Continuable fractions

Number Theory Level 1

Let $X$ and $Y$ be positive integers such that the first few digits of the fractions $\dfrac{29}{{\color{#D61F06}{70}}},\ \dfrac{{\color{#D61F06}{70}}}{{\color{#3D99F6}{169}}},\ \dfrac{{\color{#3D99F6}{169}}}{{\color{#20A900}{408}}},\ \dfrac{{\color{#20A900}{408}}}{{\color{#624F41}{X}}},\ \dfrac {{\color{#624F41}{X}}}{{\color{grey}{Y}}}$ are all equal to 0.4142. What is ${{\color{#624F41}{X}}}+{{\color{grey}{Y}}}?$

Note : The fractions are not equal.

The answer is 3363.

13 solutions

Lucas Chaves Meyles
Feb 25, 2018

A useful trick for approximating the fractional part of $\sqrt{2}$ is the following sequence $\frac{p}{q}\implies \frac{q}{p+2q}$ Clearly, the sequence in question follows this sequence as shown below $\frac{29}{70} \implies \frac{70}{29 + (2)(70)} = \frac{70}{169} \implies \frac{169}{70 + (2)(169)} = \frac{169}{408}$ Therefore, the sequence containing $X$ and $Y$ is as follows $\frac{169}{408}, \frac{408}{985}, \frac{985}{2378}$

Thus, $X + Y = 985 + 2378 = \boxed{3363}$

My recursion was something like "the denominator of the first fraction becomes the numerator of the second and so on..."

Aren't recursions usually stated using not p and q, but natural numbers n, n+1, m?

My mental operation would be something like: 1. Take the multiplicative inversive of the fraction. 2. Multiply the now numerator times 2. 3. Add it to the denominator of the last step.

Don't know, how to formalize this...

Julia Seidel - 3 years, 3 months ago

You're probably right in that defining my sequence as a recursion is not quite correct, so I fixed it! I could address the sequence in words; however, I feel as if the cases shown provide ample example as to what the p and q sequence means. I'm not too adept in formal mathematical notation, but I hope the point comes across well enough. Thanks for your comment!

Lucas Chaves Meyles - 3 years, 3 months ago

Did it the same way, nicely explained!

Peter van der Linden - 3 years, 3 months ago

Dankuwel! or thank you!

Lucas Chaves Meyles - 3 years, 3 months ago

But the question never said it is root 2. A lot of irrational numbers can start with 0.4142 and give different answers. How can we say that they all will be same?

Satvik Golechha - 3 years, 3 months ago

If we disregard the fact that it approaches root 2, the pattern with the p's and q's presented still holds among the fractions in the question. Knowing that this sequence approaches the fractional part of root 2 is an extra, essentially trivial, piece of information.

Lucas Chaves Meyles - 3 years, 3 months ago

This is a nice trick, but the question is not really set up in such a way to lead one to discover the trick. The other solutions give the very simple arithmetic required to answer the question.

Stephen Beck - 3 years, 3 months ago

This solution generalizes the sequence of fractions in question. Generalizing the sequence tends to become much more useful in understanding the problem and making it clear for others to interpret, and it provides a wholesome approach to the problem. A simple arithmetic solution, albeit quicker, may not provide adequate reasoning behind the process of receiving the answer.

Lucas Chaves Meyles - 3 years, 3 months ago

Why spend the time trying to find the relationship between the values in the fractions, when a simple division yields the same solution. On a competition quiz without a calculator, recognizing the root would be faster, but for an online quiz where simple calculations are encouraged, this is awkward. Also, as I said in response to another answer, if this question had gone one more step, the answer would have been ambiguous, meaning that the pattern doesn't hold indefinitely. Competition question writers beware!

Chris Brown - 3 years, 3 months ago

@Chris Brown – Interesting. Well, duty noted. Thanks for your input.

Lucas Chaves Meyles - 3 years, 3 months ago

How did you know it was the fractional part of sqrt(2)?

Aviv Silman - 3 years, 3 months ago

It's like knowing that the area of an ellipse is ab(pi). It's essentially a trivial piece of knowledge that might (or might not) assist in a math competition context. I directly learned this from the Art of Problem Solving series of books.

Lucas Chaves Meyles - 3 years, 3 months ago

Concluding with the comments below, this is a bad question. Thanks for sharing though.

