Questions on fractions, which are expressions (not values) of the form $\frac pq$.

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1
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0answers
18 views

Stern-Brocot Tree and sum of coefficients of continued fraction

Suppose we are given a continued fraction $$\frac{p}{q}=a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\cdots}}}$$ I am trying to find an expression, possibly asymptotic, for the sum of the ...
3
votes
3answers
55 views

Basic algebra problem: $ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $

Basic algebra problem I can't seem to figure out: $$ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $$ $x,y \in \mathbb{R}, x^2 \neq y^2, xy\neq0$. Now I know the result is: ...
2
votes
3answers
54 views

Why Doesn't $2^{1/n}= 1/(2^n)$

Take $2^{1/n}$. Since $1/n$ can be simplified as $n^{-1}$, the original term can become $2^{n^{-1}}$. The exponents can then be multiplied to result in $2^{-n}$ which is $1/(2^n)$. However it is ...
0
votes
4answers
37 views

Simple Fraction needing explanation

$$\frac{x}{x^{-1/2}} = x^{3/2}$$ How? I don't see what is going on here. What rule is being used to achieve this amount?
3
votes
1answer
56 views

Repeating decimal notation of 1/53 on WolframAlpha vs notation on Wikipedia

WolframAlpha shows for 1/53 $0.0\overline{1886792452830}$ as the repeating decimal. Why is it not $0.\overline{0188679245283}$ instead? For example, Wikipedia shows for 1/81 ...
1
vote
0answers
37 views

Is this a mistake on my part or theirs?

I'm not sure if I'm the one making the mistake, or my math book. It looks like the negative sign completely disappeared. $$\frac{3x^2}{-\sqrt{18}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = ...
1
vote
3answers
30 views

How do you solve B and C for $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$?

How do you solve B and C for $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$ ? $A = \left.\frac{s^2-s}{s^2+1} \right\vert_{s=-1} = \frac{1-(-1)}{1+1}=1$
19
votes
9answers
2k views

How to find irrational numbers between rationals. (And is my method correct?)

I have a question from an A-level revision book: Find an irrational number which lies between $\frac34$ and $\frac78$. What is the correct method for doing this? Here is my method: Square ...
3
votes
1answer
39 views

Simple formula for the $n$-ary version of $(x,y) \mapsto \frac{x+y}{1-xy}$

Let $x * y = \frac{x + y}{1 - xy}$. I want a single formula for $x_1 * x_2 * \ldots * x_n$, for all natural $n$. In order to generate plausible candidates, let's see what happens at small values of ...
-1
votes
2answers
38 views

World Problem Math Algebra Fraction

The denominator of a fraction in simplest form is greater than the numerator by $3$. If $7$ is added to the numerator, and $5$ added to the denominator, then the fraction itself is increased by ...
0
votes
1answer
37 views

Partial fraction integration with unclear roots

Let's look at a simple example like $\frac{1}{x^3+2x+1}$ here. We know that the denominator has a real root between $0$ and $-1$ (could go closer, but that's not the point). By the concept of slope of ...
0
votes
2answers
39 views

Partial fraction in two variable problem

How to write partial fraction of $$\frac{12m-n-3mn+7}{5m-2n-2mn+5}$$ I just write first and second denominator: $5-2n$ and $m+1$.
0
votes
1answer
14 views

Ratio of sums vs sum of ratio

Is anyone aware of any general (or perhaps not so general) relationship (inequality for instance) relating $A(x,y)= \frac{\sum_z f(x,y,z)}{\sum_z g(y,z)}$ and $B(x,y)= ...
3
votes
3answers
56 views

How to articulate where the extra 1 came in this easy question

"Albert owns 5/9ths of the stock in the North West Chocolate Company. His sister, Rena, owns half as much stock as Albert. What part of stock is owned by NEITHER Albert nor Rena?" The answer is ...
-1
votes
1answer
61 views

The name of the sum $\sum_{i=0}^n \frac{1}{m-i}$

Sorry for the vague question name, since I am looking for the name of the series. Also this might not be a "series" by the strict definition of a series.. anyways here it is: Choose some $m$ and $n$ ...
2
votes
1answer
26 views

Is it correct? Prove that any fraction can be reduced

I want to know if my prove is correct. My goal is proving: Hypothesis: $a,b \in \mathbb Z$ and $a,b \notin \{-1, 0, 1\}$. Thesis: for all $ a, b$, exist $a',b'\in \mathbb Z$ that verify ...
2
votes
1answer
20 views

About a largest integral value of this sum of reciprocal numbers.

