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 an integral involving the bessel function

Prove: $$\mathscr{I}_{\space\text{n}}:=\int_0^\infty\text{n}\cdot x\cdot\exp\left(\text{n}\cdot x\cdot i\right)\cdot\mathscr{J}_0\left(x\cdot\sqrt{1-\text{n}^2}\right)\space\text{d}x=-\frac{1}{\...
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55 views

Is there a method to calculate a dyadic fraction for given irrational number with least error?

For example $1/\sqrt{2}$ can be represented as $23/32$ or $181/256$? Is there a standard method to generate dyadic fractions for given irrational number to approximate them? Dyadic fractions are ...
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38 views

What 2 and 4 decimal number has the largest difference against its double IEEE754 representation?

We know that fractional numbers that are powers of 2 like 0.25 (2^-2) or 0.0625 (2^-4) can be represented exactly in a floating point or double type variable. Any fractional number that is also an ...
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38 views

Can I cut matrix $W$ in the fraction $\frac{W}{WH}$ when $WH$ is matrix multiplication?

$W_{ik}$ is matrix with $i$ rows and $k$ columns, $H_{kj}$ has $k$ rows and $j$ columns. $\frac{W_{ik}}{W_{ik}H_{kj}}$ is the element-wise division of matrix but $W_{ik}H_{kj}$ is matrix ...
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135 views

How to find the numerator values of simplified fractions in $T$ to convert it to sets defined on coprime numerators and denominators?

Consider set $T$, a subset of $\mathbb{Q}$, structured as functions in the form of a quotient. Specifically, $$T=\left\{\frac{N(m)}{D(n)}\right\}$$ Where $N$ and $D$ are functions defined on ...
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37 views

How do I prove that half of length n is $\lceil\frac{n}{2}\rceil$

Lets assume there are n objects, ordered in a list. How do I prove that the first half (including the middle element) of length n is $\lceil\frac{n}{2}\rceil$? And how do I prove that the last half (...
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35 views

Why there is at most one fraction $d/r$ with $r<n$ and $q \geq 2n^2$ such that $|\frac{c}{q}-\frac{d}{r}|\leq \frac{1}{2q}$

I am reading a research paper on 'Factoring Algorithms for Quantum Computation' by Peter W. Shor. In this paper there is written somewhere :- (n,c,d,q,r are all positive integers) $$\frac{-r}{2} \...
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185 views

Integer solutions to fraction equation

I am having trouble finding all solutions (or at least proving I have all the solutions already). The equation is $$\frac{1}{a}+\frac{1}{b}+\frac{2}{c}=1$$ *a,b,c are positive I tried to base it on ...
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31 views

On common repeating expansions of coprime pair?

Let $a,b>0$ be coprime and not divisible by $2$ or $5$. Now $$\frac1a=0.00\dots00r_100\dots00r_100\dots00r_1\dots$$ $$\frac1b=0.00\dots00r_200\dots00r_200\dots00r_2\dots$$ where $r_1$ and $r_2$ ...
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93 views

How to expand a fraction to 100 as the denominator.

I'm working with some code that calculates point values based on computer hardware. On one computer as an example, the computer ends up with 29 possible points. Now lets say that I run the same code ...
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252 views

The numerators in partial fraction decomposition involving repeated factors

So I'm confused about the partial fraction decomposition of fractions that involved repeated factors in the denominator. From what I've been taught, when you are breaking your fraction into partial ...
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1answer
424 views

Estimate the value as a whole number or as a mixed numeral where the fractional part is 1/2

I got into a math problem, where it asked the following question: I’m not asking for the answer, but I want to know what it means by: “where the fractional part is 1/2”
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1answer
35 views

Can you solve for the value of x?

I don't remember where I saw this equation but it was difficult to solve. The equation was: $$\frac{\tan(x)}{1+\sin(x)} = q$$ where $q$ is a constant. When I started to solve this equation it made ...
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131 views

How do I interpret [blank] per unit [blank] per unit [blank]?

For example, if I have the energy per unit time per unit area, do I formally interpret that as: $$\frac{E}{(dt)(dA)}$$ and intuitively interpret it as how much energy is in an infinitesimal area in ...
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1answer
80 views

Please help me with the wording of a question

I need to prepare a number of questions for my homework on numbers and fractions. Not being a native English speaker, I'm stuck on constructing a question. Everyday a student solves questions in the ...
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29 views

double series and integral resulting in an indetermined answer

I have the following: $$P = \sum_{j=1}^{6} \sum_{k=1}^{6} A(k,j) \int_0^h e^{(\alpha^{(k)} -\overline\alpha^{(j)})x} dx$$ Where A is a constant that depends on $j$ and $k$. Taking the integral ...
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272 views

Under which conditions maximizing a ratio of functions is equivalent to minimizing its reciprocal?

