so, in a way that disguises the fact that the answer is the long-term number of new cups, not the immediate number.)

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Ah yes it seems so simple thinking about it like that, thank you Brian and also to Dominic. – David Barker Mar 18 '13 at 16:32
@David: You’re welcome. – Brian M. Scott Mar 18 '13 at 16:34

An easier way to see it: From 6 used cups they make 1 new one, so they are essentially using up 5 only (6 come in, 1 comes out). Thus 551/5 = 110.

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oh here you be carefull, think if they have 555 cups – Dominic Michaelis Mar 18 '13 at 19:12
I was thinking this for a while, but it doesn't actually make sense. They 'use' 6 cups, despite one being produced they still use 6 so one cycle through the factory would only produce 551/6 new cups. – David Barker Mar 19 '13 at 9:47