Articles of fractions

Fractions with radicals in the denominator

I’m working my way through the videos on the Khan Academy, and have a hit a road block. I can’t understand why the following is true: $$\frac{6}{\quad\frac{6\sqrt{85}}{85}\quad} = \sqrt{85}$$

Simplify the fraction with radicals

I want to simplify this fraction $$ \frac{\sqrt{6} + \sqrt{10} + \sqrt{15} + 2}{\sqrt{6} – \sqrt{10} + \sqrt{15} – 2} $$ I’ve tried to group up the denominator members like $ (\sqrt{6} + \sqrt{15}) – (\sqrt{10} + 2) $ and then amplify with $ (\sqrt{6} + \sqrt{15}) + (\sqrt{10} + 2) $

What is $\lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$?

These are two little questions that came to mind while I was looking at this problem. What is $\displaystyle \lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$? I am fairly certain that the answer is $\infty$ because as $n$ gets closer to $\infty$ there are more terms that are very close to $1$ (if $n = 1,000,000$ then […]

Fraction raised to integer power

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a… I don’t know what you call it. Not a whole number, but something like 15/7 where you can’t reduce it any more and it’s non-integer. Can $(p/q)^n$ ever be an integer?

Expanding integers into distinct egyptian fractions – what is the optimal way?

We know that there are infinite ways to represent $1$ as a sum of distinct unit fractions (i.e. egyptian fractions). The most optimal one (the least demoninators and the least number of fractions) is: $$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$$ But how to represent other integers in the same way? The rules are as follows: we can’t use $1$ as […]

Why does the Denominator of the Denominator go to the Numerator?

Quite blindly I’ve learnt a basic rule about fractions: The Denominator of the Denominator goes to the numerator. I’m confused about it and I’ll give an example as to why. Imagine the following: 1/2/2 Now, if the denominator of the denominator went to the numerator this would be 2/2 which is 1 and that’s the […]

How to show that $\frac{x^2}{x-1}$ simplifies to $x + \frac{1}{x-1} +1$

How does $\frac{x^2}{(x-1)}$ simplify to $x + \frac{1}{x-1} +1$? The second expression would be much easier to work with, but I cant figure out how to get there. Thanks

Writing $1$ in form of $\frac{1}{t_1}+\cdots+\frac{1}{t_n}$

Possible Duplicate: Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$ Can anyone help me with this problem? It’s a little strange: Let $M$ be a natural number. Prove that we can write $1=\frac{1}{t_1}+\cdots+\frac{1}{t_n}$ such that all $t_i$’s are distinct natural numbers greater than $M$.

Faster arithmetic with finite continued fractions

I was curious about different representations of rational numbers and came across the finite continued fraction (see wp:Finite_continued_fractions ). Note: I will refer to traditional rational representation with two integers as fractional representation and to reduced fractions ($gcd(n,d)=1$, where $n$ is the numerator, and $d$ is the denominator) of this sort as reduced fractional representation. […]

Is there any toy for learning algebraic manipulation of fractions?

Is there any toy for learning algebraic manipulation of fractions? If you don’t know of any, how would you design one? What I’m imagining is something similar to a Rubik’s cube whose manipulation produces only true equations in some number of variables, for example: $\frac{a}{b} = \frac{c}{d}$ (turn a knob) $a = \frac{b c}{d}$ (twist […]