Questions on fractions, numbers of the form $p/q$ where $p$ and $q$ are integers, and $q$ is not zero.

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3answers
48 views

Irreducibility of gcd/lcm or lcm/gcd

Consider two irreducible fractions: $r_{1} = \frac{p_{1}}{q_{1}}$ $r_{2} = \frac{p_{2}}{q_{2}}$ Are these two fractions: $r_{3} = \frac{\text{gcd}\left(p_{1}, p_{2}\right)}{\text{lcm}\left(q_{1}, ...
14
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4answers
304 views

Show the identity $\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}=-\frac{a-b}{a+b}\cdot\frac{b-c}{b+c}\cdot\frac{c-a}{c+a}$

I was solving an exercise, so I realized that the one easiest way to do it is using a "weird", but nice identity below. I've tried to found out it on internet but I've founded nothingness, and I ...
1
vote
1answer
117 views

Evaluate the expression

Evaluate the expression $$\frac{1^2}{1^2-10+50}+\frac{2^2}{2^2-20+50}+\cdots+\frac{80^2}{80^2-80+50}$$ What should I do after that ? $\sum\frac{n^2}{n^2-10n+50}$ I'm not seeing anything to find ...
0
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0answers
68 views

difficult inequality to prove

I need help proving this inequality is correct for a homework assignment: $$\displaystyle \left(\frac{13}{4}\right)^{n} \leq ...
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5answers
83 views

largest fraction less than 1 [duplicate]

What is the mathematically rigorous way to answer the question: "what is the largest fraction less than 1"? (or to explain why it cannot be answered in the manner worded).
0
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2answers
22 views

Help Reducing simple fraction

This has got to be easy, but for some reason I just don't see it, probably because it's late. Solution key to a quiz we took online says that: $\frac{A^2 + A + AB}{(A + B)(A + B + 1)}$ can get ...
3
votes
4answers
170 views

Simplify this complex algebraic fraction

I'm stumped on this problem, I need to know how this answer was arrived at but my text book doesn't show this. $$\frac{\frac{1}{x+y}}{\frac{x}{y}}$$ The text book says the answer is this: ...
0
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2answers
55 views

How to simplify this fraction

It's embarrassing, but I need help solving this one... Need some refresher course for algebra. $$ \frac y{y+\sqrt y} $$
1
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3answers
89 views

Mixed number fractions vs regular fractions? $3\frac{1}{6}-1\frac{11}{12}$

I just passed Calculus 2 in college with an A and I'm rather embarrassed that I'm asking this question. My wife is taking an intermediate Algebra course in college and they gave her the below ...
1
vote
6answers
105 views

Why is $1 / x^y = x ^ {-y}$

I've known this rule for a long time, but I never got to understand why is that and how it works, could anyone explain to me how it is done? Any help is appreciated.
1
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3answers
127 views

How to work out this easy fraction?

I need help working out this fraction, I know it seems quite easy but I'm a bit stuck. The question is: $$\frac{5/2}{5/9}$$ My attempt was changing the denominators by multiplying by $2$ to make ...
0
votes
2answers
55 views

What is the formula of converting fraction into addition of fractions?

What is the formula of converting fraction into addition of fractions? $$\frac{4-s}{(s-3)(s-2)(s+1)}=\frac 1 4 \frac 1 {s-3} - \frac 2 3 \frac 1 {s-2} +\frac 5 {12} \frac 1 {s+1}$$
0
votes
4answers
61 views

Short expression $\frac{x^2-y^2}{x(x-y)}+\frac{x^2-y^2}{x(x+y)}$

I tried to short this expression: $$\frac{x^2-y^2}{x(x-y)}+\frac{x^2-y^2}{x(x+y)}$$ The result should be $2$ but I get: \begin{align} &\frac{x^2-y^2}{x^2-xy}+\frac{x^2-y^2}{x^2+xy}\\ =& ...
0
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4answers
103 views

Fraction Problem

The product of two fractions is 1/9. The larger fraction divided by the smaller fraction is 4. What is the sum of the two fractions? $\frac{a}{b}\frac{c}{d}=\frac{1}{9}$ I will assume that ...
0
votes
1answer
22 views

Scaling fractions

I have a set of values between $0$ and $1$ inclusive and I want to scale them so that they are all between $0.5$ and $1$ inclusive. What function can I use to do this? \begin{align} f(0) &\to ...
0
votes
4answers
69 views

Partial Fraction Decomposition.

