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My son came home from advanced math with this question: if someone counted by tens all the way to one-hundred, what's the sum of all the numbers he said?

I thought the answer was simply 100, being that if you're council by 10s to 100, that's essentially 10+10.....10 times.

He said the teacher said it was a trick question,. What am I missing? Does this mean a hundred times (i.e. To a thousand?)? My son said its 10+20+30....do one...but how is that "counting by tens to 100?

Any thoughts?

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  • $\begingroup$ If you count by tens up to 100 you have $10 + 20+ 30 + 40+ \cdots$. $\endgroup$ Apr 5, 2016 at 20:36
  • $\begingroup$ The numbers the counter says are $10,20,30,...,100$ and we have to sum them up. $\endgroup$
    – Peter
    Apr 5, 2016 at 20:37
  • $\begingroup$ A nice story about Gauss comes in mind to me : Gauss surprised his teacher by summing up the numbers $1$ to $100$ very fast by pairing $1-100$ , $2-99$ , $3-98$ , ... $50-51$ , so he only had to calculate $50\cdot101=5050$. Gauss was undergraded (I think 6-10 years, but I do not know) and it is believed that he was the first to recognize this method. Today it is standard. $\endgroup$
    – Peter
    Apr 5, 2016 at 20:41
  • $\begingroup$ It's misleading to call this a trick question. It would be better to say that there's a trick to calculating the answer (although you can get the right answer without using it). Here "trick" really just means "technique". $\endgroup$
    – Théophile
    Apr 5, 2016 at 21:33

2 Answers 2

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You count $10,20,30,40,50,60,70,80,90,100$ and this sums to $550$.


To compute the sum quickly, take the numbers in pairs starting from both ends : $10+100,20+90,30+80\cdots$ Every pair sums to $110$, and there are $5$ of them.

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  • $\begingroup$ I see it now! Thanks $\endgroup$
    – gracey209
    Apr 5, 2016 at 20:50
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You have $10\cdot (1+2+...+10)$. Apply the method of Gauss we pair $1-10$ , $2-9$ , $3-8$ , $4-7$ , $5-6$ to get $5\cdot 11=55$. Multiplied with $10$, we get the final result $550$.

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