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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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59 views

First International Olympiad, 1959

The problem is: Prove that $\dfrac{21n+4}{14n+3}$ is irreducible for every natural number $n$. Can anyone please give me a hint?
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2answers
27 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
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0answers
32 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}) ) $ ...
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1answer
46 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
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3answers
23 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?
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3answers
28 views

Probability in Psychometric Exam

Forgive me for anything that I Write wrong since this is still new to me and I haven’t used stackexchange in a long time... Also I’m writing this using my ipad .... A psychometric exam is an Israeli ...
0
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1answer
19 views

For which the below fraction refer to?

I have got this fraction representation :$$a=\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\cdots}}}} $$ but i can't know for which it's refer to , I mixed that with Golden ratio however the ...
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1answer
18 views

Formula Rearrangement

Hi StackExchange community, The formula has this form: $$ {-7 \pm X \over \sqrt{2} - 3}-3.$$ How can I rewrite this to be more compact ? Thank you.
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4answers
80 views

What is the limit $\lim_{x\to 3}\frac{1}{(3-x)^2}$?

Why does the limit of $\frac{1}{(3-x)^2}$ become infinity as $x\to 3$? When I simplify the expression I get $$ \frac{1}{(9-6x+x^2)}, $$ which would give the limit $-\frac{1}{18}$ as $x$ approaches $...
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1answer
22 views

What is the generic name for “decimal” type fractions?

A number in the form: 1.234 is often loosely called decimal, though the name really refers to the fact that its base is 10, and has nothing to do with the ...
3
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1answer
34 views

Egyptian fractions: does the greedy algorithm never give more fractions than absolutely necessary?

Given $n > 1$, it's obvious that $$\frac{2^n - 1}{2^n} = \sum_{i = 1}^n \frac{1}{2^i}.$$ For example, $$\frac{15}{16} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}.$$ That's not the ...
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3answers
25 views

If there is a negative sign beside the fraction bar, does that mean the numerator and denominator are both negative?

In math, sometimes I see a negative symbol beside the fraction bar. Does that mean both the numerator and the denominator are negative, or just the numerator?
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2answers
21 views

How to rectify “time lost ” by pendulum clock by alteration in its length?

question: If a clock loses $5$ seconds per day ,what is the alteration required in the length of pendulum in order that the clock keeps correct time $(a)\dfrac{4}{86400} $times its original length ...
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1answer
19 views

Php problem of the period of a fraction

Question A decimal representation of $\frac a b$ with coprime $a$, $b$ has at most period $b - 1$. What does the period of this fraction mean?
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0answers
21 views

Number of muffins baked

Mr Ali baked some muffins. (3/5) of the muffins wee chocolate and the rest were vanilla. After she sold (2/9) of the chocolate muffins and (1/2) of the vanilla muffins, 230 muffins were left. How many ...
0
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0answers
50 views

Which mathematical property prevents reducing equations to 0 = 1?

Which mathematical property prevents taking an equation, this one for example: $r = -0.5\cdot a \cdot t^2$ and moving all terms to one side to make it equal to zero... $\Longrightarrow 0 = -r - 0.5\...
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2answers
22 views

Number of chocolate chip and butter cookies

(3/5) of the cookies were chocolate chip cookies and the rest were butter cookies. After (2/5) of the chocolate chip cookies were sold, there were 24 more butter cookies than chocolate chip cookies. ...
1
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1answer
12 views

Monthly allowance spent

Devi spent (2/3) of her monthly allowance on food and (5/6) of the remaining amount on stationery. She spent $35 more on food than on stationery. How much as Devi's monthly allowance? My work: ...
1
vote
1answer
40 views

For positive integers $x$, $y$, $z$ with $\frac1{x}+\frac1{y}+\frac1{z}=\frac89$, what is smallest value of $x+y+z$?

For some positive integers $x$, $y$, and $z$, $$\frac1{x}+\frac1{y}+\frac1{z}=\frac89$$ What is the smallest possible value of $x+y+z$? I'm guessing you would have to de-simplify the fraction ...
-1
votes
2answers
30 views

Prove this fraction will increase as x and y decrease

Is there a way to prove this fraction will increase as x and y decrease: $$0<x<1$$ $$y>1 $$ $$\frac{xy}{x^2(\frac{y!}{2!(y-2)!})}$$ $\frac{y!}{2!(y-2)!}$ is the combination formula, C(y,2)
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1answer
40 views

Interpreting the “bar” in an expression like $x=2.\left[1/2+\{4-(3\times\overline{2+3}\}\right]$

