Questions on fractions, which are expressions (not values) of the form $\frac pq$.

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3
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1answer
60 views

When is $(12x+5)/(12y+2)$ not in lowest terms?

I am struggling to solve this problem and would appreciate any help: When is $\frac{12x+5}{12y+2}$ NOT in lowest terms? (x,y are nonnegative integers) I have found that it is not in lowest terms for ...
1
vote
1answer
22 views

Function that maps a rational number to its numerator and denominator

Question: Is there a simple way to represent a function $f:\mathbb Q\to \mathbb Z^2$ that maps a rational number in lowest terms $r=\frac ab$ to the ordered pair of its numerator and denominator ...
1
vote
1answer
115 views

Understanding the concepts of division and fractions

$\require{cancel}$ I'm having some issues regarding division so I will start by asking how this concept was developed throughout the ages: What was the first civilization to introduce the idea of ...
1
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1answer
48 views

Can two integer divisions be unified above one whole fraction line?

Is there a way to combine two integer divisions (i.e. division with the result rounded down to the nearest integer) into a single division operation? What I mean is that, when working with with real ...
1
vote
1answer
46 views

The elementary question on sign of Rational numbers

The under picture show that $$+\dfrac{8}{3}=\dfrac{+8}{3}$$ Similarly we can show $-\dfrac{8}{3}=\dfrac{-8}{3}.$ Now How do can show that $$-\dfrac{8}{3}=\dfrac{8}{-3}?$$
1
vote
1answer
153 views

If I add a constant $c$ to each fraction's numerator and denominator in a sequence of fractions, how is the sequence affected?

Given a sorted ascending sequence of fractions, if I add a constant $c$ to each fraction's numerator and denominator, how is the sequence affected? For example, if I have a sequence in ascending ...
0
votes
1answer
32 views

Recursive formula for partial fraction decomposition of a specific kind of fractions

I need to make a partial fraction decomposition of the following fraction : $$ \frac{1}{(x-a)^2(x-b)^2(x-c)^2(x-d)^2(x-e)} $$ The problem is that Wolfram doesn't give any answer : ...
0
votes
1answer
17 views

Second Order Approximation for a Polynomial

if I have an expression: $L=\frac{12a^3d^3-4wa^3d^2+16a^2d^2-4wa^2d+6ad+1}{12a^3d^3-4wa^3d^2-4a^2wd+16a^2d^2+7ad-aw+1}$ what is the second order approximation in $\frac{d}{w}$? I know that ...
0
votes
1answer
25 views

Is there a value for $a$ other than a factor or a multiple of $c$ in $\frac{a}{b}=\frac{c}{d}$

Suppose $a,b,c,d$ to be whatever quantities whatsoever that satisfy the proportion $\frac{a}{b}=\frac{c}{d}$. Is there a value for $a$ other than a factor or a multiple of $c$. Or, is there a value ...
0
votes
1answer
44 views

Integral exponential and fraction of powers

I am trying to solve the following integral $$ \int_0^y \frac{x^{m-1}}{(1+x)^{m+k}} \exp\left(-\frac{m}{\gamma} x \right) dx. $$ I tried to look into different books such as Gradshteyn and Prudnikov ...
0
votes
1answer
29 views

How to calculate a whole amount with fractions?

A contractor first completes $7/16$ of a building. Then he completes $1/4$ of it. And finally completes $2/5$th of the remainder of the building. If there is $36$ days left to finish the construction ...
0
votes
1answer
28 views

Basic solving for fractions

Can someone help me with this? I am a beginner to physics and in a question i need no isolate a constant to solve for another one. A/B=C/D solve for D step by step please.
0
votes
1answer
31 views

Regarding +/- fractions: what are some mental tests you can apply to uncommon fraction denominators?