The intial fraction is: $\dfrac{3x^3-x+4}{x^3+2x^2+6x}$ How should to make the decomposition for the denominator? Thanks!
0
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4answers
95 views

Why does $ \frac {\frac {1}{\sqrt{x}}}{x} = \frac {\sqrt{x}}{x^2} $?

A homework question recently asked for me to simplify: $\frac{1}{\sqrt{7}} \div {7}$ It's easy to see that this becomes $\frac{1}{7\sqrt{7}}$ But according to wolfram alpha this is also equal to ...
0
votes
1answer
54 views

simplify cube root of a fraction

I asking this question here to just check the work and that I have simplified it correctly. Given the formula $\Large \sqrt[3]{\frac{12m^4n^8}{5p^4}}$ We need to rationalize the denominator by ...
0
votes
1answer
122 views

Dividing an amount proportionately among unequal amounts

I have \$100000. I want to divide the amount proportionately to 300 employees. I want the money divided inversely from the top down. So, the top earner get \$0 and the lowest paid employee(s) get the ...
5
votes
3answers
750 views

What's the algebraic property where you can flip the fractions in an equation?

Earlier in algebra, we spent over 20 minutes trying to figure out $$ \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e} \,\,\,\, \text{solve for }R_2 $$ when the teacher said "What you start out with is ...
2
votes
2answers
70 views

For how many integers $a$ is $\frac{2^{10} \cdot 3 ^8 \cdot 5^6}{a^4}$ an integer?

In Mathleague $11316$ Target #$4$, the question is: For how many integers $a$ is $$\frac{2^{10} \cdot 3 ^8 \cdot 5^6}{a^4}$$ an integer?
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2answers
40 views

Finding modular of a fraction

Im really into cryptography and to find the private key of a message I need to use modular arithmetic. I understand how modular arithmetic using a clock with whole numbers. But I get really stuck when ...
1
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1answer
46 views

GRE - Percentage Question

A full glass of juice is a mixture of 20% grape juice and 80% apple juice. The contents of the glass are poured into a pitcher that is 200 percent larger than the glass. The remainder of the ...
0
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2answers
85 views

Algebraic Manipulation question - trying to get alternate form

I'm currently working on algebraic manipulation, changing algebraic fractions into a chosen alternate form but I've hit a brick wall. I'm trying to get: $$\frac{2(3^x - 2^x)}{3^{x+1} - 2^{x+1}}$$ ...
0
votes
3answers
45 views

Proof of rational division…

I know that this how it's done but i used to know a simple proof which i have forgotten...so please if you know let me know.. $$\frac{\frac{a}{b}}{\frac{c}{d}} =\frac ab \frac dc.$$ When you ...
1
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3answers
60 views

Rationalizing a denominator.

The question instructs to rationalize the denominator in the following fraction: My solution is as follows: The book's solution is which is exactly the numerator in my solution. Can someone ...
0
votes
0answers
21 views

I need to find $n$ that $\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$

$\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$ So what I did is this: $(n+1)\ln(n+1) > 1000 \Rightarrow n>190$ When I put it back I see that $\frac{1}{192 \cdot ln(192)} \not < 10^{-4}$. ...
0
votes
1answer
71 views

Algebraic Fractions, no I'm not kidding..

I know most of the people on this forum/site are very advanced, and I'm just sitting here wondering how in the world you do this equation, or "simplify" it. I know how to do equations like these when ...
1
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3answers
32 views

Equation with fractions

If $P=\frac{h}{1-h}$ then $h$ is equal to? Answer is: $\frac{P}{1+P}$ I understand that $\frac{P}{1+P}$ is the right answer for when I replace $\frac{P}{1+P}$ for h the answer solves the equation, but ...
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2answers
31 views

Fractions and invitations

Diane sent out 25 invitations to her birthday party. If 1/6 of the invitations were for her family, about how many invitations were for her family? Please show your work and explain.
1
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2answers
58 views

Simple algebra question - separating fractions

How does the following come about? I'm completely lost. Can anyone help me fill in the steps in between? $$ \frac{s+2}{s(s+1)} = \frac{2}{s} - \frac{1}{s+1} $$ I figured that $$ \frac{s+2}{s(s+1)} ...
0
votes
0answers
20 views

Simplified Form for a Series

Is there a simplified form for the following sum: $Z = \frac{r_1}{c}+\frac{r_2}{c+r_1}+\frac{r_3}{c+r_1+r_2}+...+\frac{r_n}{c+r_1+...+r_{n-1}}$ I need to express it if possible in a way that I can ...
1
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2answers
53 views

How to derive duration of unemployment?

