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

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5
votes
1answer
42 views

Multiplication of algebraic fraction not giving desired result

I am having a try at solving this: that supposed to return: but I get stuck at: which can be written as
1
vote
5answers
166 views

Proving that $\frac{a}{b} = \frac{c}{d}$ if and only if $ad = bc$

I was working on a problem which asked: Prove that $ \frac{a}{b} = \frac{c}{d} $ if and only if $ad=bc$, provided $c,d \neq 0$. Is it sufficient to manipulate $ \frac{a}{b} = \frac{c}{d} $ via ...
0
votes
2answers
38 views

How do we compare fraction without changing to a similar denominator?

This is Singapore Mathematical Olympiad 2015 Grade 8/Secondary 2 Junior Round 1 Question 1. 1.Among the five numbers, $\frac{5}{9},\frac{4}{7},\frac{3}{5},\frac{6}{11}$ and $\frac{13}{21}$, which ...
3
votes
1answer
54 views

$\frac{a(a+b)}{4a^2+ab+b^2} + \frac{b(b+c)}{4b^2+bc+c^2} + \frac{c(c+a)}{4c^2+ca+a^2} \leq 1$

I've got stuck at this problem: Let $a$, $b$, $c$ be real numbers. Prove that $$\frac{a(a+b)}{4a^2+ab+b^2} + \frac{b(b+c)}{4b^2+bc+c^2} + \frac{c(c+a)}{4c^2+ca+a^2} \leq 1$$ Firstly, I've ...
1
vote
1answer
33 views

How to derive this simple equality?

Let us define $L_i\triangleq \log \left( \dfrac{Prob(x_i=+1) }{ Prob(x_i=-1)} \right)$ $E\{x_i\} \triangleq Prob(x_i=+1)-Prob(x_i=-1)$ I need to show that \begin{equation} E\{x_i\} = \tanh(L_i/...
-5
votes
1answer
65 views

Can't calculate math problem [closed]

Can anyone help me with this math problem? I would require step how you came to the result.
3
votes
6answers
210 views

What is the value of $\frac{\sin x}x$ at $x=0$?

On plotting graph for $\frac{\sin x}{x}$ using Wolfram|Alpha and Google, got that : also, I can get the value of $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ using squeeze theorem and as illustrated ...
10
votes
1answer
82 views

Prove that $abc$ is a cube of some integer.

Given three integers $a$, $b$ and $c$ such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer too, prove that the product $abc$ is a cube. By the way: Merry Christmas! ;)
1
vote
4answers
81 views

What is the simplest way to find $\frac{n}{7}th$ of a line with mathematical proof?

I'd like to know if it is possible to find out the simplest way to get $\frac{1}{7}$, $\frac{2}{7}$, $\frac{3}{7}$, $\frac{4}{7}$, $\frac{5}{7}$ and $\frac{6}{7}$ of a line in 2-dimensional geometry. ...
2
votes
3answers
55 views

How is this limit being solved? I can't grasp it

I am going over limits for my finals as I notice this example in my schoolbook discribing limits of the undefined form $0\over0$ in the shape of an irrational fraction. $$\lim\limits_{x \to 1} {\...
0
votes
4answers
40 views

Dividing factorials

I'm told that $\dfrac{(n+1)!}{(n+2)!}$ simplifies to $\dfrac{1}{n+2}$, but I dont understand how this works. Could someone explain the theory of how to divide factorials like this?
1
vote
1answer
41 views

What rule is used for this simplification?

$$ \frac{8}{(s+1)^2 + 2^2} \times \frac{1}{s} = \frac{8}{5} - \frac{1}{s} + \frac{16}{10}\times \frac{s+1}{(s+1)^2 + 2^2} + \frac{8}{10}\times \frac{2}{(s+1)^2 + 2^2} $$
0
votes
3answers
26 views

Help Solving Fraction Math Question

Im stumped by this question on a practice ACT math test. If $\frac 1x + \frac 1y = \frac 1z$ then $z =$? The correct answer is $\frac {xy}{x + y}$ How do you arrive at this answer? I don't know how ...
1
vote
1answer
22 views

Prove that $\frac{n-a}{n} < \frac{n+1-a}{n+1}$?