In the mathematics book of 6th grade, there is a chapter about fractions and grouping operators like (), {}, [] and fourth - the horizontal bar ________ which is placed over a math expression like the ...
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0answers
12 views

Bounds of fractional tetration

I know about Kneser, but if we take a simple recursion $$^{1/d}b=c, a(0)=b, a(n)=b^{\frac{1}{^{d-1}(a(n-1))}}$$ so $$\lim\limits_{n\to\infty}a(n)=c$$ and we can quickly find $c$ for positive ...
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1answer
28 views

Filling in fractions

I am trying to solve the problem in the below photo My try: Let $S_N$ be the set $\{N,2N,3N\}$. These numbers will be the deominators of the fractions. $S_{11},S_{12},\cdots, S_{15}$ would not work ...
0
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4answers
64 views

Does increasing both variables increase $\frac{y}{x-y}$?

I have this fraction with positive variables, x and y: $$\frac{y}{x-y}$$ $$x-y<1$$ Does increasing x and y always increase the whole fraction? I think it is true because any number divided by a ...
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2answers
44 views

Rewriting $\frac{1-x}{1-y}$ in terms of $\frac{x}{y}$

Is it possible to rewrite $\dfrac{1-x}{1-y}$ in terms of $\dfrac{x}{y}$? That is, the rewritten form includes $\dfrac{x}{y}$?
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0answers
20 views

Straight forward question about fraction — re-arrange fractions and percentage

Let's say I want to interview $123$ students (years $1$ to $6$) and I'll find $40\%$ of them in the first year of the medical school, and $10\%$ of them in the second year, $20\%$ in the third year ...
0
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2answers
599 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 ...
2
votes
1answer
23 views

How to tell when a fraction does not end? [duplicate]

Is there a way in math / programming to tell if a fraction (reciprocal in particular) does not end? For example, 1/3 is 0.33 repeated, but 1/2 is just 0.5 Is there a way to find if 1/n for any ...
0
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1answer
14 views

Fraction simplification in polynomial

$\frac{-\frac{y-g}{x-k}-g+y}{\left(x^2-k^2\right)-\frac{x^2-k^2}{x-k}}$ Simplyfing fraction I arrived at this form: $\frac{-\frac{y-g}{x-k}-g +y}{(x-k) (x+k)-(x+k)}$ I know it can be further ...
0
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1answer
13 views

How do I rewrite a fractional expression with a numerator that is a variable raised to an exponent…as an exponential?

For example: $(x^2)/2 = x^w$ where w stands in for the value I'm trying to find.
3
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0answers
39 views

Why can't we treat $\text{d}y/\text{d}x$ as a fraction? [duplicate]

Disclaimer - I'm not a mathematician, I'm a dirty physicist. My work often involves performing calculus on various things without thinking about what I'm doing too much (I leave the proof of various ...
1
vote
1answer
34 views

Tips on additional methods to solve inequality involving multiple variables

So I want some feedback about my proof of the following implication: Let $a,b,c,d\in\mathbb{Z}$ where $a,b,c,d>0$. Prove that if $\frac{a}{b}<\frac{c}{d}$, then $$\frac{a}{b}<\frac{a+c}{b+d}...
2
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2answers
77 views

Prove that if $x$ is irrational, then $\frac{x+1}{x-1}$ is irrational

I have to prove that if $x$ is irrational, then $$\frac{x+1}{x-1}$$ is irrational too, but I'm not sure where to start from. Could someone give me a clue?
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3answers
80 views

prove $\frac{x}{(ay + bz)} + \frac{y}{(az + bx)} + \frac{z}{(ax + by)} ≥ \frac{3}{(a+b)}$ [closed]

Let a, b, x, y, z be positive real numbers. How is it possible to prove the following inequality: $$\frac{x}{(ay + bz)} + \frac{y}{(az + bx)} + \frac{z}{(ax + by)} ≥ \frac{3}{(a+b)}$$
4
votes
2answers
97 views

Find $x$: $\sqrt[3]{1+\sqrt{x}}+\sqrt[3]{1-\sqrt{x}}=\sqrt[3]{5}$ [closed]

I've seen another equation that I have to solve for $x$. $$\sqrt[3]{1+\sqrt{x}}+\sqrt[3]{1-\sqrt{x}}=\sqrt[3]{5}$$ Hint me how I must simplify it and then solve it. I don't have any ideas! :(
2
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3answers
44 views

solve for $x$: $(x^4-13x^2+36)^4+|x^2+x-6|+\sqrt{x^3-7x+6}=0$

There is an equation that I think it is complicated ,a little! $$(x^4-13x^2+36)^4+|x^2+x-6|+\sqrt{x^3-7x+6}=0$$ Actually we must solve for $x$ here. I want you to hint me how can I simplify the ...
2
votes
4answers
84 views

Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$

Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $$\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$$ I have attempted multiple times in this question and the only method that I ...
2
votes
1answer
110 views

Common fraction for :$\frac{1}{x^2-2x+2}+\frac{2}{ x^2-2x+3}=\frac{6}{ x^2-2x+4}$

Hi guys please help me with this equation: $$\frac{1}{x^2-2x+2}+\frac{2}{ x^2-2x+3}=\frac{6}{ x^2-2x+4}$$ My problem is with finding common fraction for denominators ($x^2-2x+2$ and $x^2-2x+3$and $x^...
2
votes
2answers
47 views

Name and proof of the identity $c=\frac{a_1}{b_1}=\frac{a_2}{b_2}$ then $c=\frac{a_1+a_2}{b_1+b_2}$

I was shown in a textbook (though not a mathematics one) the following identity: If $$c=\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}=\dots=\frac{a_n}{b_n}$$ then $$c=\frac{a_1+a_2+a_3+\dots+a_n}{...
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votes
2answers
28 views

$2/3$ of a tank contains $112$ litres of water. What will it contain when it is $3/4$ full?

$2/3$ of a tank contains $112$ litres of water. What will it contain when it is $3/4$ full?
0
votes
0answers
14 views

On maximizing sum of fraction nominators after removing some fractions

I am looking for a formula to calculate something like the title (perhaps) suggests. I am having problem formulating my question rigorously so let me give the example. I have four fractions $\frac{a}{...
0
votes
1answer
14 views

Need understanding on the strategy used in a one gre question

The question is below Suppose x and y are 2 integers and that $0< x < y < 10 $. The tenths digit of the decimal representation of $x/16$ is $5$. What is the hundredths digit of $17/y$ ? I ...
5
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4answers
596 views

Prove $\ 1<\frac{1}{n+1}+\frac{1}{n+2}+ … + \frac{1}{3n+1}<2 $ using Cauchy-Schwarz

How do I prove this inequality $$\ 1<\frac{1}{n+1}+\frac{1}{n+2}+ ... + \frac{1}{3n+1}<2 $$ I've tried to prove that $$\ \frac{1}{n+1}+\frac{1}{n+2}+ ... + \frac{1}{3n+1} \ $$ is less than $$\...
2
votes
1answer
43 views

Simplifying $\frac{\cos(4a)-1}{\sin(a)-\sin(3a)}$

Got some questions from my niece who is studying for her exams. This one, I couldn't figure out: Simplify the following: $$\frac{\cos(4a)-1}{\sin(a)-\sin(3a)}$$ I'm stuck at the $4a$ and $3a$... ...
0
votes
1answer
13 views

Multiplying reciprocal fractions with exponents

The question is as follows: If we multiply a fraction with its reciprocal then we get 1. So by this logic this should be equal to 1^(a/a+b). But, the solutions I have been provided with states the ...
2
votes
4answers
84 views

Computing the value of $\frac{1}{3^2+1} + \frac{1}{4^2+2} + \frac{1}{5^2+3}…\infty$=?

I have tried converting this series into a telescopic sum whose terms could cancel out but haven't succeeded in that effort. How should I proceed further?
1
vote
1answer
27 views

How to find limit of this fraction using L'Hopital's rule

I need to find the limit of these two fractions as $P$ goes to infinity. $$v_0 = \frac{-\frac{\mu_k}{P} + \frac{\mu_k^2 c_k^2}{P^2} + s_A^2}{\sqrt{s_A^2 + \frac{\mu_k^2 c_k^2}{P^2}}}$$ $$v_1 = \frac{...
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votes
2answers
33 views

How do you find the bigO of a fraction?

I am given $$f(n) = \frac{\log (n^n)}{n^6 - 1}$$ I am told to find the least integer $k$ such that $f(n)=O(n^k)$. I am completely stuck. All I know to try is big-oh of the top over big-oh of the ...
0
votes
0answers
79 views

write as a single fraction in its simplest form.

I've been given this problem to solve and a little confused on how to go about it. Any help would be greatly appreciated. Here's the problem I need to solve: $$55+\frac{x^5}{(x-6)(x+1)}-\frac{7x}{x+...
0
votes
5answers
31 views

Which is right way to calculate percentage?

A student gets the following marks. 50 out of 100 120 out of 150 30 out of 50 In first method : I calculate the percentage as (sum of obtained marks) / (Total marks) * 100. Hence [(50 + 120 +...