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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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1answer
22 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 ...
0
votes
3answers
11 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 ...
0
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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
1
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1answer
21 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 $0.255_{10} = \overline{\text{4147AE}_{_{16}}}$ ...
0
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1answer
22 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 ...
0
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3answers
35 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
22 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 $\frac {b^2+2a} {a^2-2b} = m, m \in \mathbb{Z} \tag 2$ ...
0
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2answers
25 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/...
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2answers
46 views

Explain this fraction equivalence [closed]

Can you explain why $1-\frac{a}{b}$ is the same as $\frac{b-a}{b}$?
1
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1answer
54 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
36 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 ...
3
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2answers
136 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 ...
0
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3answers
27 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\...
1
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0answers
24 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
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1answer
52 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
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1answer
20 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
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1answer
23 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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3answers
55 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
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4answers
65 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 ...
1
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3answers
67 views

AMC 1834 - If $a, b $ and $c$ are nonzero numbers such that (a+b-c)/c=(a-b+c)/b=(-a+b+c)/a and x=((a+b)(b+c)(c+a))/abc and x<0, then x=?

I am getting stuck on this question: If a, b, and c, are nonzero numbers such that $\frac{a+b-c}{c}$=$\frac{a-b+c}{b}$=$\frac{-a+b+c}{a}$ and x=$\frac{(a+b)(b+c)(c+a)}{abc}$ and x<0, then x=? What ...
0
votes
1answer
39 views

A and B can do a job together in 80 days, B and C in 20 days.

A does the job for 5 days, then B comes over and does it for 15 days, then C does it for 18 days and the job is finished. How much time does C need to do the job alone? Attempt: Let their job ...
-1
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1answer
36 views

A fraction proof problem [closed]

Known: $$N=\frac{A+B}{A+C}$$ $$A>1$$ $$\frac{B}{C}>1$$ Does $N$ increase/decrease with respect to $A$ (In other words, will increase $A$ always decrease $N$?
0
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1answer
59 views

How to split up a fraction with a sum in the denominator?

How would you split up the fraction $$\frac{x}{a+b}$$ (or$$\frac{1}{a+b}$$) so one fraction has $x$ and $a$ in it, only and another one has $x$ and $b$, only?
0
votes
1answer
38 views

How to rationalise the denominator when it comes out undefined?

$$\frac{3-{\sqrt 5}}{\sqrt 5 + 5}$$ This is probably ridiculously straightforward but I need to get to the answer $$ 1-\frac{2}{5}{\sqrt 5}$$ and can't figure out how to rationalise the denominator ...
1
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1answer
67 views

Trail Mix Packaging problem part 1.1

I apologize in a advanced, I have tried looking at the other steps to what could be similar problems on stackexchange and could not understand the concepts. I am trying to teach myself math from this ...
0
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2answers
69 views

How to solve the inequality with logarithm?

The inequality is: $$\frac{1}{\log_{(x-1)}\frac{x}{20}}\ge-1$$ I made a plot of the function $f(x)=\frac{1}{\log_{(x-1)}\frac{x}{20}}+1$ and it looks the answer is $x\in(1,5]\cup(20,+\infty)$. Using ...
0
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1answer
17 views

A Central Limit Theorem simple example

A disscusion in the book: Let $(X_n)_{n=1}^\infty$ a sequence of i.i.d random variables such that $\mathbb{E}[X_n]=60, \operatorname{Var}[X_n]=25$. Let $S_N= \sum_{i=1}^NX_i$. By the central limit ...
0
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0answers
25 views

Field of fractions and rational functions

When I have a field $k$ and take the ring of polynomials in the variables $x1, x2, ..., xn$, and subsequently take the quotient field of these polynomials, I was asking myself, is this in 1:1 ...
0
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1answer
25 views

A random variable in a denominator

For some random variable $X$ & $Y$, I need to calculate the pdf of $Z=\frac{X}{Y}$. I've managed to calculate $$F_Z(t)=\mathbb{P}\left(\frac{X}{Y}\leq t\right)=\mathbb{P}(X\leq t\cdot Y)\,,$$ but ...
0
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1answer
20 views

Why $\lim_{x\to0^+}{\frac bx\left[\frac xa\right]}=0$?, where $[x] = \sup\{n \in N, n \leq x\}$

I ask this question here, and I was told that using squezze theorem I could solve it, but using this idea in this limit I end with. $\frac{b}{a} \leq \lim_{x\to0^+}{\frac bx\left[\frac xa\right]} \...
3
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2answers
113 views

Proving fraction is irreducible

Example: The fraction $\frac{4n+7}{3n+5}$ is irreducible for all $n \in \mathbb{N}$, because $3(4n+7) - 4(3n+5) = 1$ and if $d$ is divisor of $4n+7$ and $3n+5$, it divides $1$, so $d=1$. I want to ...
0
votes
3answers
31 views

Is the any math function that change the power of denominator of an input fraction?

I'm seeking for a function that gets a fraction as input and change its denominator power in the log function like follow: $f(\frac{a}{b}) = log(\frac{a}{b^k})$ Is it possible to find a function ...
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3answers
25 views

Absolute value less than some value [closed]

This is a noob question. If, $$\biggm| \frac{1}{2} - e \biggm | \le n$$ Then how do I get the following? $$e \le \frac{1}{2} + n$$
4
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3answers
63 views

Simpler ways to show that $n^2$ divides a polynomial?

