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

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2
votes
2answers
62 views

Proper decimal fraction for $\frac{4n+1}{n(2n-1)}$

Assume I have a function $f(n) = \frac{4n+1}{n(2n-1)}$ with $n \in \mathbb{N} \setminus \left\{ 0 \right\}$. The objective is to find all $n$ for which $f(n)$ has a proper decimal fraction. I know ...
0
votes
1answer
49 views

Is there an abstract algebraic way to analyze the rational numbers between 0 and 1 (inclusive)? [closed]

I'm wondering if the rationals between $0$ and $1$, have been studied in a systematic manner using abstract algebra. Is there any interesting theory behind this set?
1
vote
3answers
253 views

If the sum of two irreducible fractions is an integer, then the denominators are equal

I have to show the following:"If the sum of two irreducible fractions with positive denominators is an integer, then the denominators are equal." $$\frac{a}{b}+\frac{c}{d}=k, \text{ where k an integer ...
8
votes
4answers
168 views

Primary/Elementary Pedagogy: What is the rationale for the absent '+' in mixed fractions?

Why are elementary students taught to represent one and a half as 1 1/2 rather than 1 + 1/2? This mode of expression seems standard throughout at least North America. I think it is bad pedagogy for a ...
1
vote
1answer
52 views

Simplify this expression $w\left(\cfrac Q {1+\frac w r}\right)^2 + r\left(\cfrac Q {1 + \frac r w}\right)^2$

I can't figure out how to do the algebra to simplify this expression! $$w\left(\cfrac Q {1+\frac w r}\right)^2 + r\left(\cfrac Q {1 + \frac r w}\right)^2$$ It's supposed to turn out to $\cfrac {Q^2} ...
-2
votes
3answers
127 views

Explanation of series for sin(x) and cos(x) [closed]

Can anyone explain me what is this equation telling us? I need to implement it in my computer program. I do not need a proof of these, but an explanation of notation used here. AND
1
vote
1answer
43 views

Help with load balancing math based on fractional capacity

I'm looking to create an algorithm that allows me to select a number(index) from a list based on it's fractional weight component. It's for load balancing, I'll give an example below of what I mean. ...
7
votes
2answers
220 views

Rationalizing the denominator 3

It is a very difficult question. How can we Rationalizing the denominator? $$\frac{2^{1/2}}{5+3*(4^{1/3})-7*(2^{1/3})}$$
3
votes
3answers
124 views

What is $\lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$?

These are two little questions that came to mind while I was looking at this problem. What is $\displaystyle \lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$? I am fairly certain that the ...
1
vote
3answers
162 views

Finding the sum of fractions with increasing denominator and decrease numerator for n iterations?

Considering something like this: $ \frac{10}{10} + \frac{9}{11} + \frac{8}{12} + ...$ Where denominator increases each iteration while the numerator decreases. Is there a simple way to find the ...
0
votes
2answers
28 views

Fractional Power Interpretation

I have a following query in my mind. It has been in my mind since i was a kid. I know that 2^3 means that multiply 2 three times,3^-2 means multiply (1/3) two times.What does 2^(0.22) means. multiply ...
1
vote
2answers
34 views

Simplified way of writing summation of $1$ to $n$ in fractions

I have the following expression that I'm trying to simplify: $$\frac{1}{3} + \cdots+\frac{1}{3^{n-1}} +\frac{1}{3^n}.$$ This looks like a summation of $1$ to $n$ but in different terms. Can someone ...
5
votes
8answers
360 views

If $\,\,x+\dfrac{1}{x}=5,\,\,$ find $\,\,x^5+\dfrac{1}{x^5}$.

If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$. Is there any other way find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ Thanks
1
vote
1answer
17 views

Calculating the time it takes to send a file

The time of sending a file is calculated: $$\textrm{time} = \frac{\textrm{file size}}{\textrm{link capacity}}.$$ In this example, $\textrm{file size} = 4492643566$ bytes, and $\textrm{link capacity} = ...
1
vote
3answers
52 views

Simplifying long fractions

How would I go about simplifying long fractions, such as the likes of this: $((8+\frac{3}{4}) + (3\frac{2}{3}))$ / $((4+\frac{2}{5}) - (1\frac{7}{8}))$ The correct answer is ($4 + \frac{278}{303}$) ...
2
votes
1answer
57 views