Assume two non-linear functions, $f(x)$, $g(x)$ respectively, both positive and monotone non-decreasing, $f(x)$ is concave, $g(x)$ is convex. I am trying to maximize their ratio, $\frac{f(x)}{g(x)}$, ...
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2answers
44 views

Proof that sum of combinatorics and fractions equals $1$

For one of the questions in my statistics textbook, the last step of the solution is $$ {m \choose x} (pr)^x (1 - rp)^{m-x} \sum_{t=0}^{k} {k \choose t} \bigg( \frac{r - rp}{1 - rp} \bigg)^t \bigg( \...
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45 views

suggest simplest way to compare strange fraction.

I am doing certain data-interpretation questions. In data-interpretation I am consistently encountering the percentage calculation. sometimes calculations are simple but most of the time they are time ...
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36 views

Fractions not equaling when applied to two sets of data.

I am an accountant with a university mathematics degree. Although its been some years since I became heavily involved in Mathematics, I should be able to understand most things, however I can't get my ...
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39 views

substract functions

Im trying to substract a set of functions $$f(x)-g(x)-h(x)-i(x)$$ where $$ f(x)=\frac{1}{(x+3)(x+4)^2(x+5)^3} $$ $$ g(x)=\frac{-\frac{1}{2}}{(x+5)^3} $$ $$ h(x)=\frac{-1}{(x+4)^2} $$ $$ i(x)=\...
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1k views

How to split the denominator of a fraction into another fraction plus the rest of the denominator?

I know the title is a badly formulated, but I have an example, that should show my Problem better: ${y^2\over(2c+2)} = {y^2\over 2c} * x$ Through Brute-Force I found out that x should for example ...
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1answer
53 views

$\frac{r! - 1}{r! (7-r)!} , 7 \ge r $ [Simplest form required]

I am unsure of the possibility of finding the simplest form of this fraction while $r$ is unknown but if it is, would it also be possible to find the simplest form of the fraction $\frac{r! - 1}{r! (n-...
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1answer
51 views

When is $\frac {t^a - 1} {t^b -1} $ an integer?

Given three positive integers $t$, $a$ and $b$, I'm interested to know when the fraction $$\frac {t^a - 1} {t^b -1} $$ is an integer itself. Excluding any trivial cases, by trying out different values ...
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60 views

Converting sum of individual fraction to a combined fraction

I have following scenario. I want this combined fraction $\frac{A+B+C}{X+Y+Z}$ But Some How I am a little unfortunate and have results in the form of $\frac AX$, $\frac BY$ and $\frac CZ$. How can ...
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139 views

Is there a method to avoid long division?

Arbitary $4$ digit numbers $a$ and $b$ are given. The object is to calculate the fraction $\frac{a}{b}$ without electronic help and without the usual division method. The result should be correct ...
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1answer
137 views

Euler's Continued Fraction Theorem for fractions

How can I use Euler's Continued Fraction Theorem to find the continued fraction expansion for a (ordinary, finite) fraction via its (terminating or recurring) decimal expansion, rather than via the ...
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2answers
53 views

Please help me understand this simple fraction

I got started learning about fractions a few days ago, The tutorial I'm using for this, is limited to fractions like this Now as I'm trying to find further exercices, I keep stumbling upon fraction-...
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56 views

Why is (x-xi)^n still a linear factor (Partial Fraction Decomposition)?

When we perform a Partial Fraction Decomposition and one of the solutions of the denominator is a multiple solution (let's say quadratic), we write: $$\frac{A_{1}}{(x-x_{i})} + \frac{A_{2}}{(x-x_{i})^...
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147 views

Simplifying MATLAB fraction to make numerator equal to 1

I have a function which returns this fraction, which is not in the needed format. $$\frac{36893488147419103232*z^2}{36893488147419103232*z^2 - 672282507639892864*z + 6656262451880127}$$ I need it to ...
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22 views

Term for “Remainder in the Whole”

If I have a proper fraction I want to know what the name is for the amount remaining in the whole. So given $\frac1 3$ I want the name of the term $\frac 2 3$.
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84 views

How to find the common denominator with multiple variables

Find $\frac{zf^{'}(z)}{f(z)}$, where $-1 \leq \alpha \leq 0 $ and $0< v < 1$ Given: $f(z)= \frac{1}{\pi}(-\log (1-vz)+ \alpha \log(1-vz^{-1}))$ and $f^{'}(z)= \frac{1}{\pi}\left(\frac{v}{1-vz} ...
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147 views

Approximate ratio with a small fraction so that numerator multiplied by denominator give enough rectangular area?

I would like to layout given number of objects (like plots) into rectangular area (like computer operating system window on screen). I would like to calculate the width and height of the window (in ...
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2answers
742 views

How to cancel out a negative in a denominator?

