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
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2answers
48 views

Simplifying Multiple Summations for worst case analysis

I'm figuring out a worst case analysis on a function. After converting it to a set of summations, and changing the sigma notations into summation formuale I ended up with: ...
0
votes
0answers
21 views

Calculating summary with variable multiplication factor

I have a formula of thermal conductance heat transfer rate. Here it is: $$ Q = \lambda{S (T_1 - T_2) \over L} \Delta t $$ For my calculations I have got some constant values available $$ Q = 0.58{1 ...
4
votes
1answer
82 views

Correctness of proof of division in ceiling function

My professor asked me to present a proof to my fellow students tomorrow that $$\left\lceil\frac{n}{2^k}\right\rceil = \left\lceil\frac{\left\lceil\frac{n}{2^{k-1}}\right\rceil}{2}\right\rceil$$ I ...
1
vote
1answer
136 views

$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$ is true for any $n,N\in\mathbb N$?

Is the following true for any $n,N\in\mathbb N$? $$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$$ Motivation : I've known the $N=3$ case. ...
0
votes
3answers
61 views

simplify fraction with radical in numerator, first steps?

$(\sqrt x−2)/(x−4)$ Looking to simplify a fraction with radical in the numerator.
3
votes
1answer
140 views

About two 'negative' continued fractions whose sum equals $1$

Letting $a_1,a_2,\cdots,a_r$ be integers which are larger than or equal to $2$, let us define $$[a_1,a_2,\cdots,a_r]=\frac{1}{a_1-\frac{1}{a_2-\frac{1}{\ddots-\frac{1}{a_r}}}}$$ (Note that the ...
1
vote
1answer
81 views

Any 'odd unit fraction' whose denominator is not $1$ can be represented as the sum of three different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is 1 'odd unit fraction'. Then, here is my question. Question : Is the following ...
0
votes
3answers
67 views

Fractions and roots

I have this problem: $$\frac{\sqrt{18}+\sqrt{98}+\sqrt{50}+4}{2\sqrt{2}}$$ I'm able to get to this part by myself: $$\frac{15\sqrt{2}+2}{2\sqrt{2}}$$ But that's when I get stuck. The book says ...
12
votes
3answers
193 views

Any 'odd fraction' can be represented as the finite sum of different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is $1$ 'odd unit fraction'. Then, here is my question. Question : Is the ...
0
votes
4answers
41 views

Reducing a fraction with exponents

How do I reduce $$ \frac{(k+1)^2}{k^2} $$ to simplest terms? My algebra is really rusty... Thanks!
2
votes
1answer
117 views

About the set of all the solutions $\mathbf x=(x_1,x_2,\cdots,x_m)$ to $\sum_{j=1}^m\frac{1}{x_j}=\frac1n$

Let $m,n$ be natural numbers, and let $S_{m,n}$ be the set of all the natural number solutions $\mathbf x=(x_1,x_2,\cdots,x_m)$ to the following equation : $$\sum_{j=1}^m\frac{1}{x_j}=\frac1n.$$ ...
3
votes
2answers
92 views

Calc 101 Question on simplifying a fraction

$$\lim_{h \to 0} \left(\frac 1h -\dfrac{1}{h^2+h} \right).$$ What do I do about the denominators?
4
votes
2answers
140 views

Fraction raised to integer power

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a... I don't know what you call it. Not a whole number, but something like 15/7 where you can't reduce it any more and it's non-integer. Can ...
0
votes
3answers
77 views

How to solve for x if it is on the top of the fraction?

so you have the equation: $$0.0850= \frac{x}{0.125} $$ How do you solve for x?
2
votes
1answer
76 views

Percentage of an amount?

I'm totally confused, we were doing a question in class and there are two answers but I'm not sure why one works and the other one doesn't. For example; there are 6000 pandas now and over 10 years ...
6
votes
1answer
128 views

How to best understand Euclid's definition of equal ratios? How does it relate to Dedekind cuts?

This is something I've been wondering about. When I think of "ratios" $x/y$ and $z/w$ as being "equal", with $x$, $y$, $z$, and $w$ being real numbers, this means the results of dividing the real ...
1
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1answer
55 views

What is a “constant fraction” of a total?

