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If $p$, $q$ and $r$ are positive integers and $p + \displaystyle\frac{1}{q + \displaystyle\frac{1}{r}} = \frac{129}{31}$ then what is the value of $p + q + r$?

I tried getting a common denominator, but nothing seems to work as the correct answer is an actual number

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    Hint: $p = \lfloor 129/31 \rfloor\,$. – 2017-01-17
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    Second hint: if the answer is a unique number, and $(p,r,q)=(4,5,6)$ does work, then we are done. – 2017-01-17
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    how do u know p = 129/31? @dxiv – 2017-01-17
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    @Shahad $p$ is an integer, and $1 / (q + 1/r) \lt 1$. – 2017-01-17
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    is this process considered to be part of calculus or algebra? As I have never came across this before, and I need to learn it now. – 2017-01-18

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The fraction $$\frac1{q+\dfrac1r}$$ is lesser than $1$, so $$p=\left\lfloor\frac{129}{31}\right\rfloor=4$$ (The brackets $\lfloor\quad\rfloor$ mean "integer part").

Now, $$\frac1{q+\dfrac1r}=\frac{129}{31}-4=\frac5{31}$$ Therefore $$q+\frac1r=\frac{31}5$$

Proceed similarly to find $q$ and $r$.

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    is this process considered to be part of calculus or algebra? As I have never came across this before, and I need to learn it now. – 2017-01-18
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    now we know that p=4, i used that same technique for q and r but i did not arrive to the correct answer: – 2017-01-18
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One Liner: $$\frac{129}{31} = 4+\frac{5}{31} = 4+\frac{1}{\frac{31}{5}} = 4+\frac{1}{6+\frac{1}{5}}$$ Fundamentally, this just comes down to writing fractions in simplest form, where denominator exceeds the numerator

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    +1, Nice, although it is not totally clear that the found values are unique. – 2017-01-17
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    @ajotatxe fair enough. I was more going for brevity :) Uniqueness shouldn't be hard... clearly $p$ must be unique because everything after it is less than $1$, and $q$ must be unique for the same reason. Given this, the fact that $r$ is unique is trivial. – 2017-01-17
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    is this process considered to be part of calculus or algebra? As I have never came across this before, and I need to learn it now. – 2017-01-18
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    @Shahad the first and third steps are just grade school mathematics. I've often seen it called "writing a fraction in simplest form" in the United States (where it would probably be written as a "mixed number", e.g.. $\frac{129}{31} = 4\frac{5}{31}$. Note the lack of a addition sign, although it's implicitly there). All you are doing in these steps is seeing how many times the denominator goes into the numerator completely (I.E. the greatest multiple of the denominator less than the numerator) and then separating this from the remainder – 2017-01-18
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    For example, on the first step I would note that $31\cdot 4 =124$. I can't do $31\cdot 5$ because I exceed numerator, so I know that I can write the numerator as $\frac{124+5}{31}$. Now just split the fraction and simplify – 2017-01-18
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    @Shahad the second step isn't a super common technique, but in essence I did it to put the fraction in the form you provided. I realized we needed a $1$ in the numerator and that $\frac ab = \frac{1}{\frac ba} $ – 2017-01-18
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    @Shahad admittedly, this is often considered more arithmetic than algebra. It's definitely not at all related to calculus. I would review your elementary algebra (called Pre-Algebra in United States) a lot. – 2017-01-18