The question was to make $y$ the subject in $x=5-3y$ (i.e. solve for $y$). My working was this: $$\begin{align*} x&=5-3y \\ -3y&=x-5 \\ y&=(x-5)/(-3) \end{align*}$$ But this was ...
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1answer
40 views

Simplifying this expression, trigonometry

I have been having trouble understanding how $$6-6\cos\left(\frac{\pi}{4}\right) = 3\sqrt{2}.$$ My main problem is the conversion of the two separate terms into one.
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1answer
109 views

Fractions and decimals

Ten companies sponsored a tournament and decided to give M rupees collectively.Two companies dropped and remaing agreed on paying their share equally.What was the increase in share of each company? ...
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1answer
287 views

Choosing suits of cards in a row

Three cards from a standard deck are dealt. What is the probability that the first is a heart, the second is a spade, and the third is another heart? I have figured out so far that you can use ...
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1answer
39 views

How to calculate $n$th term in terms of constants?

The expression is $$\large t_n=\frac{(x\times t_{n-1})^2}{((x-t_{n-1}\times y)^2+4\times x\times t_{n-1})\times t_{n-2}}$$ where $x$ and $y$ are constants. $t_0$ , $t_1$ , $t_2$ , $t_3$ and $t_4$ ...
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59 views

Can This Expression Be Simplified? (Involves Square Roots)

I started with the expression $$ \frac{4mlt(1-\sqrt{1-\frac{v^2}{c^2}})c^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ and have ended up at: $$ \frac{4mlt(c^2 - c \sqrt{c^2-v^2})}{\sqrt{1-\frac{v^2}{c^2}}} $$ ...
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37 views

Calculating enrichment

My question concerns how enriched something is as im trying to combine several lists of uneven group size and the answer is escaping me. So basically, I have 6 groups and I want to compare them with ...
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52 views

Long division for multipolynomial expression, little o notation

I have this expression: $$\mathrm{Exp}=\frac{d^3(-12a^4)+d^2(4a^4-16a^3)+d(4a^3-6a^2-a)}{d^3(-12a^4+12a^3)+d^2(4a^4-20a^3+16a^2)+d(4a^3-11a+7a)+(1-2a+a^2)}$$ Is there any way I can take the second ...
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1answer
219 views

Equivalent forms of expressions with complex numbers

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$ Select all that apply. $ {18i\over{(81−z)^8}}$ $ {−18i\over{(81+z)^8}}$ $ {18i\over{(81+z)^8}}$ $ {−18i\over{(81−...
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1answer
77 views

Second Order Approximation for a Polynomial

if I have an expression: $L=\frac{12a^3d^3-4wa^3d^2+16a^2d^2-4wa^2d+6ad+1}{12a^3d^3-4wa^3d^2-4a^2wd+16a^2d^2+7ad-aw+1}$ what is the second order approximation in $\frac{d}{w}$? I know that $(\frac{d}...
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1answer
28 views

Is there a value for $a$ other than a factor or a multiple of $c$ in $\frac{a}{b}=\frac{c}{d}$

Suppose $a,b,c,d$ to be whatever quantities whatsoever that satisfy the proportion $\frac{a}{b}=\frac{c}{d}$. Is there a value for $a$ other than a factor or a multiple of $c$. Or, is there a value ...
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1answer
47 views

How to calculate a whole amount with fractions?

A contractor first completes $7/16$ of a building. Then he completes $1/4$ of it. And finally completes $2/5$th of the remainder of the building. If there is $36$ days left to finish the construction ...
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1answer
44 views

Regarding +/- fractions: what are some mental tests you can apply to uncommon fraction denominators?

When adding and subtracting fractions: what if there is no uncommon factor (for example 4=2,2 and 6=2,3). Does that always mean to use the LCM? What if the LCM is too big or time consuming to ...
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1answer
30 views

Solving $\frac12 (3y+2)-\frac58=\frac34y$ for $y$ using LCD method

I am solving $$\frac12 (3y+2)-\frac58=\frac34y$$ for $y$ using LCD method. Can't figure out what I did wrong! The answer in the back of the book is $-1/2$. PS: In the first line that is a $1/2$ in ...
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1answer
50 views

Solving function in difference quotien equation

I have the problem Find the difference quotient $\frac{f(2 + h) - f(2)}{h}$ for $f(x) = \frac{1}{x^2}$. The answer they gave is $\frac{-(4 + h)}{4(2 + h)^2}$ So far I've done: $$\frac{[1/(2 + h)^2 ...
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1answer
253 views

Deradicalization of denominators

Task: Develop a fraction equivalent to $$ 1\over{\sum\limits_{i=0}^{n-1}c_in^{i/n}} $$ in which the denominator is rational.
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1answer
436 views

Rearranging algebraic formula when subject is on both sides

I have run into some difficulty with a question on making a variable the subject of an equation where the variable is on both sides. I am really struggling to find a method for making "a" the ...