What is it mean to say that some quantity is a "constant fraction" of another quantity?
0
votes
0answers
12 views

If $d/dx_t ({\dot y}_t/y_t) > 0$ and $dy_t/dx_t < 0$ what can I then say about the sign of $d{\dot y}_t/dx_t$?

Assume that the rate of change in $y_t$ over time is ${{{{\dot y}_t}} \over {{y_t}}} = {x_t}$, where $x_t >0$. The derivative of this expression with respect to $x_t$ will be positive (well, it ...
1
vote
0answers
53 views

Does there exist an operation which partitions any fraction into the sum of the minimum number of unit fractions?

Motivation : I've been interested in finding an operation which partitions a fraction into unit fractions. The following is one of the operations which I've found. Let's start a rational number $q_0$ ...
0
votes
2answers
48 views

Simplifying two fractions on top of a third fraction

How would I go about simplifying this fraction: $$\frac{\frac{1}{x} - \frac{1}{a}}{x - a}$$ I've looked at similar questions such as this one but still can't seem to figure this one out. Any help ...
1
vote
4answers
126 views

Pre calculus fraction simplify question

Simplify: $$\frac{\frac{16x^4}{81} - y^4}{\frac{2x}{3} + y}$$ Wolfram alpha confirms the answer from the answer sheet: Wolframalpha answer
2
votes
1answer
28 views

Proportion Problem

I came across a problem about proportion as follow: There are 5 girls for every 12 boys, if the total number of children is 5200, how many are boys and girls? I tried following solution but the ...
15
votes
2answers
574 views

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$?

What is the max of $n$ such that $$\sum_{i=1}^n\frac{1}{a_i}=1$$ where $a_{i}\ (i=1,2,\cdots,n)$ are integers which satisfy $2\le a_1\lt a_2\lt\cdots\lt a_n\le 99$ ? Also, I need how to prove that ...
1
vote
1answer
150 views

Compute operations with fractions using calculator

I have a CASIO fx-350MS, and I need to make fraction computations, like $\frac{7}{2} \cdot \frac{4}{5}$ for an exam where I have to compute a lot matrix multiplications (programmable calculators ...
-1
votes
1answer
32 views

How to create a series of fractions, such that each subsequence slice is smaller than the previous?

I need to develop a grading system for a course. The first assignment is evaluated based on only one criteria, a. The second assignment based on the first criteria, a, plus one new criteria, b, the ...
2
votes
2answers
89 views

Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge?

Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that $\prod_i (1+\frac{1}{p_i})$ diverges ?
3
votes
4answers
177 views

Can't simplify this fraction: $ \frac{1+x^6}{1+x^2}$

I've been having trouble simplifying this fraction : $$ \frac{1+x^6}{1+x^2} $$ Can anyone explain step by step on how to solve this? Thank you.
5
votes
2answers
129 views

REVISTED$^2$: Fraction Existence Proof

Question 1: I'm asked to prove that there exists an $n\in\mathbb{N}$ such that $$\frac{1}{n+1}\leq\frac{a}{b}<\frac{1}{n},$$ where $0<\frac{a}{b}<1$. Here $\frac{a}{b}$ is a fraction in ...
2
votes
2answers
87 views

When is the class of a fraction, the set of multiples of the fraction?

Let $D$ be an integral domain and $F=\mathrm{Frac}(D)$ be the field of fractions of $D$. We will look at $F$ as a set of equivalence classes from the equivalence relation $\sim$ on $D\times ...
1
vote
1answer
93 views

change values by a percentage factor

I have a problem in that i have a camera with a zoom range values between 1,000 and 30,000. The problem is i am accessing these values through a slider bar with values starting 250 to 750. How can i ...
0
votes
3answers
79 views

Ratio problem to find the woman weekly salary

A woman spend $5/8$ of her weekly salary on rent, and $1/3$ of the remainder on food, leaving $40 available for other expenses. What is the woman's weekly salary ? I have tried , i am really confused ...
4
votes
1answer
152 views

Algebraic structure of a set of Egyptian fractions of a positive rational?

It is said that every positive rational number can be represented by infinitely many Egyptian fractions (defined as the sum of distinct unit fractions). I am struggling to understand in a formal way, ...
1
vote
2answers
125 views

Bar Notation Problem

everyone! I came across a problem in math that dealt with bar notation. Does anyone know how, for instance, 1.234(with a bar notation over the 34) is expressed as a fraction? I know already how ...
4
votes
2answers
84 views

Why does my intuition for “order of divergence” for algebraic fractions fail?

