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I was solving this interesting, yet challenging word problem and it was very hard for me. This type of word problem has many different ways to be solved, one of them by solving it backwards. (It was translated to English).

The king summoned the knights of the castle and decided how they would divide their reward: "The first knight takes one gold piece and one-seventh of the rest, the second takes two gold pieces and one-seventh of the new rest, and so on. The Y knight takes Y gold and a seventh of the remaining amount of gold, as long as there are any. Thus, it was possible to distribute all the gold, while all knights received the same amount. How many knights shared the reward?

So far, I have tried solving it from the back. I have given knights the letter k And I have given the rest the letter r

$$ k - 1 + \frac 17 r = k $$

From here I solved that $\frac 17$*r=1 and therefore r = 7 (the answer isn't that 7 knights shared the reward)

But that is all that I was able to solve. I would appreciate any help and also if these kind of word problems have some kind of specific name, so I would be able to do some research to learn more about them.

Rodrigo de Azevedo
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MaxE
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1 Answers1

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The last knight to draw gets $k$ gold pieces. His predecessor gets $(k-1)+\frac 17 r$ and these two values are equal.

$k-1 + \frac 17 r = k$

subtract $k-1$ from both sides. $\frac 17 r = 1$ or $r = 7$

After the second to last knight takes his coins there will be $k$ left over.

$r - \frac 17 r = k$
$k = 6$

There are 6 knights, each gets 6 coins. There were 36 coins initially.

And, as David Lui points out, there is the case k=1.

user317176
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