When adding and subtracting fractions: what if there is no uncommon factor (for example 4=2,2 and 6=2,3). Does that always mean to use the LCM? What if the LCM is too big or time consuming to ...
0
votes
1answer
38 views

Integration with partial fractions help please

I'm trying to work in my partial fractions chapter and some were easy but for whatever reason, I'm stuck now: ∫ (x-3) / (x2+2x+4)2 What I tried: since my denominator is of higher order and a ...
0
votes
1answer
20 views

Solving $\frac12 (3y+2)-\frac58=\frac34y$ for $y$ using LCD method

I am solving $$\frac12 (3y+2)-\frac58=\frac34y$$ for $y$ using LCD method. Can't figure out what I did wrong! The answer in the back of the book is $-1/2$. PS: In the first line that is a $1/2$ in ...
0
votes
1answer
27 views

Solving function in difference quotien equation

I have the problem Find the difference quotient $\frac{f(2 + h) - f(2)}{h}$ for $f(x) = \frac{1}{x^2}$. The answer they gave is $\frac{-(4 + h)}{4(2 + h)^2}$ So far I've done: $$\frac{[1/(2 + h)^2 ...
0
votes
1answer
147 views

Deradicalization of denominators

Task: Develop a fraction equivalent to $$ 1\over{\sum\limits_{i=0}^{n-1}c_in^{i/n}} $$ in which the denominator is rational.
0
votes
1answer
80 views

Rearranging algebraic formula when subject is on both sides

I have run into some difficulty with a question on making a variable the subject of an equation where the variable is on both sides. I am really struggling to find a method for making "a" the ...
0
votes
1answer
75 views

Why is the word “of” equivalent to multiplying in fraction word problems?

I know this is a very easy problem, but I'm having a hard time getting my head around this concept, consider this example from a book. *Jerry bought a pie and ate 1⁄5 of it. Then his wife Doreen ate ...
0
votes
1answer
24 views

Comparing Fractions that contain epsilon

Given $\epsilon$ a constant s.t. $0<\epsilon<1$, and $n,p$ positive integers, $n >= 2p$, is the following true: $\frac{(1+\epsilon)n}{(2+\epsilon)p} \geq \lceil\frac{n}{2p}\rceil$
9
votes
0answers
251 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 ...
3
votes
0answers
38 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
3
votes
0answers
65 views

All those unit fractions add to 1?

Consider $$S(n)=\{x \mid x=(a_1 ,a_2,a_3 \cdots a_n) \text{ where } \sum_{r=1}^{n}\frac{1}{a_r} =1 \}$$ Now let $|S(n)|$ denote the cardinaly (order) of set $S(n)$. Thus: $S(1)= \{(1)\} \implies ...
2
votes
0answers
97 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} = ...
2
votes
0answers
594 views

Four candle problem: Using candles as timers

The candles each take one hour to burn completely. Cutting off bits of the candles is forbidden, but the candles are placed on a raft of fork handles so they may be burnt at both ends (e.g. to time ...
1
vote
0answers
31 views

How many elements are in the following set?

The set is $$\{ x \in Q:x^2 =64/25 \} $$ I thought the answer was $\{ \frac{8}{5}, -\frac{8}{5} \}$ but I am told there are in fact 4 distinct elements: $$\{ \frac{8}{5}, \frac{8}{-5}, \frac{-8}{5}, ...
1
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0answers
17 views

Extracting a function of a variable from an expression

I have this expression: $\frac{d+2wd}{2w+3wd-3d-w^2-1}$ Is there anyway I can write it just as a function of f(d)? [To me this looks like it is already a function of d, but I want to confirm if ...
1
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0answers
40 views

exponential integration with fractional powers

I am trying to solve the following integral $$\int_{-\infty}^a \frac{\beta_1 \beta_2}{y^2(c-y)^2} e^{-\beta_1/(c-y)} e^{-\beta_2/y} \, dy$$ where $a<0$, $c>0$, $\beta_1>0$, $\beta_2>0$ I ...
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0answers
23 views

MultiEquations (with fractions)

Can you please help me solve these equations i don't understand how to solve them with fractions. 1=n-2/15 151/20 =2a+1 3/4 -3/5 -2 1/5k = - 26/25
1
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0answers
64 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 $ ...
1
vote
0answers
26 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 ...
1
vote
0answers
58 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 $q_0$ ...
1
vote
0answers
85 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 ...
1
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0answers
36 views

Computability of division of large numbers

What is the largest computable mathematical division in terms of the number of digits that can be handled by a typical desktop computer using the best available big number libraries, assuming input is ...
1
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0answers
764 views

