# Questions tagged [fractions]

Questions on fractions, i.e. expressions (not values) of the form $\frac ab$, including arithmetic with fractions. Not to be confused with the tag (rational-numbers): fractions denote rational numbers, but the same rational number may be written in different ways as a fraction.

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### Prove that $x^xy^y+x^yy^x \le 1$ if $x+y=1$ and $x,y$ are positive [duplicate]

Given $x+y=1$, $x>0$, $y>0$, prove that $$x^xy^y+x^yy^x \le 1.$$ I first tried using lagrangian, $\mathcal{L}=f(x,y)-\lambda(g(x,y)-c)$ where $f(x,y)=x^xy^y+x^yy^x$ ,$g(x,y)=x+y$ and $c=1$, ...
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### Expressing $\frac{2}{n}$ as the sum of two unit fractions

Consider fractions such as $\frac{2}{5}$ and $\frac{2}{7}$ expressed as the sum of two unit fractions. Respectively, they can be expressed as $\frac{1}{3}+\frac{1}{15}$ and $\frac{1}{4}+\frac{1}{28}$. ...
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### How can i do the following partial decomposition?

I need to prove that: $$\frac{1}{(x-a)(x-b)} = \frac{1}{(b-a)(x-b)}- \frac{1}{(b-a)(x-a)},$$ and I must note that I need to go from the left expression to the right (because of the exercise). So, I ...
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### Is the following mixed fraction equation correct?

I am wondering if the plus sign is understood in mixed fractions. So it is not written.
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### Method to difference between two continuous fractions and keep continued fraction form

I warmly welcome an approach on differencing between two continuous fractions Without applying the appropriate algebra to reduce the continuous fraction. I cannot find a way without completing the ...
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1 vote
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### Finding the number of fractions with unique values where the range for the denominator and numerator is in $\{1,2,\dots, n\}$

Imagine that you have a fraction, where both the numerator and denominator take values in the set $\{1,2,\dots, n\}$. Let us assume that the fraction is smaller then and or eqal to 1. The question is ...
1 vote
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### The Equivalence Transformation (??) for Generalized Continued Fractions

The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/...
1 vote
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### Continued Fraction Representation of sin(x)

To provide context, the continued fraction in the form $\frac{a_0}{1-\frac{a_1}{1+a_1-\frac{a_2}{1+a_2-...}}}$ evaluated to the $n$th denominator equals $\sum_{k=0}^{n}\prod_{j=0}^{k}a_j$. If one ...
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### What is the formula for converting an improper fraction to a mixed number [closed]

There are methods for converting improper fractions to mixed numbers, but I am interested on finding a formula to which I can input the numerator and denominator of an improper fraction and get an ...
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### Is this example of scaling fractions correct?

Suppose we have $\frac{5\times2}{3\times5}$. If we consider it as $\frac{5}{5}\times\frac{2}{3}$, then $\frac{5}{5}$ is being scaled down and $\frac{2}{3}$ is neither being scaled down, nor being ...
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### How to prove a circle in the complex plane tangent to the real axis at the origin is invariant by the function $\frac{\omega}{1-\omega}$? [closed]

Let $C$ be a circle in the complex plane $\mathbb{C}$ whose centre is $(0,ir)$ (where $r$ is a non-zero positive real number) and is tangent to the real axis at the origin $(0,0)$ (the radius of the ...
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### Simplifying a complex fraction

Simplify the complex number: $\frac{(\sqrt{3} + 9i)^{181}}{(-12 + 48\sqrt{3}i)^{90}}$ I've tried a few things such as turning the top and bottom of the fraction into their trigonometric form or trying ...
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### What is the Difference between these two expressions: $3$ $\frac {1}{7}$ and $3 + \frac{1}{7}$?

What is the difference between $3$ $\frac {1}{7}$ and $3 + \frac{1}{7}$? In the first expression, $3$ seems to be multiplied by $\frac {1}{7}$ using juxtaposition, but while doing the calculation, we ...
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### How to calculate amortized cost of the push operations for a resizing stack

In Algorithms (4th ed.) by Robert Sedgewick and Kevin Wayne at page 199 it's illustrated how the cost of resizing a stack (based on a dynamic array) is amortized among the most recent push operations: ...
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