In a test , I was asked to solve the following question : If $a_1,a_2,a_3, \cdots ,a_n$ are $n$ distinct odd natural numbers and not divisible by any prime number greater than equal to $7$ . Then the ...
2
votes
3answers
58 views

Solve fractions multiplication

I believe this is a very simple one, but I simply can't figure it out. How to solve? $$\frac12\cdot\frac34\cdot\frac56\cdots\frac{17}{18}\cdot\frac{19}{20}$$
3
votes
2answers
64 views

Laurent expansion of $\frac{1}{z^2}$

I need to find a Laurent expansion of $\frac{1}{z^2}$ with centre in $z_0 = 1$ and $P(1, 2014, 2015)$. If it was $\frac{1}{z}$, I'd rewrite the fraction like this: $$ \frac{1}{(z-1) +1 } $$ But ...
1
vote
4answers
80 views

How do I show that $-\frac{1}{e^x + 1} + 1 = \frac{e^x}{e^x + 1}$?

The expression is $$-\frac{1}{e^x + 1} + 1 = \frac{e^x}{e^x + 1}$$ I would like help to get from the left side to the right side.
-1
votes
2answers
46 views

If the chance of an event was $1/128$ and increased by $20\%$, what is the new chance?

So I have something that has a 1/128 chance of occurring, let's say. Suddenly, the chances of that thing happening are increased by 20%. How is that fraction written? Would you multiply 1/128 by ...
3
votes
3answers
42 views

How to properly set up partial fractions for repeated denominator factors

I was just trying to solve a problem that had the following item which I needed to split into separate generating functions: $$\frac{x}{(1-2x)^2(1-5x)}$$ I had assumed I needed to split it into: ...
0
votes
1answer
25 views

Find original cost based on fractional purchase

How would I go about finding the original cost of bitcoin knowing that $20 purchased .0531401 of bitcoin? I would like to know what the cost of 1 bitcoin was at the time of purchase? ...
3
votes
1answer
32 views

Unit Fraction Addition

My teacher challenged us: "Can you express the fraction $55\over 108$ as the sum of two unit fractions$?$" I figured out that ${1\over 54} + {1\over 2} = {56\over 108}$ but I could not figure out a ...
4
votes
1answer
31 views

Multiplication of algebraic fraction not giving desired result

I am trying to solve this: that supposed to return: but I get stuck at: which can be written as and I don't know how to procede from there....
1
vote
5answers
161 views

Proving that $\frac{a}{b} = \frac{c}{d}$ if and only if $ad = bc$

I was working on a problem which asked: Prove that $ \frac{a}{b} = \frac{c}{d} $ if and only if $ad=bc$, provided $c,d \neq 0$. Is it sufficient to manipulate $ \frac{a}{b} = \frac{c}{d} $ via ...
0
votes
2answers
32 views

How do we compare fraction without changing to a similar denominator?

This is Singapore Mathematical Olympiad 2015 Grade 8/Secondary 2 Junior Round 1 Question 1. 1.Among the five numbers, $\frac{5}{9},\frac{4}{7},\frac{3}{5},\frac{6}{11}$ and $\frac{13}{21}$, which ...
3
votes
1answer
54 views

$\frac{a(a+b)}{4a^2+ab+b^2} + \frac{b(b+c)}{4b^2+bc+c^2} + \frac{c(c+a)}{4c^2+ca+a^2} \leq 1$

I've got stuck at this problem: Let $a$, $b$, $c$ be real numbers. Prove that $$\frac{a(a+b)}{4a^2+ab+b^2} + \frac{b(b+c)}{4b^2+bc+c^2} + \frac{c(c+a)}{4c^2+ca+a^2} \leq 1$$ Firstly, I've ...
1
vote
1answer
33 views

How to derive this simple equality?

Let us define $L_i\triangleq \log \left( \dfrac{Prob(x_i=+1) }{ Prob(x_i=-1)} \right)$ $E\{x_i\} \triangleq Prob(x_i=+1)-Prob(x_i=-1)$ I need to show that \begin{equation} E\{x_i\} = ...
-5
votes
1answer
54 views

Can't calculate math problem [closed]

Can anyone help me with this math problem? I would require step how you came to the result.
3
votes
6answers
200 views

What is the value of $\frac{\sin x}x$ at $x=0$?

On plotting graph for $\frac{\sin x}{x}$ using Wolfram|Alpha and Google, got that : also, I can get the value of $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ using squeeze theorem and as illustrated ...
10
votes
1answer
79 views

Prove that $abc$ is a cube of some integer.

Given three integers $a$, $b$ and $c$ such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer too, prove that the product $abc$ is a cube. By the way: Merry Christmas! ;)
1
vote
4answers
77 views

What is the simplest way to find $\frac{n}{7}th$ of a line with mathematical proof?