I come across this identity once in a while but I actually never grasped it: $$\frac{2}{1-x^2}=\frac{(1-x)+(1+x)}{(1+x)(1-x)}=\frac{1}{1+x}+\frac{1}{1-x}$$ I'm surprised by it because I would ...
10
votes
0answers
217 views

To how many decimals is $\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$ correct?

Consider: $$\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$$ This is, as far as I'm able to check with my software, correct to at least 167 decimals. If anyone has the ...
0
votes
1answer
79 views

Finding percentage which is less

The number that is 50% greater than $60$ is what percentage less than the number that is 20% less than $150$ ? My try : I considered a number is 50% of $130$ which is greater than the $60$ and 20% ...
1
vote
0answers
76 views

Turn a number $x$ into a fraction with a denominator with no more than $k$ digits

Is there a function for turning any number $x$ into a fraction with a denominator that has a maximum of $k$ digits? (I'm sure there is, since Excel has one built in, I just can't figure out what it ...
3
votes
2answers
103 views

Question involving approximation, taylor series and proving

Question: Consider the approximation $$\ln(2)\approx 2\left ( \frac{1}{3}+\frac{1}{3\times 3^{3}}+\frac{1}{5\times 3^{5}} \right )$$ Prove that the error in this approximation is less than ...
1
vote
4answers
79 views

Simplifying compound fraction: $\frac{3}{\sqrt{5}/5}$

I'm trying to simplify the following: $$\frac{3}{\ \frac{\sqrt{5}}{5} \ }.$$ I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I ...
-2
votes
2answers
83 views

Finding modulus of two numbers

Let $\frac xy$ be a fraction in reduced form, $y\geq x$, and $m=y \mod x$. How do you find $x$ mod $m$?
0
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2answers
753 views

Numbers that cannot be expressed as fractions

What are Numbers that cannot be expressed as Fractions called?
1
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2answers
139 views

How to solve a ratio question

Studying for the GRE. In the GRE guide, it says that If the ratio is $2x:5y$, and this equals the ratio $3:4$, what is the ratio of $x:y$? I tried cross multiplying but I don't get the answer. ...
2
votes
0answers
93 views

Trigonometric functions of rational fractions of pi

Consider rational numbers $\frac{m}{n}$ and $\frac{m'}{n'}$, where $0<\frac{m}{n}, \frac{m'}{n'} <1$. Then $$\sin^2 (\tfrac{m}{n} \pi) = 2 \sin^2 (\tfrac{m'}{n'} \pi)$$ When $\frac{m}{n} = ...
0
votes
1answer
332 views

Ceiling to Floor Function Conversion Proof

I am working on a proof to convert a ceiling of a fraction to a floor of a fraction. I found this: \begin{aligned} q=\left\lceil \frac{n}{m} \right\rceil \;&\Leftrightarrow\; \frac{n}{m} \leq q ...
3
votes
2answers
75 views

How do we know $p/q$ can be expressed as a terminating fraction in base $B$ only if prime factors of $q$ are prime factors of $B$?

On cs.stackexchange I asked a math question: How to demonstrate only 4 numbers between two integers are multiples of .01 and also writable as binary. Yuval Filmus answered with a explanation ...
0
votes
2answers
244 views

Ratio - Basic Question

If ratio of A:B = 1:2 if it is doubled , should it be not 2:4 i see many problems where they are simply multiplying numerator by 2 please can some one explain
2
votes
3answers
69 views

breaking up fractions

I have these two fractions ${11 \over 31 }+{-11 \over 61}$ Adding them gives $330 \over 1891$ But how do I go back to the two fractions, once I've added them? I can get the denominators just by ...
1
vote
3answers
171 views

Reverse percentages

My mothers recently started doing a distance learning course. And is struggling with her mathematical questions. I'm trying to explain to my mother how to answer the following question. Despite my ...
1
vote
2answers
89 views

Fractional overlap of 1/2 and 1/3

Given a subset of the natural number sequence (positive integers starting from 1) we could say that $\frac12$ of the numbers in the set are divisible by 2. e.g if the set were ${[1,2,3,4,5,6,7]}$ we ...
17
votes
11answers
3k views

Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...