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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17 views

The prime $p$ divides the numerator of $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{p-1}$ [duplicate]

Let $p \ge5$ be a prime number, and let $a$ and $b$ be natural numbers. If $\sum_{k=1}^{p-1}\frac{1}{k}=\frac{a}{b}$, prove that $p|a$.
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1answer
38 views

Find the solutions of second derivative [on hold]

If $$f(x)=\frac {2x}{x^{2}-3x+2}$$ find the solutions of $$f''(x)=0$$ This problem is from a test. Is it possible to solve this without a calculator and without hard work? Maybe this fraction can be ...
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3answers
41 views

Technicality at multiplying fractions when solving inequalities?

I have a question when multiplying fractions in this case (assuming): $$x>0$$ 1.$$\frac{x+4}{3x+2}>\frac{1}{x}$$ $$\frac{x+4}{3x+2}-\frac{1}{x}>0$$ $$\frac{x^2+x-2}{x(3x+2)}>0$$ 2.$$\...
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1answer
33 views

Probability mass function of sum of random variables

Let $\mathbb P(L=k) = \frac{140}{223(k+2)}, k = 0, 1, ..., 5.$ Credit portfolio consists of 8 independent loans, where $L_i, i = 1, ..., 8$ is the number of “physically” possible defaults of one loan....
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0answers
21 views

Is it correct to say 5 is divided in equal parts by 6 [closed]

If we write 16/2, we say 16 is divided into two equal parts, likewise is it correct to say 5 is divided into 6 equal parts?
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2answers
666 views

Fibonacci sequence and other metallic sequences emerged in the form of fractions

The Fibonacci sequence $P_n = P_{n-1}+P_{n-2}$ is $$1,1,2,3,5,8,13,21,34,55,89,144,233,377, 610, \cdots $$ I learnt that the fraction $1/89$ contains all the numbers in the sequence. $$\begin{align} \...
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0answers
27 views

Integral of a squared sum of sin + exponential

What is the best way to solve this integral? $$\int_{0}^{\pi/2}\bigg(\frac{\sin^{2}\left(x\right)}{\sin^{2}\left(x\right)+c_{1}}e^{m\frac{\sin^{2}\left(x\right)}{\sin^{2}\left(x\right)+c_{1}}} + \...
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3answers
50 views

When sum of fraction is the same as the fraction made by the sum of numerators and sum of denominators

My students naturally want to add fractions adding numerators and denominators. I say many times it does not rule like this, but is there a (small) set of integers which this rule work? That is Where ...
3
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1answer
35 views

Finding the inverse Laplace transform of a physics related problem

I'm trying to find the inverse Laplace transform of: $$\frac{as+b}{s(cs+d)+g}\tag1$$ First of all I can expand the fraction: $$\frac{as+b}{s(cs+d)+g}=a\cdot\frac{ s}{s(cs+d)+g}+b\cdot\frac{1}{s(cs+...
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0answers
13 views

How to rewrite the following equation as x in terms of y?

How to convert the following function y of x $y=f(x)=\frac{x}{x+a}+\frac{x}{x+b}$ into a function x of y $x=g(y)=?$ in its simplest form. This is part of an integration problem so having no ...
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1answer
70 views

Diophantine $\frac{a^2 + b^2}{ab + 1} = \frac{c^2}{d^2} $

Consider the Diophantine equation with $a,b,c,d > 0$ : $$\frac{a^2 + b^2}{ab + 1} = \frac{c^2}{d^2} $$ For the case $d=1$ , this is a Classic ; we know that Diophantine $\frac{a^2 + b^2}{ab + 1}...
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3answers
21 views

Simplifying fraction with nested radicals and fractions

This is my first question here and on a stack exchange in general. I hope my question is precise enough. I have spent a good 15min searching the forum but didn't manage to understand the below. I am ...
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3answers
42 views

How to simplify this expression with fractions?

$$\frac{1}{a(a-b)(a-c)} + \frac{1}{b(b-a)(b-c)} + \frac{1}{c(c-a)(c-b)} $$ I tried to get everything to the same denominator, and then simplify numerators first but it is very complicated and long if ...
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1answer
44 views

Find a general expression for $\frac{p}{p+1 - \frac{p}{p+1 - \frac{p}{p+1 - \ldots}}}$ $n$ times for any value of $p \in \Bbb R$ .

Find a general expression for $\frac{p}{p+1 - \frac{p}{p+1 - \frac{p}{p+1 - \ldots}}}$ $n$ times for any value of $p \in \Bbb R$ . Obs: Consider $n=1 : \frac {p}{p+1}$ and $n=2: \frac {p}{p+1 - \...
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3answers
41 views

How to transform a continued fraction so that all denominators are positive?

I came across this quote: "Every continued fractions $a_1, a_2, ..., a_n$ can be transformed to a unique canonical form $\beta_1, \beta_2, ...., \beta_m$, where all $\beta$ 's are positive or all ...
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1answer
25 views

Doubt in Fraction and percentage

There's a question in a book, 16 2/3% of 600 gm - 33 1/3% of 180 gm I was solving it like regular method which i use to find x% of a number, for eg. ...
4
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3answers
515 views

Why is the reciprocal used in fraction division?

