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

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0
votes
2answers
60 views

Rationalising the Surds

Please help me rationalise and simplify: $$ \frac{1}{\sqrt[3]{2} - 1} \ - \ \frac{2}{\sqrt{3} - 2} \ . $$ I have tried using the cube of the denominator and the square of the denominator on the ...
0
votes
4answers
124 views

How to simpify this?

How to simplify following fraction? I have tried everything, but nothing seems to work... $$-a^3 (c^2 - b^2) + b^3 (c^2 - a^2) - c^3 (b^2 - a^2)\over (c-b)(c-a)(b-a)$$
8
votes
2answers
92 views

Integrating $\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$

I am a little bit lost with integral: $$\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$$ I have already worked on in and done substitution $x = \sin(t)$: This brings me to: ...
1
vote
2answers
114 views

Dividing amount unequally in tournament winners?

Hi I am trying to write the algorithm to calculate the winning amount of the winners. My problem is as below, 1) I have open ended tournament in which participants can be in ratio of 2 (i.e 2,4,8,16 ...
1
vote
6answers
253 views

Solution of $\dfrac{a}{b}=\dfrac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$

Let $\dfrac{a}{b}=\dfrac{a'}{b'}$ , $a,b,a',b' \in \mathbb{N}$ s.t. $a$ and $b$ have no common factors. How can we show that the only solution to this equality is $a'=na$ and $b'=nb$, $n$ is a natural ...
0
votes
1answer
59 views

Integration involving $\log_2(x)$

Having a hard time going about this problem: $$\int{\frac{\ln(2)\log_2(x)}{x}}$$ I believe $\ln(2)$ would be considered a constant, so than the equation would then changed to: ...
1
vote
0answers
61 views

Prove that there exists a subset with sum >=1 such that the remaining integer sum reduces by 1

let $ n \in \mathbb{N} $ and $ \frac{1}{w_1},\ldots, \frac{1}{w_n} $ for some (not necessarily distinct) $ w_1,\ldots,w_n \in \mathbb{N} $ and $ w_1,\ldots,w_n \ge 2 $ be given. Assume that $ ...
0
votes
0answers
33 views

How to find maximum and minimum of (x+y+z)/(ax+by+cz) where 0≤x≤y≤z≤1 for given positive real numbers a,b,c

How to find maximum and minimum of $$\frac{x+y+z}{ax+by+cz}$$ where 0≤x≤y≤z≤1 for given positive real numbers a,b,c? I guess those are one of $\frac{3}{a+b+c}$ or $\frac{2}{b+c}$ or $\frac{1}{c}$, ...
0
votes
1answer
73 views

How to solve a inequality with fractions and roots in denominator and numerator

The inequality is like that: $$ \sqrt{\frac{3x+1}{2}}>1 $$ I have no idea how should i begin with it.
0
votes
2answers
162 views

Counting four-digit numbers with repeating digits

Of all the four-digit positive integers containing only digits from the set $\{2,4,6,8\}$, what fraction of them have at least one of their digits repeating? Express your answer as a fraction. ...
8
votes
1answer
153 views

$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$

I have a pretty simple straightforward question. Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$ Instinctively, I do the quickest thing I know how to ...
0
votes
2answers
85 views

Solve algebraically: $\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$

$$\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$$ The professor says we can't use l'hopital's rule and must solve algebraically.
7
votes
2answers
112 views

Why does partial fraction decomposition always work?

Say you have a function $p(x)/q(x)$ for some polynomials $p(x)$ and $q(x)$ and $p$ has a lower degree than $q$. Say $q$ has degree three and $p$ has degree two. If you partially decompose it, you'll ...
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
50 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
267 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
169 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
134 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. $$ \sin x = ...
1
vote
1answer
45 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
225 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
166 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
29 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
65 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
73 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
91 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
64 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
102 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
44 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
277 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
340 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
418 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 ...