I want to show that $n^2 \mid P(n)$, where $$P(n) = \frac{n^2(n+1)^2(n+2)(n+3)}{48}$$ for every odd positive integer $n$. The approach I took involved showing that $\cfrac{P(n)}{n^2}$ is always an ...
0
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0answers
18 views

decaying fractions and make them closer

My problem is actually algorithmic. I have two request queues, each holding its average wait time. For each queue I am holding x = total_wait_time, y = number_of_requests. ...
0
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1answer
22 views

Sum of fraction series

I have been trying to solve this question but I am not sure how to, and would appreciate some help. Thanks. Simplify: $$\frac{1}{4} +\frac{1}{4^2} +\frac{4^2}{4^3} +\frac{4^2}{4^4} +\frac{4^4}{4^...
2
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2answers
50 views

Conjecture: if $a$, $b$ and $c$ have no common factors, dividing each of them by their sum yields at least one irreducible fraction

Let $a$, $b$ and $c$ be $3$ integers with no common factors. I conjecture that at least one of the three fractions: $$\frac{a}{a+b+c},\quad\frac{b}{a+b+c},\quad\frac{c}{a+b+c}$$ is ...
0
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2answers
27 views

Solving Linear Equations, with 3 unknowns

$$6x-8y=24$$ $$\frac{-2}{3}x+ \frac{8}{9}y = m$$ I want to solve for all three of the variables. I did plot this into my calculator and the answer is $$\frac{-8}{3}$$ I want to know the process in ...
0
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2answers
13 views

How to divide and set an increment within a total?

I have total=90. I want to divide this number between 5 employees. For each employee from the second, I want to add an increment of 30% but the sum must remain 90. ...
0
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4answers
38 views

Having issues understanding fraction division when applying quadratic formula

I'm trying to apply the quadratic formula, and having trouble understanding how: $$\frac{-3 ± 3\sqrt{41} }{-18}$$ evaluates to $$\frac{-1 ±\sqrt{41}}{-6}$$ and not $$\frac{1}{6}±\frac{\sqrt{41}}{...
0
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1answer
45 views

How to calculate the sum of the multiples of two fractions in order?

$\frac{a}{b_1}, \frac{a}{b_2} = \frac{1}{3}, \frac{1}{5}$ $x_1 \in \{0, ..., b_1-1\} = \{0, 1, 2\}$ $x_2 \in \{0, ..., b_2-1\} = \{0, 1, 2, 3, 4\}$ $$\frac{a_1}{b}, \frac{a_2}{b} = \frac{a_1b_2}{...
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1answer
30 views

How can I simplify this step by step? [closed]

I want to simplify this : $$\frac{x^2}{a^2} + \frac{(x\tan \theta)^2}{b^2} = 1$$ The answer is supposed to be this : $$x = \frac{ab}{\sqrt{b^2+a^2\tan^2 \theta}}$$ I’d like to understand this step ...
6
votes
3answers
182 views

An Inconsistency in Numerical Approximation

Consider the expression $$ 10^5 - \frac{10^{10}}{1+10^5}. $$ Using the elementary properties of fractions we can evaluate the expression as $$ 10^5 - \frac{10^{10}}{1+10^5} = \frac{10^5 + 10^{10} ...
0
votes
1answer
29 views

Explain this algebraic manipulation

The proof of the product rule for derivatives in a book I have goes like this: $$\begin{align} (f \cdot g) (a) &= \lim_{h \to 0} \frac{ (f \cdot g)(a + h) - (f \cdot g)(a) }{ h } \\ &= \lim_{...
0
votes
1answer
52 views

Expanding $\frac{a}{a+b}$ into two terms

I'm unsure how I arrived at this assumption but I assumed : $$ \frac{a}{a+b} = \frac{a}{a} + \frac{a}{b} $$ testing with values $a = 3$ and $b = 4$ this is not true as $$ \frac{3}{3+4} \neq \frac{...
0
votes
1answer
22 views

Infinite non-periodic binary fraction

I have an infinite non-periodic binary fraction. For example: $frac_1=0.101111011100110010001001010010000001001...$ Is it always true that $1-frac_1$ = non-periodic binary fraction?
4
votes
5answers
115 views

General solution or approximate solution

Is there a known general or approximate explicit solution for $\xi$ in $$(1+\xi)^m (1-\xi)^n = C$$ where $m$ and $n$ positive fractions and $C$ being constant?
2
votes
0answers
174 views

Why is $\inf g \sup g = \frac{9}{16} $?

Consider this question here : Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $? Call that conjecture about $\frac{5}{4} $ conjecture $1$. Let $g(n) = \prod_{i=0}^n (\sin^2(n) + \frac{9}{16}) ) $ ...
1
vote
1answer
51 views

Can I cancel the factor $\ n-1$'s in $\frac{n-1}{n(n-1)(n-2)}$?

In the equation $$\frac{n-1}{n(n-1)(n-2)!} = \frac{1}{n(n-2)!} $$ can I cancel out the factors $(n-1)$'s in the numberator and denominator, so the equation is equal? I've learned that you can't ...
1
vote
3answers
30 views

Recurring fraction

Changing $x=1.0345454545...$ into a fraction $$x=\frac{10345}{10000}=\frac{2069}{2000}=1.0345$$ but it is missing the recurring 45. It should work, but why it is wrong?