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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.

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2answers
26 views

Simplification of a complicated fraction

I am going over a physics text and I have difficulty to see how one can go from $$2A = (1+ \frac{\alpha}{ik})(1+\frac{ik}{\alpha})\frac{Fe^{ika}e^{-\alpha a}}{2} + (1- \frac{\alpha}{ik})(1-\frac{ik}{...
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1answer
21 views

How to break long expressions into multiple lines?

Say I want to write a long radical expression like this $\sqrt{a + b - c + d - e + f - g + h - i + j…}$ on paper or blackboard. It cannot fit in one line. How should I break it into two lines? ...
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2answers
79 views

Help me simplify $\left(\frac{a^2-ab}{a^2b+b^3}-\frac{2a^2}{b^3-ab^2+a^2b-a^3}\right)\cdot\left(1-\frac{b-1}{a}-\frac{b}{a^2}\right)$ [on hold]

Supposedly solution is $\frac{a+1}{ab}$, but both problem and solution could be erroneously defined because a book I retyped problem from has a few misprints here and there. There's no need to post ...
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0answers
29 views

What's with Fractional Subtraction as an action on a number

This a simple (sort of stupid) arithmetic based question that may require just the littlest bit of work. Consider $\cfrac {x}{a}$ I know that division is the number of times I'll have to remove a ...
1
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1answer
50 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)^{...
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2answers
72 views

Calculate the integral $\int \frac{2-3x}{2+3x} \sqrt{\frac{1+x}{1-x}}dx$ [duplicate]

I have to calculate the integral $\int \frac{2-3x}{2+3x} \sqrt{\frac{1+x}{1-x}}dx$. I tried the following substitutions: $x \rightarrow \frac{1+t}{1-t}, x \rightarrow \frac{1-t}{1+t}, x \rightarrow \...
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1answer
31 views

asinh from fraction

If I have $$ \mathrm{asinh}\left(\frac{x}{2.8\cdot10^{-10}}\right) = 15 $$ How can I calculate $x$? Should I use $\mathrm{asinh} \, x = \ln(x+\sqrt{x^2+1})$ Or something else?
2
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3answers
49 views

Stumped by a pretty basic fraction division

I'm self-studying through Stroud & Booths's amazing "Engineering Mathematics", 7th Edition, and am still on the "Arithmetic" section. Even though I've gone through the whole chapter and a lot of ...
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1answer
28 views

Rationalising the denominator problem

I need help rationalising this $$\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1) }$$ I'm so stuck with this problem!
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1answer
114 views

Estimation of fractional expression

We define $$\displaystyle f(x,y)=\frac{1}{x^{2y}-\frac{1}{4^y}}+\frac{1}{(1-x)^{2y}-\frac{1}{4^y}} \text{ for } (x,y) \in \left[0,\frac{1}{2}\right) \times \left(\frac{1}{2},1\right]$$. A study ...
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3answers
48 views

Understanding Square Root rules to understand an equation [closed]

So this is an equation from one of the solutions in my textbook that I am trying to understand as part of solving a cholesky-factorization problem: $$\sqrt{18-(\frac{a}{\sqrt2})^2} = \sqrt{\frac{36-a^...
1
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1answer
32 views

why componendo and dividendo is failing in this question

If a positive fraction numerator and denominator is increased by 2 the fraction increases by $\frac{1}{24}$ find the difference between the numerator and denominator, given the sum of Numerator+...
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1answer
16 views

Difference between rationals in a certain set is at least a certain amount

Define $A = \{\frac{p}{q} \in \mathbb{Q} \mid q \in \mathbb{N}, q < n, gcd(p,q) = 1\}$. I am trying to prove that the difference of any 2 distinct elements of this set is greater than $\frac{1}{n}$....
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1answer
45 views

Adding to the denominator of two fractions

Say I am working with one fraction larger than another and only in the domain of positive fractions.For example, let's say I have $\frac{20}{20}$ and $\frac{50}{80}$. Clearly $\frac{20}{20}$ is the ...
2
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3answers
30 views

Confusion Dividing A Fraction with a Whole Number…

In the lesson I am doing, I divide fractions. Here is my problem: 28/55 / 7 I had to look up how to do this problem. According to Math Is Fun, you divide the ...
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2answers
34 views

Method for finding a sequence's limit

i'm trying to find the following limit, if it exists, $$\lim_{n→ ∞} \frac{(n+7)^{n-5}}{n^n}$$. Now, I've tried division like $$\lim_{n→ ∞}\frac{|n+1|}{|n|}$$, or dividing by the highest power, but I ...
1
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2answers
26 views

Fractions swaped

I am following an electronics tutorial (slide 23 here) and at some point, he shows the formula: $\frac{Vs}{Ri} + \frac{Vo}{Rf} = 0$ Vs, ...
2
votes
2answers
101 views

is there a faster method to calculate $1/x$ ($x$ an integer) than this?

