Fractions Definition and 606 Threads

Is there a better way to get fractions to display like ## \rm{(a+b)/(c+d)}## instead of ## \frac{a+b}{c+d}##? Sometimes I'd like LaTex to do formatting "inside" the fraction; let's say with an integral symbol, for example.

For this problem (b), The solution is, However, I don't understand how they got their partial fractions here (Going from step 1 to 2). My attempt to convert into partial fractions is: ##\frac{2s + 1}{(s - 1)(s - 1)} = \frac{A(s - 1) + B(s - 1)}{(s - 1)(s - 1)}## Thus, ##2s + 1 = A(s - 1) +...

I know this problem can be done as follows. P(1 boy and two girls) = (C(2,1)*C(5,2))/C(7,3) = 20/35 My question can this be written as probility fractions? Meaning Lets say if that (2/5*3/7+1/2*1/7)/(3/7) But that doesn't give same result? what am I doing wrong?

I know of two reasonable ways to represent a complex fraction: \dfrac{ \left ( \dfrac{a}{b} \right ) }{ \left ( \dfrac{c}{d} \right ) } ##\dfrac{ \left ( \dfrac{a}{b} \right ) }{ \left ( \dfrac{c}{d} \right ) }## and \dfrac{ ^a / _b }{ ^c / _d } ##\dfrac{ ^a / _b }{ ^c / _d }## What I am...

There appears to be a simple isomorphism between continued fractions and nested radicals. Does anybody know more about this?

*Kindly note that i created this question (owned by me). My Approach, ##\dfrac {(x+y)(4x+6y)}{(5x-5y)}##=##-1## ##(x+y)(4x+6y)=-5x+5y## ##\dfrac {4x+6y}{-5x+5y}##=##\dfrac {1}{x+y}## to get the simultaneous equation, ##4x+6y=1## ##-5x+5y=x+y## ... ##4x+6y=1## ##-6x+4y=0## giving us...

So the part in italics "an operation performed on one quantity which when performed on unity produces the other." I do not understand. Can anyone help me understand what this means? I know how to multiply fractions, but this explanation is confounding to me.

Let $$y=\frac {1+3x^2}{(1+x)^2(1-x)}= \frac {A}{1-x}+\frac {B}{1+x}+\frac {C}{(1+x)^2}$$ $$⇒1+3x^2=A(1+x)^2+B(1-x^2)+C(1-x)$$ $$⇒A-B=3$$ $$2A-C=0$$ $$A+B+C=1$$ On solving the simultaneous equations, we get ##A=1##, ##B=-2## and ##C=2## therefore we shall have, $$y=\frac {1}{1-x}+\frac...

I don't understand the logic behind why derivatives can be treated like fractions in solving equations: ## \frac {du}{dx} = 2 ## simplified to ## du = 2dx ## I keep seeing this done with the explanation that "even though ## \frac {du}{dx} ## is not a fraction, we can treat it like one". Why...

now a bit confusing here, i always use Bodmas in that case, ##9\frac {1}{5}##-[##3\frac {3}{10}##+##2\frac {2}{3}]##...[1] is this correct and what if i re arrange to ##9\frac {1}{5}##+##2\frac {2}{3}##-##3\frac {3}{10}##...[2] input guys...cheers i have seen my mistake [1] is wrong...

see summary

A paper out today in Nature might interest some folks in this forum: https://www.nature.com/articles/s41586-021-03229-4 Permanent citation: Nature volume 590, pages67–73(2021) The authors used machine learning to generate a large number of continued fraction expressions for fundamental...

So the original question is from Control Theory, and the topic is the inverse z-transform. This is a part from the solution I just can't understand. The reason it has to be in this form (##z^{-1}##) is because that's the form used in the z-transform table. The question essentially is, how do you...

I am trying to get one step further with my search for \sum_{n=1}^{\infty}\frac{1}{n^{2s+1}} . Part of the way is to calculate some algebraic expressions containing fractions with really huge numbers (as in (\frac{1}{5^{9}}+\frac{1}{7^{9}}-\frac{1}{17^{9}}-\frac{1}{19^{9}})\div...

Question 1; a. sin θ=√3/2 θ=arcsin √3/2 θ=π/3 rad sin √3/2=60 degrees 60 degrees *π/180=π/3 rad. To find the other solutions in the range, sin θ=sin(π-θ) π-π/3=2π/3 The solutions are π/3 and 2π/3 in the range 0 ≤θ ≤2 π b. cos2θ=0.5 2θ=arccos 0.5 2θ=π/3 rad Divide both sides by 2; θ=π/6 rad...

