Fraction Definition and 653 Threads

As I understand, a ratio is a comparison between two or more quantities. Ratios involve two or more numbers. Whereas a fraction is a single real number. Why are ratios and fractions the same when ratios involve two or more different numbers whereas fractions represent only ONE real number like...

I would like to confirm my understanding between conjugates and complex conjugates. When rationalizing the denominator (when dealing with square roots) it is shown that we can multiply the fraction by its conjugate. I have seen the conjugate being defined as one of the signs of the binomial...

My first method to simplify the fraction is to to I flip ##\frac{5}{3}## up I get ##2 \times \frac{3}{5} = \frac{6}{5}## Method 2: if I flip 3 up I get ##\frac{2}{5} \times \frac{1}{3} = \frac{2}{15}##. Method 3: I could use it multiply ##\frac{3}{3}## since this is the same as mutlipying by...

How did they get y^7 in the bottom fraction? I got their answer except I had y^6. Would some please be able to help? Many thanks!

For this problem, I don’t understand why the bottom fraction cannot equal zero since we are not plugging in any numbers it just in terms of variables. Would anybody else be able to provide with an explanation? Many thanks!

When working with fractions and when we have a fraction or equation with fractions like this one for example ##\frac{x}{x-1}+\frac{x}{x+1}=\frac{9}{4}## do we always need to set boundaries? Like, do we always need to write that x can't be a number that would give the denominator 0? In this...

The first part of the question asked me to calculate the mean and standard deviation for the number of remain votes in the simple binomial model consisting of total sample size of 2091 people. I believe this is fairly straightforward, it was simply ##E(X) = \mu = 2091(0.5) = 1045.5## votes and...

I can do the question using brute force. First I multiply both the numerator and denominator by ##\sqrt{5} + \sqrt{3} - \sqrt{2}## then I simplify everything and rationalize again until no more square root in the denominator. I want to ask if there is a trick to reduce the monstrous calculation...

Say you want to find the following Integrals $$\int \frac{1}{(x-1)(x+2)} (dx)$$ $$\int \frac{1}{(x-1)(x^2 + 2)} (dx)$$ The easiest way to solve them will be by using partial fraction decomposition on both the given functions. Decomposing the first function, $$\frac{1}{(x-1)(x+2)} =...

Hello! Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...

Problem : Let me copy and paste the problem statement as it appears in the text, as shown above. Attempt : I can sense there is an "elegant" way of doing this, but I don't know how. I show below my attempt using ##\text{Autodesk Sketchbook}##. I hope am not violating anything. Ok so I have...

So my final equation is: ##\frac {1} {2700} + \frac {1} {3930n^2} + 10^{-5}## I need to boil this down, the learning materials has the following working, but I can't seem to get it $$\frac {1} {2700} + \frac {1} {3930n^2} + 10^{-5}$$ $$\frac {3930n^2+2700+2700*3930n^2*10^{-5}}...

Hi, I have given the following, which I would like to show that this estimation is correct, where ##|\theta| \leq \frac{\pi}{^2}## and ##M \geq 1##: $$\frac{1}{M^2}\frac{\sin^2(M\theta)}{\sin^2(\theta)} \geq \frac{4}{\pi^2}$$ I would approach an estimation of the denominator via ##\sin(x)...

Problem statement : Let me copy and paste the problem as it appears in the text : Attempt : I am afraid this looks like a very difficult problem, despite being at the elementary level (high school). My glance through the text shows that the authors have gone about reducing the first set of...

I am a little lost on the last step of this problem. I get that we want to know how much time elapses for the capacitor to reach 2/3 of its final charge. That is why 2/3Qf is equal to Qf(1-e^-t/RC). I don't understand how we make the jump to e^-t/Tau is equal to 1/3? and then somehow e^-t/Tau...

Can we make sense out of the formula of entropy like we do for density (like "quantity of mass per unit volume")? What's the sense of Q/T? Couldn't it be something else? Of course it probably is a 'me-problem', but I haven't studied Thermodynamics deeply yet and was wondering what Entropy...