I gave this stackexchange a second go. Is there a faster way to calculate $1/x$ than the following: Calculate $100/x$ (.or other arbitrary positive power of $10$) with remainder Write multiplier in ...
3
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1answer
61 views

Compute: $\frac3{7\cdot2}+\frac3{7\cdot12}+\dots$

Compute: $$\frac3{7\cdot2}+\frac3{7\cdot12}+\frac3{17\cdot12}+\dots+\frac3{2017\cdot2012}$$ I couldn't really find the pattern in this one. I tried evaluating the first two terms which was $\frac14$, ...
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4answers
470 views

How can $\frac{4}{3} \times 3=4$ if $ \frac{4}{3}$ is $1.3$? [duplicate]

Ok use your closest calculator, and type $\frac{4}{3}$, which is $1.3333333333$,and then multiply it with $3$ which is $3.9999999999$ but then type $\frac{4}{3} \times 3=4$ how?. How can it be $4$ if $...
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2answers
49 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 ...
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1answer
35 views

Could infinity have a numerical value?

For example, $$\frac{1}{1}=1\quad \frac{1}{2}=0.5\quad \frac{1}{3}=0.\overline3\quad \frac{1}{10}=0.1$$ so the larger the denominator is, the smaller the number is. Would this mean that $\frac{1}{\...
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1answer
30 views

How to handle an identical zero pole when expanding to a Laurent Series?

I have a function given as $f(z)=\frac{2(z-3)}{z^2-8z+15},$ which is clearly the same as $f(z)=\frac{2(z-3)}{(z-3)(z-5)}$ when $z\neq3$. Typically I would break these into two separate fractions using ...
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1answer
27 views

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $ I feel like the products between different variables (i.e. not x^2, y^2, z^2) give this inequality the $ \gt $ sign and I don't ...
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votes
2answers
40 views

Top Base Fractional Number

For example: $ 2^3 = 2 \cdot 2 \cdot2 $ But what about a fractional power such as: $$ 2^{2/3} = ? $$ 1) How would I explain this? 2) How would I find value? 3) How would computers calculate ...
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1answer
50 views

Is $9.\overline{9} = 10$? [duplicate]

If you divide $10$ by $3$, you get $3.\overline{3}$ but $3.\overline{3}\times{3}=9.\overline{9}$ Does this make $9.\overline{9}=10$?
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11answers
3k views

How to prove that adding $n$ to the numerator and denominator will move the resultant fraction close to $1$?

Given a fraction: $$\frac{a}{b}$$ I now add a number $n$ to both numerator and denominator in the following fashion: $$\frac{a+n}{b+n}$$ The basic property is that the second fraction is suppose ...
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0answers
21 views

If $\frac{Y(s)}{U(s)} = \frac{NUM}{DEN}$ does $Y(s) = NUM$ and $U(s) = DEN$

In a course I am currently taking, my professor performed the following steps in a derivation for the reachable canonical form of a transfer function. Given $$H(s)=\frac{Y(s)}{U(s)}=\frac{b_1s^2+b_2s+...
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votes
3answers
128 views

Prove that $1+b+(1+c)+1/c+1+a \ge 3$ if $a, b,$ and $c$ are positive real numbers.

Let $a, b, c$ be positive real numbers. prove that $$ \frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)}\ge\frac{3}{1+abc}, $$ and that equality occurs if and only if $a = b = c = 1$. What I tried: $...
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1answer
29 views

Simplifying an infinite sum which might involve a power series

I have the following expression: $$ \frac{\frac{\bar{N}^N}{(N-n)!}(1-q)^{N-n}}{\bar{N}^n\sum_{k=0}^\infty \frac{\bar{N}^k}{k!}(1-q)^k}$$ where $n,N,\bar{N}$ are positive integers, $n\leq N$, and $0\...
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1answer
20 views

Simplifying compound fraction not producing answer provided by book

I am working on a problem in a textbook(Precalculus Mathematics for Calculus, by James Stewart) and the answer in the back of the book for the problem(1.4 #67) is ...
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2answers
25 views

Is there any way to simplify the following fractions? Thank you!

Is there any way to simplify the following fractions? $[n^{3}-1]/[n^{4}-1]$ $n!/n^{n}$
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1answer
27 views

how to convert continued fractions into normal fractions?

i couldnt find anything on google so i just tried opening it normally and recording each step. so i got: [d,c,b,a] = ((((a)*b+1)*c+A)*d+B)/C. [e;d,c,b,a]=(((((a)*b+1)*c+A)*d+B)*e+C)/D. etc.. (X ...
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3answers
12 views

Mixed Fractions and Multiplication (with Variables)

I stumbled over this expression: $3 \frac{1}{x^3}$. How should you interpret something like that? While you could see that as implicitit multiplication ($3 * \frac{1}{x^3}$), you could also argue ...
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1answer
15 views

reducing fractions with combinations and powers of large integers

For example, I'm trying to reduce $\frac{49 \choose6}{49^6}$. Is there anyway to reduce this fraction further? Thanks
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1answer
32 views

Correct way of displaying different base equality

Given any fractional number in any base, for example $0.5_{10} = 0.8_{16}$, pretty simple as this gives no recurring fractional numbers. However, given $0.255_{10} = \overline{\text{4147AE}_{_{16}}}$ ...
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1answer
25 views

Elementary schools summation in denominator?

after being lazy for a long time and being away from any fraction and equations, I am confused with a seriously ridiculous math problem, and I want to confirm my answer: the equation is pretty simple ...
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3answers
37 views

Is “three ten-millionths of an inch” the same as “thirty millionths of an inch”?