I have this math question that I'm kind of stuck on. Prove that $\frac{n-a}{n} < \frac{n+1-a}{n+1}$ So far I have that: $\frac{n-a}{n} < \frac{n+1-a}{n+1} = n-a < \frac{n(n+1-a)}{n(n+1)}...
1
vote
1answer
38 views

How to calculate remainder value of a fraction

Question: Four brothers split a sum of money between them. The first brother received 50% of the total, the second received 25% of the total, the third received 20% of the total, and the fourth ...
1
vote
2answers
39 views

KhanAcademy nested fractions problem

So I was given a problem at KA today. They offer you a choice among possible simplified versions of this expression: $$\frac{1+\frac{x}{y}}{\frac{x}{y}}$$ My solution was: $$(1+\frac{x}{y})\div\...
0
votes
2answers
49 views

How to do rational expressions

Never was much of a math student but I am brushing up on my arithmetic and algebra for college. I am using sample questions from Accuplacer and then using video lectures and practice on Khan Academy....
2
votes
1answer
38 views

Find the sum of the integers in the continued fraction

Find the sum of integers $a,b,c,d,$ and $e$ if $\dfrac{2011}{1990} = a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{d+\dfrac{1}{e}}}}$. I could simplify the big fraction on the RHS, but I don't see how that ...
0
votes
0answers
50 views

How to simplify an expression?

I have tried to simplify this expression for quite a long time now but I can't find how to do it. Can someone help me with it.
3
votes
2answers
32 views

Checking convergence of series

My task is to check whether the series $$\sum_{n=1}^\infty {\frac{(-2)^n+n^2}{n2^n}}$$ converges or not. So I tried expanding the fraction like that(should equal to original form): $$\sum_{n=1}^\infty ...
1
vote
1answer
29 views

Showing this fraction is less than one?

And also, what are some techniques to show such a thing. That is, what are standard techniques for showing a fraction is less than one? I'm just trying to refresh/improve/add to my toolbox. The ...
0
votes
1answer
25 views

What does the word rational in rational fraction refer to

I understand that a rational fraction is an element in the fraction field $F(X)$ and that a fraction $f$ is represented by a quotient of two poloynomials $A\over B$ under the relation ${A\over B}={C\...
1
vote
1answer
85 views

How to explain aspects of derivatives to little brother?

So as the title suggests, my $12$ year old little brother loves math. Since he is a bright kid, I started teaching him derivatives. The issue is he always keeps asking weird questions that are above ...
0
votes
4answers
59 views

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$ Sorry for the really dumb question but I'd like to see the process of how this is achieved.
2
votes
4answers
90 views

Simple fraction calculation

I'm trying to write this expression as one fraction: \begin{align} \frac{5}{y+1}\cdot\frac{1}{5}+\frac{y}{\frac{y+1}{3}} &= \frac{5y}{5y+5}+\frac{3y}{y+1} \\ &= \frac{5y^2+5+15y^2+15y}{5y^...
1
vote
5answers
82 views

How to raise a fraction to a fractional exponent?

I am trying to simplify this but I'm not sure how to approach it.
6
votes
5answers
151 views

What is $\frac{x^{10} + x^8 + x^2 + 1}{x^{10} + x^6 + x^4 + 1}$ given $x^2 + x - 1 = 0$?

Given that $x^2 + x - 1 = 0$, what is $$V \equiv \frac{x^{10} + x^8 + x^2 + 1}{x^{10} + x^6 + x^4 + 1} = \; ?$$ I have reduced $V$ to $\dfrac{x^8 + 1}{(x^4 + 1) (x^4 - x^2 + 1)}$, if you would like ...
1
vote
1answer
13 views

How to simplify equations with a constant times a polynomial to a power

I've started doing integration in class and I often come across parts like this towards the end of my integration: $\frac{2(6x^2+6x+9)^{3/2}}{3} +C$ And I get stuck. How should I go about ...
0
votes
1answer
43 views

Simplifying an expression.

I've been trying to simplify this expression for quite a long time but I just can't get to the result given in the book. The result in the book says that this expression should be simplified to (a*b; ...
-4
votes
2answers
87 views

Find all values of $x$ such that $\frac{x-4}{2x-3}$ is an integer. [closed]

Find all values of $x$ such that $\frac{x-4}{2x-3}$ is an integer. I tried many ways to do it, like by setting it to a fraction and what not, but it could just come for me, I hope you can help. ...
0
votes
0answers
33 views

Inverse Laplace Transform by Partial Fraction Expansion

I've been trying to solve this partial fraction for a Laplace transformation but I can't. Is there any way to solve it? $$\frac{(s-t)^2}{((s-t)^2-1)((s+1)^2+4)}$$ Could somebody help, I've been ...
2
votes
2answers
39 views

What is a field with bounded whole and unbounded fractional part called?

Given numbers of the form: $W + \frac n d$ where $d \gt n \ge 0, W \ge 0$, and all are integers, when defining addition and multiplication on these numbers, I want $W$ to be bound by a positive ...
0
votes
2answers
53 views

Simplification of $\sum_{k=1}^n \frac{1}{4k^2-1}$

So I want to simplify this expression: $$\sum_{k=1}^n \frac{1}{4k^2-1}$$ and Wolfram Alpha tells me it can be simplified to two forms: $$\frac{n}{2n+1}$$ and $$\frac{1}{2}-\frac{1}{2(2n+1)}$$ The ...
0
votes
1answer
42 views

Factors that Impact a Weighted Average the Most

I am trying to determine how to calculate the factors that have the most significant impact on a weighted average. For example, let's say I am reviewing the number of patients that responded to a ...
3
votes
4answers
100 views

Why Does $f(x) = x\sqrt{x+3}$ Only Have One Critical Point?