I'm typing up answers to the exercises in my Insight article "A Path to Fractional Integral Representations of Some Special Functions", this problem is from section 1 (Gamma/Beta functions). I need a bit of help with this one: The problem statement: 1.9) Use partial fraction decomposition to...

I really can t find my mistake here 17/20 is the proper answer Thanks

I use 2x -4 as the LCD and turn 8/(x - 2) - (13/2) = 3 into 16 - 13x - 4 = 3, I then get 12 - 13x = 3 which leads me to 13x = -9 so x = -9/13 which is the wrong answer. Where did I make a mistake?

My method is converting 14 into ##\frac {42} {3}, = \frac 7 3x## Then $$ \frac {126} { 21}, = 6$$

$$c'\begin{pmatrix}u \\ v \end{pmatrix}=\begin{pmatrix}\frac{\partial f^1}{\partial u} & \frac{\partial f^1}{\partial v} \\\frac{\partial f^2}{\partial u} & \frac{\partial f^2}{\partial v}\end{pmatrix} $$ There is a problem with the first line of the matrix, but I am not too sure what it is.

Determine the value of \frac13+\frac16+\frac1{10}+\frac1{15}+\frac1{21}+...+\frac1{231} I know that it means \frac1{1+2}+\frac1{1+2+3}+\frac1{1+2+3+4}+\frac1{1+2+3+4+5}+\frac1{1+2+3+4+5+6}+...+\frac1{231}, but how do I answer? It's from a student worksheet for 7th graders, so they haven't...

Evaluate the integral. $$\displaystyle \int_2^4 \dfrac{x+2}{x^2+3x-4}\, dx$$ W|A returned these partial fractions but I don't know where the the A=2 B=3 and 5 came from $$ \dfrac{x+2}{(x+4)(x-1)} =\dfrac{2}{5(x+4)}+\dfrac{3}{5(x-1)}$$ the book...

They are mysterious to me, they do not make any intuitive sense. What is the context for them? I found this website. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html It tries to show the history of how they were discovered geometrically. So you start off with a...

I was doing this problem from Griffith's electrodynamics book and can't figure out how to do this integral. The author suggested partial fractions but the denominator has a fractional exponent which I have never seen for partial fractions, and so, I am unsure how to proceed. The integral I am...

Let $a_1,a_2, ... , a_n$ be positive numbers. Let $i_1,i_2, ... , i_n$ be a permutation of $1,2,...,n$. Determine the smallest possible value of the sum: $$\sum_{k=1}^{n}\frac{a_k}{a_{i_k}}$$

Homework Statement For a commutative ring ##R## with ##1\neq 0## and a nonzerodivisor ##r \in R##, let ##S## be the set ##S=\{r^n\mid n\in \mathbb{Z}, n\geq 0\}## and denote ##S^{-1}R=R\left[\frac{1}{r}\right]##. Prove that there is a ring isomorphism $$R\left[\frac{1}{r}\right]\cong...

Hello all, So I've been trying to write some basic equations out like 1/2 but would like it to appear in a horizontal fashion (like 1 over 2). I have been reading threads on how to use latex, I've tried to look at others equations, right click and have the latex code shown to me so i can...

Homework Statement (x+3)(x-2)/x2-2x Homework EquationsThe Attempt at a Solution (x+3)(x-2)/x(x-2) = (x+3)/x What I don't understand is why I can't simplify this further for instance the x's cancel to give 1: (1+3)/1 = 4/1 = 4 Is it because there is no x next to the 3? Many thanks :)[/B]

Homework Statement Prove that ##\forall n \in \mathbb{N}## $$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$ Homework Equations Peano axioms and field axioms for real numbers. The Attempt at a Solution Okay so my first assumption was that this part...

Dear all, I am trying to solve this inequality: \[\frac{2}{x^{2}-1}\leq \frac{1}{x+1}\] I've tried several things, from multiplying both sides by \[(x^{2}-1)^{2}\] finding the common denominator, but didn't get the correct answer, which is: \[2<x<3\] or \[x<-1\]How to you solve this one ?

How does computer technology add fractions with different denominators?

Meiling baked some cookies. 5/7 of the cookies were butter cookies and the rest were green tea cookies. She gave away 3/4 of the butter cookies and 1/4 of the green tea cookies. After she ate 9 green tea cookies, she had an equal number of butter cookies and green tea cookies left. How many...