Cyclic rearrangements of periods of certain periodic numbers

A student of mine observed the following \begin{align} \frac{1}{7}=0.\overline{142857} &\qquad \frac{2}{7}=0.\overline{285714} &\qquad \frac{3}{7}=0.\overline{428571} \\ ...
2
votes
2answers
50 views

Determine a quartic equation

I am working on a puzzle from Popular Science from the 1980's. This was a puzzle that existed before pocket calculators or programmable computers. It results in one needing to solve a seeming quartic ...
2
votes
1answer
8 views

Convert hexadecimal fraction to hexadecimal

I have been trying to convert a hexadecimal fraction to a hexadecimal value. 3/4(hex) = 0.C(hex) What are the steps needed to convert?
1
vote
1answer
64 views

Starting velocity by distance, time, and friction

I am writing a game in Javascript, and I just got a big math problem, where $\text{friction} = 0.97$. This is what is being looped every $1000$ / $60$ milliseconds, to make the projectile move all ...
0
votes
2answers
67 views

solve ratio word problem without algebra

Four gallons of yellow paint plus two gallons of red paint make orange paint. I assume this makes six gallons. So the ratio is 4:2, or 2:1. Question: how many gallons of yellow paint, and how many ...
0
votes
3answers
50 views

How do I compute the individual terms of a polynomial to the power of -1?

If my polynomial $p$ is: $x+1$, obviously $p^{-1} = \frac{1}{x+1}$. Is it possible for me to split $\frac{1}{x+1}$ into a sum of two terms? In other words, is there an algorithm to write $p^{-1}$ as ...
0
votes
2answers
46 views

Simplifying algebraic fractions.

I cannot simplify the following expression. Please need help. $\large \frac{3}{x+1}-\frac{1}{x+3}+\frac{3}{1-x}-\frac{1}{3-x}$ Thanks in advance. Regards !
1
vote
0answers
51 views

Name of rational numbers of the form $p/q$ with $p,q$ prime

For the life of me, I cannot think of whether there is a name for fractions of the form $\frac{p}{q}$, where $p,q$ are both prime. Fractions such as $\frac{4}{5}$ are sometimes said to be in "reduced ...
1
vote
1answer
21 views

Magnitude relation of fractions for four real numbers with a condition

I would like to know if the following proposition holds or not. For all $a, b, c, d$ such that $a, b, c, d$ are positive real numbers, IF $a < b$ and $a - c > b - d \geq 0$ THEN $\frac{c}{a} ...
1
vote
2answers
59 views

Formula help with this equation

I don't know what the answer to this formula is, can someone please help me. I've tried lots of things but getting no where. If $x=\dfrac56+\dfrac{15}{18}-\dfrac{10}{12}$, then $(x-1)3=$ ?
0
votes
1answer
53 views

Linear Equation Problem

Evaluating the expression below: $\displaystyle \frac{2(6-x)}{3} = \frac{9(x+5)}{6} + \frac{1}{3}$ The answer is $-\frac{23}{13}$ but to obtain this answer what specific method do you use?
0
votes
0answers
88 views

If I have $\lfloor\frac{E}{K}\rfloor =\lfloor \frac{E}{K + m}\rfloor$, what is the upper limit of 'm' in terms of 'E' and 'k'

Given that E, K, m > 0, then is there a way to find out value of m in terms of E and ...
1
vote
2answers
83 views

Converting fractions / decimals to percentages

Can someone show me the methodology on how to convert 1.2794 to a percentage form? If I multiply it by 100, I get 127.94%, which is the correct answer. I'm not really sure 'why' I multiplied it by ...
0
votes
1answer
62 views

formula to apportion cost of transport among three people in a liftshare

I share lifts with Sed and Awk to work every day. We tally journeys owed on a spreadsheet. A week might look like this: ...
1
vote
2answers
36 views

Help with solving this fraction

Solve for $h$: $$125=\pi \left (\dfrac{5}{\sqrt[3]{\pi}} \right )^2h$$ This is from an optimization problem, in which the volume is the constraint of $125$. I've done everything in the ...
0
votes
4answers
58 views

Recognizing the proper polynomial factorization to solve an indeterminate limit

I had to solve the $\lim_{x \to 3} \frac{x^3-3x^2-x+3}{x^2-x-6}$ that is indeed an indeterminate form ($\frac{0}{0}$). The approach I adopted was to factor the polinomials so that I can deviate from ...
0
votes
1answer
100 views

period of recurring decimals

The period of a recurring decimal fraction $1/d$ is equal to the multiplicative order of $10$ mod $d$. For fractions with even period, the digits sum to 9 i.e. $1/7 = 0.(142857)...$ $$142\\ 857\\ ...
0
votes
2answers
58 views