Contribution (weighted average) of change in rate over time

I'm trying to determine the weighted average impact of one customer's change in rate on the total change in effective rate. Let's say I have two customers and two time periods: ...
0
votes
0answers
24 views

Equivalent forms of expressions with complex numbers

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$ Select all that apply. $ {18i\over{(81−z)^8}}$ $ {−18i\over{(81+z)^8}}$ $ {18i\over{(81+z)^8}}$ $ ...
0
votes
0answers
21 views

How to calculate the Integer portion of a fraction using only +, -, $\div$ and *?

I made something in excel that calculates the days left until a given date, and from that how many weeks were left. I had it so that 9 days displayed as 1.2 using this formula: ...
0
votes
0answers
16 views

Separate terms of different orders from fractional polynomial

I have an expression: $\frac{1}{1-A}+\frac{-12A^4D^3 + 4A^4D^2 -16A^3D^2+4A^3D -6A^2D - AD}{- 12A^4D^3 + 4A^4D^2+12A^3D^3 -20A^3D^2 +4A^3D +16A^2D^2 -11A^2D +A^2 +7AD -2A + 1}$ How do I write it as ...
0
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0answers
18 views

GCD and fraction problem

If x/y = 1/a + 1/b + 1/c and GCD of a , b and c is 9 then find a) minimum of x and y which do not cause x/y repeating decimal b) the best of x and y that cause x/y nearly to 3/10 many ...
0
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0answers
12 views

What can we say about the function $f(x)$ in this case?

Alright, I'm little bit confused about what's happening here to the function $f(x)$, i thought that the formula of $f(x)$, have nothing to do with its behavior or domain. there are two or many ...
0
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0answers
15 views

proving a fraction with 2 parameters to be small

Hi I have a fraction as below $$\frac{1.623x^4+0.434x^4\sum_iy_iz_i^2+(0.014x^2+0.0027)\sum_iy_iz_i^4}{1.645x^2+(0.083-0.329x^2+0.435x^4)\sum_iy_iz_i^2+0.014\sum_iy_iz_i^4}$$ where $x\in[0, 0.5]$, ...
0
votes
0answers
56 views

L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
0
votes
0answers
26 views

Identity for fractional summation

I would like to know if there's an identity to represent the following summation $\sum_{i=0}^{n}\frac{x_i}{y_i}$ Where x and y are non integer values. The result of this is being calculated using ...
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0answers
46 views

How to find maximum and minimum of $\frac{x+y+z}{ax+by+cz}$ where $0\leq x \leq y \leq z \leq 1$ for given positive real numbers $a,b,c$

How do I find the maximum and minimum of $$\frac{x+y+z}{ax+by+cz}$$ where $0\leq x \leq y \leq z\leq 1$ for given positive real numbers $a,b,c$? I guess those are one of $\frac{3}{a+b+c}$ or ...
0
votes
0answers
118 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 ...
0
votes
0answers
46 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 ...
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0answers
20 views

Question on passing from rational to exponet

it's not a question itself, but I'd like to check if am I doing this passage from rational numbers to the exponent form right: From: $\sqrt{a\sqrt{a}}$ Evaluete to: $\sqrt{a*a^{\frac{1}{2}}}$ = ...
0
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0answers
105 views

Point me the primordial and intuitive concepts about this operations on physics

Warning: Layman question. Treat me as a 10 years old child The question was based on this page. I could write this on the physics channel, but despite the context, my problem is intrinsically ...
-1
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0answers
24 views

Freshman sum related question.

I want to prove the following: If $\frac{u_1}{d_1} > \frac{u_2}{d_2} > \frac{u_3}{d_3} > \frac{u_4}{d_4}$ and $d_1 + d_3 > d_2 + d_4$ and $u_1 > u_2$ then $u_1 + u_3 > u_2 + ...
-3
votes
0answers
33 views

show some sum is exactly 1

Consider a given set of rationals of the form $\frac{1}{j*2^a}$, where $j$ is a fixed odd positive integer and $a$ is a positive integer that may vary per rational. A number doesn't have to be unique ...