Junhao and Bala both collect stamps. 1/3 of Junhao's stamps is equal to 3/5 of Bala's stamps. Junhao has 76 more stamps than Bala. How many stamps does each of them have? My answer: Number of stamps Junhao have =J Number of stamps Bala have = B We know that Junhao has 76 more stamps than Bala...

Just like the title says. Is this due to roundoff?

Homework Statement $$\frac{1}{z}+\frac{1}{2-z}=1$$ Homework Equations Quadratic-formula and algebra The Attempt at a Solution Been struggling with this one.. I keep getting the wrong answer, but that isn't the worst part, I can live with a wrong answer as long as the math behind it is...

Homework Statement For the solution to a given problem, in the second to last step I had: ##-\frac{\sqrt 6}{4} + \frac{\sqrt 2}{4}## I stated next that the solution was ##-\frac{\sqrt{6}+\sqrt{2}}{4}## I was told this was incorrect and that the correct solution is...

Homework Statement y(w)= 3/(iw-1)^2(-4+iw) Homework Equations N/A The Attempt at a Solution 3/(iw-1)^2(-4+iw) = A/iw-1 + B/(iw-1)^2 + C/-4+iw for B iw = 1 B=3/-4+1 = -1 for C iw = 4 C= 3/(4-1)^2 = 1/3 I know the answer for A should be -1/3 however I am unsure how to obtain this as if the...

im a bit confused about partial fractions If we have something like x/((x+1)(x+2)) we could decompose it into a/(x+1) +b/(x+2) If we had something like x/(x+1)^2 we could decompose it into a/(x+1) + b/(x+1)^2 We use a different procedure when there is a square in part of the polynomial in...

Josh says that 2/3 is always the same as 4/6. Meghan says that 2/3 and 4/6 are equivalent fractions, but they could be different amounts. Which student is correct? I say Josh.

Hello! I am new to Mathematica and I need some help with the code I attached. Can someone tell me how to pull the variables out of the fraction i.e. instead of ##\frac{7 p_1 p_2}{2 \cdot 5}## I would like ##\frac{7 }{2 \cdot 5} p_1 p_2## (I need this to make it look better for when I import it...

I´m not sure, whether this little challenge has been posted before. I have searched the forum and didn´t find it. It might still be a duplicate though ... Find the sum of fractions $$\frac{2}{3\cdot5}+\frac{2\cdot4}{3\cdot5\cdot7}+\frac{2\cdot4\cdot6}{3\cdot5\cdot7\cdot9}+...$$

Suppose you have the fraction 1/1. If you can make a fraction x/y, you can also make y/(2x). Also, if you can make x/y and a/b where GCD(x,y)=GCD(a,b)=1, you can make (x+a)/(y+b). Which fractions can you make?

how do I work out this problem?19/20-(1/2-3/10)

Homework Statement Please see attachment. Homework Equations I don't know how to get the final product on the ones with the question marks (textbook answers written next to them). I've gotten to the last step (except for # 29 but don't mind that one, I haven't exhausted all ideas). I've...

Hi PF! When minimizing some fraction ##f(x)/g(x)## can we use Lagrange multipliers and say we are trying to optimize ##f## subject to the constraint ##g=1##? Thanks

$\tiny{242 .10.09.8}\\$ $\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$ \begin{align*}\displaystyle I&=\int f \, dx = \int\frac{\sqrt{16+5x}}{x} \, dx \end{align*} \begin{align*}\displaystyle f&=\frac{\sqrt{16+5x}}{x}...

Hi everyone, I am stuck on a problem. I need to give a partial fraction of 1/N(k-N). I have tried every method so far ( plotting roots, systems of equations). I think I found A=1/k but I have no clue how to find B value. I would really appreciate any help as I am a desperate student trying to...

$\tiny{206.07.05.88}$ \begin{align*} \displaystyle I_{88}&=\int\frac{1}{(x+2)\sqrt{x^2+4x+3}} \, dx \\ &=? \end{align*} would partial fractions be best for this?

I wish i had one book or pdf to learn about different types of fractions . Please help

Trouble here in the below partial fraction (Bug) $\frac{5x^2+1}{(3x+2)(x^2+3)}$ One factor in the denominator is a quadratic expression Split this into two parts A&B $\frac{5x^2+1}{(3x+2)(x^2+3)}=\frac{A}{(3x+2)}+\frac{Bx+c}{(x^2+3)}$...