I don't know if this is a basic question or whatever, but I can't seem to find an answer. As far as I understand the reciprocal of a number the inverse of that number, that still doesn't clarify why ...
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1answer
17 views

How do I convert fraction percentage into mixed fraction?

I know how to convert a fraction into decimal but I want to make my calculation faster so I want to learn how to convert between fractions and percentage. I found this on a book: ...
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1answer
28 views

moving fraction into denominator

Hello Mathematics Stackexchange I had a quick question. I do sincerely apologize if this type of question was asked before. Im having trouble simplifying this fraction specifically I am not sure how ...
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1answer
26 views

How to I simplify this to a single fraction?

I don't know how to fully simplify this and get rid of the seven at the end. If anyone could help I would greatly appreciate it.
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1answer
50 views

How to simplify $\frac{e^x}{1+e^{x}}$ to $\frac{1}{1+e^{-x}}$?

The two are equivalent, as a check with wolfram alpha shows. I can also solve $\frac{e^x}{1+e^{x}} = A+ \frac{1}{1+e^{-x}}$? and I get that $A=0$. But is there a way that I can directly simplify $\...
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3answers
92 views

Number of possible integer values of $x$ for which $\frac{x^3+2x^2+9}{x^2+4x+5}$ is integer

How many integer numbers, $x$, verify that the following \begin{equation*} \frac{x^3+2x^2+9}{x^2+4x+5} \end{equation*} is an integer? I managed to do: \begin{equation*} \frac{x^3+2x^2+9}{x^2+4x+5} ...
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1answer
42 views

Pell's Equation and Continued Fractions [closed]

For each of the following equations, determine whether there are no solutions, finitely many solutions, or infinitely many solutions with $x, y$ justify your answers. $$x^2-5y^2=3 \\\ x^2+7y^2=...
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1answer
31 views

How do I get 1/15 of something, only by divide with 2 or 3 and add the result back together?

I'm currently playing the game Satisfactory, where I need to balance the conveyor belts to ensure a 100% efficient factory. To help me in this job I have Merger and Splitter. The Splitter can split ...
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1answer
37 views

How do expressions of the form $\frac{a}{b}+\frac{c}{d}+…$ compare to $\frac{a+c+…}{b+d+…}$, for positive $a,b,c,d…$?

Is the question too general to answer? I'm talking about when the former expression will be greater (or smaller) than the latter? Here's an extension: Compare expressions $$p\frac{a}{b}+q\frac{c}{d}...
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2answers
54 views

Help me solve this precalculus algebra expression!

Expression and my attempt at solution: $$\frac{3ab}{c^{-1}}:\left(\frac{b}{c^{-1}}+\frac{a}{c^{-1}}-\frac{a}{b^{-1}}\right)-\frac{(a-1)a^{-1}+(b-1)b^{-1}+(c+1)c^{-1}}{a^{-1}+b^{-1}-c^{-1}}=$$ $$\frac{...
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1answer
42 views

Simplify $9((x^2-15x+50)/84)-12((x^2-8x-20)/-35)+33((x^2-3x-10/60)$

Hi I am trying to simplify the following I found online $9\left(\dfrac{x^2-15x+50}{84}\right) + -12\left(\dfrac{x^2-8x-20}{-35}\right) + 33\left(\dfrac{x^2-3x-10}{60}\right)$ to $= x^2 -6x -7$ ...
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3answers
78 views

Is the following result true? Or Is there any known result about fractions like this?

Is the following result true? Or Is there any known result of fractions like this? Let $n$ be fixed. There are infinitely many integer solutions for $$\sum_{i=1}^n \frac{1}{x_i} = 0,$$ where $x_i \...
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2answers
234 views

Why does this procedure terminate? Or are there any numbers for which it doesn't?

I don't really have good formal education in theoretical mathematics, so please don't be upset if this is obvious question, but on the other hand I don't believe I am the first one to think of such ...
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1answer
39 views

simplifiy complex expression [closed]

How to simplify the following expression: $$\frac{z-1}{z+1}~, \quad \text{where} z\in \mathbb{C}\setminus \{-1\}$$ There is just nothing i can come up with, neither in cartesian, nor in polar.
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1answer
18 views

Find whole numbers of which's average is given

I was trying to calculate something, when I came across with something I couldn't solve. I needed to "reverse an average", find whole numbers, of which's average is 0,625. I feel like this is a thing ...
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1answer
34 views

How do I simplify $\frac{(bS)^{2}}{(bS)^{2} + y}$ to $\frac{1}{1+\left[\frac{y}{bS}\right]^{2}}$?

As the above mentions I have the fraction $\frac{(bS)^{2}}{(bS)^{2} + y}$ and the next step in the equation I am following simply states "it works out to equal" $\frac{1}{1+\left[\frac{y}{bS}\right]^{...
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1answer
53 views

I tried answering this question but my answer is in decimal. How can a number of muffin be in decimal?