The average monthly flow out of unemployment pool of $7.0$ million people each month is $3.1$ million. Put another way, the proportion of unemployed leaving unemployment equals $\frac{3.1}{7.0}$ or ...
0
votes
0answers
37 views

Transformation of fractions

I have a problem with a certain transformation of a fraction. This is part of a reudctio ad absurdum to show that there are infinit prim numbers. $\mathrm{P} = \prod_{i=1}^{n} p_i$ as the amount of ...
3
votes
1answer
38 views

Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition: ...
1
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3answers
42 views

Reducing a fraction, divisibility and indeterminate symbol

Quick question about validity, just to make sure. When I have a fraction in a form: $$\frac{3a + 3b}{a+b}$$ and I extract the common factor 3 out to get: $$\frac{3(a+b)}{a+b} \;=\; 3\frac{a+b}{a+b}$$ ...
1
vote
2answers
55 views

Calculating percentage to compensate for percent discount.

Missing something very basic here and cannot pin point it. We need to charge a client \$100 for a product. Let's say our payment processor charges us 10% on every transaction. We make this ...
0
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7answers
269 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
0
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2answers
55 views

Is the product of two numbers both less than one less than one

I'm bad at mathematics, and I wanted to know something. Say there are two numbers $a$ and $b$ where $a, b \in \Bbb R$ $-1 < a < 1$ and $-1 < b < 1$ Is it necessary that $a \times b < ...
1
vote
1answer
92 views

Simplifying square root with fraction

I'm not sure about this equality $$4(-3+\sqrt {15})/4)^2 = (9-6 \sqrt{15} +15)/4$$ Hope some one can enlighten me. I will be facing more of such fractions, please guide me on how to solve/simplify ...
1
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0answers
21 views

Rational exponentiation?

Consider the following operation: $\left(\frac{a}{b}\right)^\frac{n}{m}$ where $a, n\in\mathbb{Z}$ and $b, m\in\mathbb{N^*}$. My question is: when the result is a rational number, how (formula or ...
0
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1answer
29 views

If $\sum_{k=1}^{p-1}\frac{1}{k}=\frac ba$ where $\frac ba$ is an irreducible fraction, then $b$ can be divided by $p^2$?

Question : Is the following true for any prime number $p\ge 5$ ? If $\sum_{k=1}^{p-1}\frac{1}{k}=\frac ba$ where $\frac ba$ is an irreducible fraction, then $b$ can be divided by $p^2$. ...
5
votes
1answer
131 views

$\frac{\prod_{i=1}^n (1+x_i)-1}{\prod_{i=1}^n (1+x_i/\delta)-1} \stackrel{\text{?}}{\le} \frac{(1+x_n)^n-1}{(1+x_n/\delta)^n-1} $ .

Let $x_1 \le x_2 \le \cdots \le x_n$. Let $\delta>1$ be some positive real numbers. I assume that $0\le x_i <1$, for $i=1,\ldots,n$ and $x_n >0$. Does the following expression hold? $$ ...
0
votes
1answer
80 views

Fractional-Recursive Sequence

Here from the fraction set we have a really hard question to be answered...suppose that a sequence is defined as a(n) = a(n-1) - 1/a(n-1), where a(0) is given. ...you already know what I'm asking you ...
0
votes
1answer
40 views

How to work out these fractions?

$\dfrac{-x^4 + 4x^2 + 6}{x}$ $\dfrac{7x^8 - 5x^5 + 9x^3 + x^2}{x}$ I have no idea how to do this. I was first thinking of doing $-x$ or collecting up the $x's$ but I'm not sure as I haven't dealt ...
0
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1answer
52 views

Calculating a Fraction's Reciprocal

Is there any way or equation that allows me to calculate the reciprocal of any fraction? I mean if i have 5/6 and i need it's reciprocal by using a formula or an equation to calculate it. Is there or ...
0
votes
1answer
44 views

Embarrassingly-basic fraction question

I'm trying to do a calculation at work to figure out what the average # of pages a visitor is viewing. I am given: 47,000 visits 12% of visits do not bounce (that is, 12% navigate the site at least ...
1
vote
4answers
110 views

How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$?

How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$? What else? Well. Well can you use long division?
11
votes
2answers
295 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
0
votes
2answers
24 views

Total Time Expression Question

How do I do this question? Thanks!