I am trying to find the critical points of the function $f(x) = x\sqrt{x+3}$, then by using the First Derivative Test, determine which ones are a local maximum, local minimum, or neither. Using the ...
3
votes
2answers
128 views

Difference between fractions at group level have different sign than difference between fractions in aggregate

I have obtained a result (perhaps incorrectly; we shall find out) that appears paradoxical. Suppose I am interested in comparing fractions between 'groups' (not in the strict mathematical sense) $\...
0
votes
1answer
16 views

Conditions present after solving for x

The question asked me to solve $\dfrac {x^2+2x-8}{x^2-x-2}=3$ The answer is $x=0.5$, which I worked out, but in the answers it says that I must state the condition that $x≠2$ to get full marks. Why ...
0
votes
1answer
34 views

Problem in solving an Integral.

I'm solving a solid of revolution problem and I'm stuck at this point. The u-subtitution doesn´t work, I don´t know what method use. $\pi\int(\frac{4x-1}{8x^4})^2$
1
vote
2answers
52 views

Tips and tricks for speedy mental division with harder fractions

I'm looking for a strategy to solve these kind of questions rapidly. Do you guys have any suggestions? $$\frac{\frac{15}{25}}{X}=\frac{14}{35}.$$ A. $\frac{3}{5}$ B. $\frac{63}{30}$ C. $\frac{14}{...
0
votes
1answer
19 views

Find a common denominator

I'm trying to integrate a function but first I need to find a common denominator for: $\frac{A}{x-8}\ $ + $\frac{B}{x+1}\ $ + $\frac{C}{x-1}\ $
1
vote
2answers
119 views

Finding average of denominator knowing average of numerator and average of fraction [closed]

Good day to you all. I have a little problem I have been banging my head on for a while. I have come to think that it is impossible, but I hope you can save me. I have a fraction, $ \frac{num_i}{...
3
votes
3answers
132 views

Proving that the sum of fractions has an odd numerator and even denominator.

I'm struggling to show that, for all $n>1$ $$ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{k}{m} $$ where $k$ is an odd number and $m$ is an even number. Proof: The proof is by ...
2
votes
1answer
30 views

why this Diophantine equation $2ks=(5t+3)(16t+9)$ has always a solution for every $k$?

I would like to solve the following Diophantine equation and show that it has always a solution; i.e. for every positive integer $k$, there exists an integer $t$ such that the fraction is an integer: $...
0
votes
1answer
26 views

Convert mibibyte-days to gibibyte-months.

I have a web host that charges $1 for every Gibibyte (1024 MiB) that I use per month. Currently I am using about 7.6 MiB (Mibibytes) per day. I tried to do this myself and ended up with hopelessly ...
5
votes
4answers
558 views

Summing reciprocal logs of different bases

I recently took a math test that had the following problem: $$ \frac{1}{\log_{2}50!} + \frac{1}{\log_{3}50!} + \frac{1}{\log_{4}50!} + \dots + \frac{1}{\log_{50}50!} $$ The sum is equal to 1. I ...
0
votes
1answer
53 views

How to cancel out a negative in a denominator?

The question was to make $y$ the subject in $x=5-3y$ (i.e. solve for $y$). My working was this: $$\begin{align*} x&=5-3y \\ -3y&=x-5 \\ y&=(x-5)/(-3) \end{align*}$$ But this was ...
4
votes
2answers
28 views

Distributing multiplication of rational functions

I am having trouble distributing with fractions. This $$ \left(\frac{1}{(x + 3)} + \frac{(x + 3)}{(x - 3)}\right)\, (9 - x^2) = -\frac{(x^2-3)}{9-x^2}(9-x^2) $$ has the answer $\left\{\left[x = - \...
0
votes
2answers
41 views

Rules for solving inequalities with negative fractions

What are the algebraic rules for solving inequalities with negative signs and fractions like: $$\frac{1}{x}<-\frac{1}{5}$$
3
votes
4answers
231 views

Write a formula as a sum of fractions with constant numerators

I'm supposed to write this formula: $$\frac {9a + 43}{a^2 + 9a + 20}$$ As a sum of fractions with constant numerators as: $$\frac {7}{a+5} + \frac {2}{a+4}$$ The first step is of course: $$\frac {...
0
votes
1answer
36 views

Rearranging exponential functions

Using two different strategies, I've derived an equation for a particular function $f(\phi)$. That equation is $$ f(\phi)=\frac{1}{1-e^{-T\gamma}}(1-e^{-T\gamma\phi}). $$ However, the paper whose ...