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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.

-3
votes
0answers
21 views

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

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?
2
votes
2answers
66 views

can we perform modulo operator on a fraction on both of it's numerator and denominator?

I want to calculate nCr (mod $10^9+1)$.so for calculating nCr we have: $$nCr=\frac{n!}{r!(n-r)!}$$ so I want to know whether it is true that I perform modulo operator to numerator and denominator ...
16
votes
2answers
573 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} \...
0
votes
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}}} + \...
1
vote
3answers
49 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 ...
1
vote
1answer
67 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}...
3
votes
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+...
0
votes
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 ...
0
votes
0answers
146 views

Best rational approximation with numerator/denominator less than 255

An old problem: I have rational numbers which I want to approximate with the best fraction where both the numerator and denominator are written on eight bits, so between $0$ and $255$. Is there an ...
0
votes
2answers
3k views

Which expression is equivalent to $\left(\frac{2}{3}-\frac{2}{x}\right) \div \frac{x-3}{x}$? (from Khan academy)

I've been trying to understand this problem for hours but not getting it. HELP!!! The correct answer is $\frac{2}{3}$, but I don't know why this is the correct answer. Thank you in advance for your ...
0
votes
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 ...
0
votes
3answers
41 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 ...
1
vote
1answer
65 views

Calculate $\int_{0}^{\frac{\pi}{2}}\frac{cos(x)^{sin(x)}}{(cosx)^{sin(x)}+(sinx)^{cos(x)}}dx$

Calculate $\int_{0}^{\frac{\pi}{2}}\frac{cos(x)^{sin(x)}}{(cosx)^{\sin(x)}+(sinx)^{cos(x)}}dx$. EDIT: By changing the variable, $x\rightarrow \frac{\pi}{2}-x$, $\int_{0}^{\frac{\pi}{2}}\frac{cos(x)^{...
2
votes
1answer
136 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 $a_i$'...
0
votes
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 - \...
0
votes
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 ...
0
votes
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 ...
4
votes
3answers
513 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 ...
0
votes
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. ...
-1
votes
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: ...
-2
votes
3answers
58 views

Fraction proportionality-linearity

If $$\frac {x_1} {y_1} =\frac {x_2} {y_2}=\dotsb=\frac {x_n} {y_n}=k$$ Then $$\frac {\alpha_1 x_1+\alpha_2x_2+\dotsb +\alpha_nx_n } {\alpha_1y_1 + \alpha_2y_2 + \dotsb + \alpha_ny_n}=k $$ Can ...
0
votes
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 ...
0
votes
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.
0
votes
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 $\...
10
votes
4answers
14k views

Finding modular of a fraction

Im really into cryptography and to find the private key of a message I need to use modular arithmetic. I understand how modular arithmetic using a clock with whole numbers. But I get really stuck when ...
2
votes
3answers
91 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} ...
1
vote
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 ...
-1
votes
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=...
0
votes
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 ...
0
votes
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}...
0
votes
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{...
8
votes
2answers
233 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 ...
0
votes
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 $= x^2 -6x -7$ ...
0
votes
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 \...
2
votes
4answers
132 views

Find $\lim_{x\to 0} \left\lfloor \frac{\tan 2x}{\sin x} \right\rfloor $

Find the limit $$\lim_{x\to 0} \left\lfloor \dfrac{\tan 2x}{\sin x} \right\rfloor $$ My try: $$ \tan 2x =\dfrac{\sin 2x}{\cos 2x}$$ $$\sin 2x =2\cos x\sin x$$ So: $$\dfrac{\tan 2x}{\sin x}=\...
0
votes
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]^{...
35
votes
12answers
3k views

How to make sense of fractions?

Can anybody explain what a fraction is in a way that makes sense. I will tell you what I find so confusing: A fraction is just a number, but this number is written as a division problem between two ...
0
votes
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.
0
votes
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 ...
2
votes
2answers
63 views

Prove the inequalitiy

If $a, b, c$ are positive reals such that: $\frac a {b + c + 1} + \frac b {a + c + 1} + \frac c {b + a + 1} \le 1$ then: $\frac 1 {b + c + 1} + \frac 1 {a + c + 1} + \frac 1 {b + a + 1} \ge 1$...
1
vote
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, ...
1
vote
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 ...
0
votes
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 $x$ ...
0
votes
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}...
1
vote
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^...
2
votes
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 ...
0
votes
2answers
74 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
votes
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 ...
1
vote
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-...
0
votes
2answers
46 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 ...