I was reading this article when I came across a fraction that was difficult for me to comprehend: three ten-millionths of an inch I thought to myself that wouldn't this be the equivalent of: ...
2
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1answer
23 views

Determine all $a$ and $b$ natural numbers such that $\frac {a^2+2b} {b^2-2a}$ and $\frac {b^2+2a} {a^2-2b}$ are whole numbers.

I proceeded in the following way: It is clear that $a \ne 0$ and $b \ne 0$. Let $\frac {a^2+2b} {b^2-2a} = k, k \in \mathbb{Z} \tag 1$ and $\frac {b^2+2a} {a^2-2b} = m, m \in \mathbb{Z} \tag 2$ ...
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2answers
27 views

Compare fractions a/b and b/a

I'm trying to implement a mathematical method for calculation of injustice. The formula depends on two variables $a>=1$ and $b>=1$ and returns a fraction $a/b$. $a/b=1$ indicates justice. If $a/...
2
votes
1answer
59 views

$(-1)^3$ has different results when evaluated as $(-1)\times(-1)\times(-1) = -1$ vs $((-1)^2)^{3/2} = 1$. Which is correct?

I know that $$(-1)^3=(-1)\times(-1)\times(-1)=-1 \tag{1}$$ but also $$(-1)^3=((-1)^2)^{3/2}=1^{3/2}=1 \tag{2}$$ So which gives the correct value of $(-1)^3$?
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0answers
38 views

If $x$ is algebraic over a quotient field $K$ of $A$, then there exists an integral element $cx$ for some $A \ni c \neq 0$.

Let $A$ be a commutative ring, $K$ its quotient field and $x$ algebraic over $K$. This means that there exists a polynomial $f(X)$ with coefficients in $K$ such that $f(x) = 0$. In other words, if ...
4
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2answers
187 views

$\frac{7x+1}2, \frac{7x+2}3, \frac{7x+3}4, \ldots ,\frac{7x+2016}{2017}$ are reduced fractions for integers $x\in(0,301)$. [closed]

BdMO 2017 junior catagory Question 7. $$\dfrac{7x+1}2, \dfrac{7x+2}3, \dfrac{7x+3}4, \ldots ,\dfrac{7x+2016}{2017}$$ Here $x$ is a positive integer and $x < 301$. For some values of $x$ it is ...
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3answers
29 views

Rounding a percentage to the nearest multiple of $\frac{1}{n}$

If I take a percentage like $60\%$ I can easily round it to a multiple of $\frac{1}{n}$ where $n=2$ like this... $$60\%\doteq50\%$$ $$50\%=\frac{1}{2}$$ ...or where $n=3$ like this. $$60\%\doteq 66\...
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0answers
25 views

Multiplying an inequality with an argument

I have a question about the following inequality: $a - \frac{2}{a} + 1 > 0$ This is obviously a polynomial of the second degree. Is this the simplest way of solving the given equation? $\frac{a}...
1
vote
1answer
67 views

why does 4r multiplied by square root of 6r becone 4r^2 times square root of 6? [closed]

Original question was to write the fraction in its simplest form: Question: 4r/[(√6r) + 9] I attempted to solve it by multiplying the denominator and numerator by the conjugates of the ...
0
votes
1answer
25 views

Having Problem changing fraction with Binomial Denominator into a Mixed Expression

Not sure if my workbook is wrong or my calculation. Probably me :) Here is the problem: $\frac {k^3-1}{k-1}$ My work: (inserted some zeros because of my lack of MathJax skills) $ \begin{array}{r} \...
0
votes
1answer
24 views

Need assistance changing a Mixed Expression into a Common Fraction

My workbook says the answer to this: $a^2 + ab - b^2 - \frac {a^3 - 2b^3}{a - 2b} $ is: $- \frac {a^2b + 3ab^2 - 4b^3}{a - 2b}$ I am continually getting this answer though: $\frac {-a^2b - 3ab^2}{...
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votes
3answers
57 views

Fraction proportionality-linearity

$$\frac {x_1} {y_1} =\frac {x_2} {y_2}=\dotsb =\frac {\alpha_1 x_1+\alpha_2x_2+\dotsb +\alpha_nx_n } {\alpha_1y_1 + \alpha_2y_2 + \dotsb + \alpha_ny_n} $$ Can someone explain why this is true. I ve ...
1
vote
4answers
75 views

How would I go about solving for $x$ in $\frac{(x-a)\sqrt{x-a}+(x-b)\sqrt{x-b}}{\sqrt{x-a}+\sqrt{x-b}}=a-b$?

The question This is a homework question. Given the following, I am to solve for $x$ in terms of $a$ and $b$: $$\frac{(x-a)\sqrt{x-a}+(x-b)\sqrt{x-b}}{\sqrt{x-a}+\sqrt{x-b}}=a-b;a>b.$$ My ...