Fraction Frenzy!

Algebra Level 2

From $999999999888888887777777666666555554444333221$ subtract half of it first, then subtract $\dfrac{1}{3}$ of the remaining number, next subtract $\dfrac{1}{4}$ of the remaining number, and so on, until $\dfrac{1}{999999999888888887777777666666555554444333221}$ of the remaining number is subtracted. What is the final resulting number?

The answer is 1.

1 solution

Venkata Karthik Bandaru
Jun 15, 2015

STRATEGY

Whenever such large numbers are given in some questions, it is better to first try to check it for smaller numbers or for generally any number.

SOLUTION

Let us substitute the large number given in the question for $x$ . Then, we observe that if we remove half of a quantity, half is remained and when $\dfrac{1}{3}^{rd}$ of a quantity is removed, $\dfrac{2}{3}$ remains, and so on. So, we can now simplify these subtractions and write the final result as a telescopic product $\left( \dfrac {x-1}{x}\dots \left( \dfrac {3}{4}\left(\dfrac {2}{3}\left( \dfrac {1}{2}x\right) \right) \right) \right) = \boxed{1}$ .

Moderator note:

You should clarify that you used telescoping product.

Karthik, please be responsible when setting the level of a problem. This problem is clearly not a level 5. If you think that this is at the difficulty of a level 5 problem, that will make me question if you do deserve to be at level 5.

Calvin Lin Staff - 5 years, 12 months ago

I am sorry sir. I wont do this again.

Yes, I do not deserve a level 5, because I got it by solving over-rated problems on brilliant, to be frank. I could never solve your Level 5 Challenge picks. I want to de-level now, then you can later see that I deserve a level 5 or not. :(. But one thing, what was the necessity that you had to comment in such a de-moralising manner? You could've just informed me that I should not be setting level 5 for such problems.

Venkata Karthik Bandaru - 5 years, 12 months ago

Thanks for understanding. We keep the ratings system in line, but it can still be abused by a problem creator who sets the level to be significantly higher than it should be. This does result in perceptions of "rating system doesn't work", which lowers the confidence that others have with regards to its relevance. As such, abuse of this system is not tolerated, nor is it appreciated by the community.

This is not the first problem of yours which has an artificially high level. We have gone through and edited several of your other problems, which is why you have received the above notice.

Calvin Lin Staff - 5 years, 12 months ago

@Calvin Lin – Why couldn't it be just a simple notice, rather than such a comment ? I am sorry for what I have done, and do de-level my profile.

Venkata Karthik Bandaru - 5 years, 12 months ago

@Calvin Lin De-level my account to level 1 please. Thank you.

Venkata Karthik Bandaru - 5 years, 12 months ago

Fraction Frenzy!

The answer is 1.

1 solution

Moderator note:

0 pending reports