Fraction Cross Multiply

Students first learning about fractions are often taught to "cross-multiply" when dealing with fraction with non-like denominators, however, in Mathematica, with the function ...
1
vote
3answers
43 views

mixed numbers subtraction vertically

In the following subtraction we are subtracting $2$ mixed numbers vertically. I know how it works except the last step. $$ 7 \frac{1}{3} - 4 \frac{1}{2} = 3 + \frac{-1}{6} = 2 + \frac{5}{6} = 2 ...
0
votes
1answer
259 views

How to solve this fraction within minute? (or trick)

I want to solve this question within minute but bcz of fraction it take more than a minute. does any one know a trick to solve this type question.plz share thanks
1
vote
1answer
325 views

How to calculate decimals of the fractional number 1/49?

I find this tricky one. How to calculate the first 50 digits/decimals of the fractional number 1/49? Two of my calculators and MatLab gives different answers so I'm curious, how this is calculated ...
8
votes
1answer
412 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
0
votes
2answers
74 views

A little explanation of square root in a fraction

I was on KhanAcademy when I ran into a problem involving a sine of a triangle, this was the solution: $$\frac{9}{3\sqrt{13}}$$ (9 being the length of the opposite and $3\sqrt{13}$ being the length ...
1
vote
4answers
137 views

change $0.684 210 526 3$ into a fraction

I am a non-mathematician who quit with math after middle school. Now I face a practical problem which I cannot solve. Suppose I want to turn $0.684 210 526 3$ into a fraction, how would I do that ...
0
votes
1answer
76 views

Can someone explain how this linear equation was solved [proof provided]

Can someone please explain to me the steps taken in the proof provided to solve the linear equation? [1]: http://i.imgur.com/2N52occ.jpg "Proof" What I don't understand is how he removed the ...
1
vote
1answer
71 views

A problem with rounding

Short version of the question: when rounded variable is used to calculate another variable, should I use rounded or unrounded value? Longer version: say I have two calculations: a / b = c and c + d = ...
4
votes
3answers
159 views

Proving that $\frac{3}{2} \sum_{k=1}^{\infty} \frac{4}{k^3+k^2} = \pi^2-6$

I'm trying to prove that: $$\frac{3}{2} \sum_{k=1}^{\infty} \frac{4}{k^3+k^2} = \pi^2-6$$ I've tried looking at the partial sums, but no luck there. I just have no idea where to begin. Knowing that ...
0
votes
1answer
75 views

Graphs of functions with fractional powers: $x^{p/q}$

How does changing the value of $\dfrac{p}{q}$ affect the drawing of the graph (domain/range/shape, etc.) How do you calculate asymptotes? Below is a question dealing with this type of function. ...
1
vote
1answer
40 views

help with simple order of operations in fractions?

first time poster here, so please be kind. :) I am working on an issue that I can't see past. I've got a worksheet with a multiple choice set of answers, but i KNOW they are all wrong! But maybe I ...
1
vote
2answers
55 views

manipulation of subtraction

I am trying to solve an induction problem and got stuck at this part. $$ 1 - \frac{n+2}{(n+2)!} + \frac{n+1}{(n+2)!} = 1 - \frac{(n+2) - (n+1)}{(n+2)!} $$ Shouldn't it be $$ 1 - \frac{n+2}{(n+2)!} ...
2
votes
2answers
96 views

The floor of a product of fractions

Evaluate: $ \displaystyle \Bigg \lfloor \prod_{n=0}^{248} \frac{33+8n}{29+8n} \Bigg \rfloor= \Bigg \lfloor \frac{33}{29} \times \frac{41}{37} \times \frac{49}{45} \times\ ...\ \times ...
2
votes
3answers
168 views

Rules for cancelling fractions with exponents

I have an expression that I need to simplify, I know the answer (wolframalpha) but I'm not sure of the rule that gets me there. $\dfrac{(\alpha) X_1^{\alpha -1} X_2^{1-\alpha}}{(1-\alpha)X_1^\alpha ...
1
vote
3answers
53 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
votes
4answers
320 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
122 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 ...