Problem Statement : If ##\dfrac{x}{b+c-a}=\dfrac{y}{c+a-b}=\dfrac{z}{a+b-c},## prove that ##\boxed{\boldsymbol{\dfrac{x+y+z}{a+b+c}=\dfrac{x(y+z)+y(z+x)+z(x+y)}{2(ax+by+cz)}}}## Attempt : Let the fractions (ratios) ##\dfrac{x}{b+c-a}=\dfrac{y}{c+a-b}=\dfrac{z}{a+b-c} = \boldsymbol{k}##...

good day I want to study the convergence of this serie and want to check my approch I want to procede by asymptotic comparison artgln n ≈pi/2 n+n ln^2 n ≈n ln^2 n and we know that 1/(n ln^2 n ) converge so the initial serie converge many thanks in advance!

Greetings. I'm having a bit of difficulty with getting from the first to the second equation. I know some basic identities, but it all just feels like a mess. My approach was just going to be to write whatever I could, but some of the terms are confusing me...

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...

Hello, I am reading the Selfish gene by Richard Dawkins. I am confused on genes. In some part of the book he gives us clues on how to compute the percentage of genes we have in common with relatives, such as parents and brothers and sisters, in those particular cases it is stated that we share...

I understand how these expressions are derived, but I don't see how to calculate successive resultant values. Should the continued fraction expression simply be replaced by 1?

Hi all! I have a question ... I have to calculate the partial pressure of O2 Ccan I calculate the mole fraction from the data of 1 ppm of O2? The total pressure is 760 torr Thank you!

I am trying to find the derivative of this problem using the four step process but keep getting stuck when it comes to the third step of f(x+h) - f(x). I do not know what to do once I reach that step. Am I canceling terms out incorrectly? How should I deal with a fraction over a fraction? Any...

Hello,This is actually a piece of a little bigger problem (convergence of a series) - you can see the ratio test ak+1 / ak That's why the (n) and (n+1) terms I have lim n->∞ of (n√n) / (n+1)√(n+1) ∞/∞ I have tried L'Hopitals rule (requiring multiple times) and I am not seeing an end...

If f(x)=\frac{3x^2-5}{x+6} then f(0) + f'(0) is ... A. 2 B. 1 C. 0 D. -1 E. -2 What I did: If f(x)=\frac{u}{v} then: u =3x^2-5 → u' = 6x v = x + 6 → v' = 1 f'(x) =\frac{u'v-uv'}{v^2}=\frac{6x(x+6)-(3x^2-5)(1)}{(x+6)^2} f(0) + f'(0) = \frac{3(0^2)-5}{0+6} +...

Afternoon all, Hopefully somebody can help me, I'm doing my final year project and it's looking at the effect of heat treatment on in17, when I run an XRD scan I found that I all the phases sort of hid behind the matrix and so can't really make them out. So I've been looking at using the SEM...

I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function I know that to solve it we must first find...

For the first part, since this is a intrisinc semiconductor, n=p= intrisinc carrier concentration. Hence free electrons and hole = ##(1.5*10^{10})## per cubic centimeter. As for part 2, here are my steps. But I'm not sure if it's correct. I first find the number of atoms of one cubic...

Moved from technical math section, so missing the homework template Summary:: Find a general formula for the nth derivative Hi everyone! How would I approach and answer a Q such as this I began by rewriting the expression in a different form, then used chain rule to each given term I...

I am looking for the expectation of a fraction of Gauss hypergeometric functions. $$E_X\left[\frac{{}_2F_1\left(\begin{matrix}x+a+1\\x+a+1\end{matrix},a+1,c\right)}{{}_2F_1\left(\begin{matrix}x+a\\x+a\end{matrix},a,c\right)}\right]=?$$ Are there any identities that could be used to simplify or...

Ok this is considered a "hard" GRE geometry question... notice there are no dimensions How would you solve this in the fewest steps?