Mrs Ho baked a total of 60 banana muffins and chocolate muffins. After she gave away 5/7 of the banana muffins and 1/2 of the chocolate muffins, she had twice as many as chocolate muffins as banana ...
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2answers
60 views

Find the minimum value without using derivative

Find the minimum value of $$f(x) = {3\over \sqrt{x}+1} - {12\over \sqrt{x}+3}$$ The domain of $f(x)$ is $x ∈ (0,∞)$. Then, using derivatives, I can find the minimum value is $f(1)=-1.5$. However, ...
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2answers
47 views

Stuck on candy bowl fraction

I am really stuck on this problem because I'm not even sure where to start. Larissa has a bowl of candies. On the first day, she eats 1/2 of the candies plus one more. On the second day, she eats 1/3 ...
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4answers
46 views

Evaluating limits in fractions

When you want to find the limit of a fraction e.g. $\frac{1-x}{1-x^3}$ as $x$ tends to $1$. Why can you not just plug in x into the numerator and denominator? Why do you have to make all the $x$ ...
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1answer
20 views

Algebraic manipulation and inequality

Given real-valued terms $a,b,c \in {R}$ with the following conditions on them: $ 0 \leq a \leq 1 $, $|b|<1$, and $|c|< 1$. And given the terms $ X= \frac{1}{1-( (1-a)b + ac )}$ and $ Y =\frac{1}...
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1answer
39 views

Basic division problem: dividing a fraction by a fraction

I thought I clearly knew how to divide fractions by fractions until I came across this problem. Please can somebody let me know where I am going wrong? Here is what we start with. $(1-2x)/(2x^1/2)/e^...
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1answer
82 views

Take two numbers x and y between 1 and 100. What’s the probability that x/y is an integer?

It was stated, as an inconsequential remark, in some lecture notes I was reading that if we are to choose two natural numbers in a certain interval and divide one by the other, that it is quite likely ...
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2answers
75 views

Why can we divide by zero in limits?

Before I ask, I want to tell you that I am beginner in limits, so you may find some problems in my understanding. Let's assume a function $f(x) = 15-2x^2$. We want to know how the function behaves ...
0
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1answer
45 views

Question for preparation for the IMC internationals [closed]

Work out the addition of: (1/(1+2))+(1/(1+2+3))+...+(1/(1+2+3+...+51)) Guys, I'm having difficulties working this one out. Using a calculator, it would have been easy, however in the internationals ...
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0answers
43 views

Does $s^{-1}x = r^{-1}0$ imply $x = 0$?

Assuming $R$ a ring and $S$ an Ore subset of $R$ we construct the ring $Q = S\times R/\sim$ where the relation is $(s,x)\sim (r,y)$ if and only if there exist $a,b \in R$ such that $as = br \in S$ and ...
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3answers
40 views

Help with fractions

I have attached an image of which I have a question about. I don’t understand how you can get from equation 1 to equation 2. Could someone please explain this? $$nRc=\left(\frac{V_{cc}-V_{BE}}{V_o-...
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2answers
48 views

How to find $\frac{146}7 \mod{7}$?

I understand that if $\gcd{(b,c)}=1$ then we can find $\frac{a}b\mod{c}$ by writing $$x\equiv \frac{a}b\mod{c}$$ $$bx\equiv a\mod{c}$$ then reducing $a$ and solving the modular equation by finding the ...
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1answer
31 views

Possible values of $z$ in this inequality problem?

If $0<x<y<z$ integers And $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{4}$ So it asks the possible values of $z$. Options goes as: A)9 B)10 C)11 D)12 E)13
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3answers
48 views

Proving $\frac{a}{b} >\frac{a+\epsilon}{b+\epsilon}$ if and only if $b<a$, for $\epsilon >0$, $a,b$ positive.

The way I use to see that this is true is to take the derivative of the LHS w.r.t to $\epsilon$. This derivative is negative if $b<a$. I am not sure how I can use this to prove the if and only if ...
0
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0answers
57 views

Is there a general formula for $\int{\big(\frac {\arctan x}{x^2+1} \big )}^{\frac1k}dx$ , with $k$ is positive integer?

I'm interested to know if there is a general formual for $$ \int\left[\arctan\left(x\right) \over x^{2} + 1\right]^{1/k}\mathrm{d}x $$ with $k$ is positive integer may present integral of fraction ...
1
vote
1answer
76 views

Solving Fractional Diophantine Equations

As my search to create an efficient factorization algorithm continues, I stumbled upon this equation for one of my test cases:$$\dfrac{3-n^2}{2n-12}=k$$ To continue, I need to know what integer values ...
1
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0answers
56 views

Unit fractions pairing

(I have asked this question on stackoverflow and received a suggestion to try posting here so hey guys!) I have been given a problem where fractions between 1/2 - 1/1000 have to be added to create ...
-2
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4answers
816 views

How to communicate $\frac{5}{2}$ to other people orally? [closed]

The term one third is unambiguously understood as $\frac 13$. Likewise, the term two fifths is unambiguously understood as $\frac 25$ and three sixths is understood as $\frac 36$. But what exactly ...