I'd like to know if it is possible to find out the simplest way to get $\frac{1}{7}$, $\frac{2}{7}$, $\frac{3}{7}$, $\frac{4}{7}$, $\frac{5}{7}$ and $\frac{6}{7}$ of a line in 2-dimensional geometry. ...
2
votes
3answers
53 views

How is this limit being solved? I can't grasp it

I am going over limits for my finals as I notice this example in my schoolbook discribing limits of the undefined form $0\over0$ in the shape of an irrational fraction. $$\lim\limits_{x \to 1} ...
0
votes
4answers
35 views

Dividing factorials

I'm told that $\dfrac{(n+1)!}{(n+2)!}$ simplifies to $\dfrac{1}{n+2}$, but I dont understand how this works. Could someone explain the theory of how to divide factorials like this?
1
vote
1answer
37 views

What rule is used for this simplification?

$$ \frac{8}{(s+1)^2 + 2^2} \times \frac{1}{s} = \frac{8}{5} - \frac{1}{s} + \frac{16}{10}\times \frac{s+1}{(s+1)^2 + 2^2} + \frac{8}{10}\times \frac{2}{(s+1)^2 + 2^2} $$
0
votes
3answers
25 views

Help Solving Fraction Math Question

Im stumped by this question on a practice ACT math test. If $\frac 1x + \frac 1y = \frac 1z$ then $z =$? The correct answer is $\frac {xy}{x + y}$ How do you arrive at this answer? I don't know how ...
1
vote
1answer
22 views

Prove that $\frac{n-a}{n} < \frac{n+1-a}{n+1}$?

I have this math question that I'm kind of stuck on. Prove that $\frac{n-a}{n} < \frac{n+1-a}{n+1}$ So far I have that: $\frac{n-a}{n} < \frac{n+1-a}{n+1} = n-a < ...
1
vote
1answer
24 views

How to calculate remainder value of a fraction

Question: Four brothers split a sum of money between them. The first brother received 50% of the total, the second received 25% of the total, the third received 20% of the total, and the fourth ...
1
vote
2answers
31 views

KhanAcademy nested fractions problem

So I was given a problem at KA today. They offer you a choice among possible simplified versions of this expression: $$\frac{1+\frac{x}{y}}{\frac{x}{y}}$$ My solution was: ...
0
votes
1answer
41 views

How to do rational expressions

Never was much of a math student but I am brushing up on my arithmetic and algebra for college. I am using sample questions from Accuplacer and then using video lectures and practice on Khan ...
2
votes
1answer
31 views

Find the sum of the integers in the continued fraction

Find the sum of integers $a,b,c,d,$ and $e$ if $\dfrac{2011}{1990} = a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{d+\dfrac{1}{e}}}}$. I could simplify the big fraction on the RHS, but I don't see how that ...
0
votes
0answers
48 views

How to simplify an expression?

I have tried to simplify this expression for quite a long time now but I can't find how to do it. Can someone help me with it.
3
votes
2answers
30 views

Checking convergence of series

My task is to check whether the series $$\sum_{n=1}^\infty {\frac{(-2)^n+n^2}{n2^n}}$$ converges or not. So I tried expanding the fraction like that(should equal to original form): $$\sum_{n=1}^\infty ...
1
vote
1answer
28 views

Showing this fraction is less than one?

And also, what are some techniques to show such a thing. That is, what are standard techniques for showing a fraction is less than one? I'm just trying to refresh/improve/add to my toolbox. The ...
0
votes
1answer
23 views

What does the word rational in rational fraction refer to

I understand that a rational fraction is an element in the fraction field $F(X)$ and that a fraction $f$ is represented by a quotient of two poloynomials $A\over B$ under the relation ${A\over ...
1
vote
1answer
83 views

How to explain aspects of derivatives to little brother?

So as the title suggests, my $12$ year old little brother loves math. Since he is a bright kid, I started teaching him derivatives. The issue is he always keeps asking weird questions that are above ...
0
votes
4answers
57 views

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$ Sorry for the really dumb question but I'd like to see the process of how this is achieved.
2
votes
4answers
82 views

Simple fraction calculation

I'm trying to write this expression as one fraction: \begin{align} \frac{5}{y+1}\cdot\frac{1}{5}+\frac{y}{\frac{y+1}{3}} &= \frac{5y}{5y+5}+\frac{3y}{y+1} \\ &= ...
1
vote
5answers
78 views

How to raise a fraction to a fractional exponent?

I am trying to simplify this but I'm not sure how to approach it.