Do anyone know how to find ##1##, ##2x - 5##, and ##2\sqrt{x^2 - 5x + 6}## in the triangle? (please see attached image) Also, how do you find ##(x - 5/2)^2 - (1/2)^2##? [Moderator's note: Moved from a technical forum and thus no template.]

I have attached the full answer in PDF file. I'm not sure about the answers. will really appreciate if they get checked

i can get to 3i+1/1-3i but no further. I take it this is the correct way to start

Find integration of: \frac {1}{(x+1)(x^2 + x -1)^\frac{1}{2}} What I did: 1. Use completing square method for the term inside the square root 2. Use trigonometry substitution (I use secan) 3. After simplifying, use another trigonometry substitution (I use weierstrass substitution) 4. Use...

2. Betty would like your help planning her garden whose rows are all of equal length. She would like to grow carrots, beans, sunflowers, tomatoes and peppers. She tells you she would like to devote twice as many rows to growing peppers as she does to beans. She only has enough tomato plants to...

I just want to clear some confusion I am having with the Fermi-Dirac distribution & density of states (DOS) of a semiconductor, which are given by Say we have a piece of Silicon in equilibrium and its Fermi level lies 0.25 eV below the conduction band edge, i.e. Ec - EF = 0.25 eV. Let us say...

I tried to do like this: \frac{((76 - 15)^4 + 324)((76 - 3)^4 + 324)((76 + 9)^4 + 324)((76 + 21)^4 + 324)}{((76 - 21)^4 + 324)((76 - 9)^4 + 324)((76 + 3)^4 + 324)((76 + 15)^4 + 324)} Then stucked Thanks

## \frac{-3x^4}{7}+x=\frac{-3x^4}{7}+10 \\x=10## I solved the equation and got x=10, My question is how can I solve for x without brute-force method( Multiply all terms by (x-13)(x-7)(x-14)(x-6) and solving for x)

Welders use a logarithmic scale to identify protective eyewear. The shade number n, is given by the equation n = 1 − 7logT/3 , where T is the fraction of visible light that glass transmits. a. What shade number should a welder use that only transmits ⅛ of the light entering the glass? b Viewing...

Although I have over the years seen reports about simulations of the the Earth's moon's origin by a variety of possible mechanisms, none of these reports have ever reported the probability of such events. What is a reasonable estimate of the fraction of solar systems that have a planet of a size...

Well doesn't ##u(x) = 0.4 x## work? Seems too easy, but the phrasing at the end "for all ##x\in I##" makes me think since ##0.4x = x## only at ##x=0##, and not all of ##I##, that this is okay. But am I wrong?

Hi all, I cannot figure out how I get to the alternate form. I really do need help here. Not sure how this is done. Thank you

I want to compute the fraction of time both particles spend outside the finite potential well. All I can get is the probability to find them outside. The wavefunction outside the potential is: $$\frac{d^2\psi}{dr^2} = -L^2 \psi$$ Where: $$L = \sqrt{\frac{2mE}{\hbar^2}}$$ Solving the...

I have a word equation i cannot figure out! anyone have any ideas? There is a large bag of red and blue counters. One red and one blue counter are removed from the bag. This process is repeated until 1/4 of the red counter and 3/5 of the blue counters have been removed. What fraction of the...

Homework Statement Express the sum as a fraction of whole numbers in lowest terms: ##\frac{1}{1⋅2}+\frac{1}{2⋅3}+\frac{1}{3⋅4}+...+\frac{1}{n(n+1)}## Homework EquationsThe Attempt at a Solution Please see attached image for my work. The reason I am posting the image rather than typing this...

<Moderator's note: Moved from a technical forum and thus no template.> Suppose that all stars in this galaxy were born in a single major-merger burst event about 10 Gyr ago. From this original burst, I want to compute the fraction of stellar mass still surviving as stars in the main sequence ...

Homework Statement Sorry for so many posts lately, hopefully this is allowed. What tangent points on ##g(x)=\frac {12} {x+1}## has an instantaneous rate of change of -3? Homework EquationsThe Attempt at a Solution [/B] I know that once I derive ##g(x)=\frac {12} {x+1}## I can set the...