Jason Tang - 3 years, 3 months ago

Yep, it's my pleasure.

Lucas Chaves Meyles - 3 years, 3 months ago

Alon Gazit
Feb 26, 2018

$\frac{408}{0.4142}$ = 985.0313858 >> X is 985

$\frac{985}{0.4142}$ = 2378.0782231 >> Y is 2378

985+2378 = 3363

This is how I solved it - also, I checked that 408/984 and 408/986 did NOT result in a decimal whose first four digits were 0.4142. I then did a similar check with 985. It is of note that if you step one more time, 5740 and 5741 both meet the criteria.

Chris Brown - 3 years, 3 months ago

This is exactly how I solved it, too.

Matthew Carreiro - 3 years, 3 months ago

Just an advice: avoid using the double greater-than sign as an implication sign because it may be confusing ;) I suggest you to use —> instead

Eduardo De La Rosa Padrón - 3 years, 1 month ago

Azadali Jivani
Feb 17, 2018

169/408 = 408/X..........X = 985
408/985 = 985/Y...........Y = 2378
985 + 2378 = 3363

This is incorrect. I did not say that these fractions are equal to each other.

Pi Han Goh - 3 years, 3 months ago

No, it is not incorrect, equalities written here should be read as approximations.

Denis Husadzic - 3 years, 3 months ago

Precisely! That is the way I did it too!

Maarten van Helden - 3 years, 3 months ago

Vien Nguyen
Mar 2, 2018

Since each number in the sequence (29, 70, 169, 408) is just the previous number (rounded) multiplied by 2.414, just take the number before X and multiply by 2.414. You get 984.912, which rounds to 985. (408/985 DOES equal 0.4142...) Then do the same with 985, so you get 2377.79, which rounds to 2378. Therefore, X=985 and Y=2378. 985+2378=3363.

(I LOVE this problem, mostly because I actually solved it on my own. :))

Good intuition! +1

Pi Han Goh - 3 years, 3 months ago

Vianney Takou
Mar 13, 2018

485/x=0.4142 therefore 485/0.4142=x which is 985 on the other side, 985/Y=0.4142 therefore 985/0.4142=Y which is 2378 then X+y=2378+985=3363

Godplay Gamer
Mar 2, 2018

(169/70x408)+((169/70x408)x(169/70))

Iskander Elderson
Mar 1, 2018

I only got it because I saw the pattern of 169=29+2(70) and 408=70+2(169) so I just continued it without thinking at all about why it worked.

A Former Brilliant Member
Mar 1, 2018

$\frac{408}{0.4142}$ = X = 985.03 (rounded off to 985)

Also, $\frac{X}{0.4142}$ = Y = 2378.15 (rounded off to 2378)

Therefore, X+Y= 3363

William Chau
Feb 28, 2018

Since 169/408 = (29+2 * 70)/(70+2 * 169), so 408/X = (70+2 * 169)/(169+2 * 408). Thus, X = 169+816 = 985, Y = 408+2X = 2378, and X+Y = 3363.

Yannick Kirschhoffer
Feb 28, 2018

A Python scripting approach:

for x in range(408, 10000):
    for y in range(x, 10000):
        frac1 = 408 / x
        frac2 = x / y

        if int(frac1 * 10000) == 4142 and int(frac2 * 10000) == 4142:
            print(x, y, frac1, frac2, x + y)

will output:

985 2378 0.41421319796954315 0.4142136248948696 3363

See https://github.com/alcibiade/brilliant/blob/master/2018-02-26/l2-problem2.py

Alvin Trixx
Feb 28, 2018

if 29/0.4142=70 then X=408/0.4142 which is equal to:985 so Y becomes=985/0.4142=2378 thus X+Y=985+2378=3363

Akram Igbariya
Feb 26, 2018

Why X=169 , Y=70 are not a correct solution?

Because 408/169 isn't equal to .4142

One of the statements in the problem is that all of the fractions are nearly equal.

Brian Egedy - 3 years, 3 months ago

Matt Miguel
Feb 26, 2018

408/.4142=985.031... 985 is the only integer that has the right first digits in quotient - try plus one and minus one (408/984 & 408/986) to confirm.

Repeat on 985/.4142=2378.078...

2378+985=3363

Continuable fractions

The answer is 3363.

13 solutions

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