Questions on fractions, numbers of the form $p/q$ where $p$ and $q$ are integers, and $q$ is not zero.
0
votes
1answer
80 views
How does this fraction simplify?
how does $$1+\cos2\theta = (2-\sqrt2)/2$$ give you $-\sqrt2/2$?
i get that you subtract 1 from the left side, but how does doing so on the right give you $-\sqrt2/2$?
6
votes
1answer
187 views
Number of triples $(a,b,c)$ of positive integers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{4}$?
What is the number of triples $(a,b,c)$ of positive integers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{4}$ is:
A) $16$
B) $25$
C) $31$
D) $19$
E) $34$
Note: ...
0
votes
3answers
300 views
What is common and widely recognized abbreviations for *Numerator* and *Denominator* terms for Anglophone mathematicians? [closed]
I have a basic notation/convention question. I'm writing a program in Pascal programming language which does computations in the rational number field (ℚ). For that i defined a data type, which ...
7
votes
1answer
147 views
Number to the exponent divided by exponent value
Can someone explain including working out how to solve this?
$$\dfrac{5^x}{x} = 79.85957$$
I know that the answer is $x = 3.5$, but how does one normalise the equation so that the x is on one side?
1
vote
3answers
315 views
Solving a literal equation containing fractions.
I know this might seem very simple, but I can't seem to isolate x.
$$\frac{1}{x} = \frac{1}{a} + \frac{1}{b} $$
Please show me the steps to solving it.
13
votes
2answers
491 views
Why is $\pi$ irrational if it is represented as $c/d$?
$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction.
Then why is $\pi$ an irrational number?
0
votes
1answer
244 views
Equivalency of percentage formulas
I know 3 methods for calculating percentages,
one example, find 70% of 50:
1) 50/100 * 70
2) 70/(100/50)
3) 70/100 * 50
I do not undertand how this 3 methods can be equivalent, also conceptually ...
1
vote
0answers
119 views
What's the difference between a stacked fraction and a small function? [closed]
MS Word has stacked fractions and small fractions. What's the difference?
18
votes
5answers
358 views
Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?
When I was in third grade, I was playing with rectangles and diagonal lines, and discovered something very interesting with fractions. I've shown several math teachers and professors over the years, ...
0
votes
1answer
154 views
limit of floor function
I can solve the question limit of function like
$$
\lim\limits_{x\to\infty}\frac{\lfloor x-3\rfloor}{x-1}
$$
but I cant solve the question like
$$
\lim\limits_{x\to n^\pm}\frac{\lfloor ...
0
votes
1answer
40 views
Basic question about fractions
I'm solving some exercises about fields and am trying to find the inverse for $a_1 + \sqrt{2}b_1$, i.e. $\frac{1}{a_1 + \sqrt{2}b_1}$. This means I need to split the fraction into something of the ...
3
votes
1answer
82 views
How to solve this System of Polynomial Equations?
I have to complete a summer packet of 90 Algebra 2 questions. I have completed 89 of them, the only one I could not get was this. I know the answer is $y = \frac {47}2$, $\frac 17$ according to ...
5
votes
1answer
89 views
2x2 Matrices and Differences of Fractions
Consider the difference of two arbitrary fractions, $\frac{a}{b}$ and $\frac{c}{d}$.
$$\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$$
The numerator is the determinant of the 2x2 matrix $$ \left( ...
1
vote
1answer
62 views
Simplifying a fraction?
$$\frac {n-2}{n} \cdot \frac {n-3}{n-1} \cdot \frac {n-4}{n-2} \cdots \frac{2}{4} \cdot \frac{1}{3} = \frac {1}{n(n-1)}$$
Why is this true? Notice the denominators and numerators cancel out, but ...
1
vote
4answers
210 views
Fastest way to compare fractions
Which is the fastest method to compare the below fractions with minimum calculation possible and finding which is greatest and which the smallest??
$$\frac{26}{686},\quad \frac{48}{874},\quad ...
1
vote
2answers
155 views
How does he get a perfect swap numerator and denominator.
I'm going through a exercise, in which all the answers are given, but the tutor makes a step and I can't follow at all. A massive jump with no explanation.
Here is the question:
$\lim_{x \to 2} ...
0
votes
1answer
91 views
Simplify this algebraic fraction
I have this algebraic fraction:
$$\frac{t^4-1}{t^2-t^6}$$
And I'm told the answer is:
$$\frac{-1}{t^2}$$
I can't for the life of me work out how to simplify it. (I'm sorry for the simple question)
...
1
vote
1answer
64 views
Value of a fraction
It it true that is ${a^2+c^2\over b^2+d^2}=1$ for $ad-bc=1$?
I tried substituting in $a={1-bc\over d}$ but it is still a mess.
(How do you ask Wolfram Alpha a question like this where we ask it to ...
1
vote
1answer
68 views
Simplifying a fraction through factoring
I have the following fraction:
$\frac{a^3-8}{a^2+2a+4}$
Because the numerator is the difference of two cubes, I've factored it like this: $(a-2)(a^2+8a+64)$.
The denumerator does not have natural ...
0
votes
2answers
225 views
Graphing Fractional Exponents
$f(x)=x^\frac{5}{3}-5x^\frac{2}{3}$
is the same as :
$f(x)=(\sqrt[3]x)^5-(\sqrt[3]{5x})^2$
Except, with the first equation, my calculator returns an error for negative values of $x$ (We are ...
2
votes
2answers
106 views
How to write inverse of integer as sum of fractions
I was reading this article about partial fractions and at the bottom of the article there was a paragraph about integers. However, I cannot seem to get it right each time.
For example:
...
1
vote
2answers
131 views
How do I manipulate algebraic fractions with an addition in the denominator?
My lecturer has given me some notes to study and I can't follow one of the steps...
I need to find the inverse laplace transform of $$\frac3{s(0.1s+1)}\;.$$ The notes do the following:
...
7
votes
2answers
221 views
Writing $1$ in form of $\frac{1}{t_1}+\cdots+\frac{1}{t_n}$ [duplicate]
Possible Duplicate:
Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$
Can anyone help me with this problem? It's a little ...
0
votes
2answers
72 views
Narrowing a Stern-Brocot tree
Say I only wanted to enumerate the rational numbers between 0 and $a$. Is there a way to "narrow" a Stern-Brocot tree to provide this? I tried keeping my left bound at "$\frac{0}{1}$" and setting my ...
0
votes
2answers
188 views
Math Database For Problem Descriptions In An App.
I am developing an app for kids and they will have a variety of problems from percentage problems, absolute value problems, negative number problems, fraction problems, etc. I was hoping to have a ...
0
votes
0answers
197 views
How to get approximate fraction numbers from imaginary numbers with MatLab
I'm not sure how to title this problem actually, but I have a clumsy PHP code that I've used to get approximate fraction numbers for imaginary numbers like pi, phi, square root of 2, 3 and so on. I'd ...
4
votes
3answers
118 views
Divide with remainder $\frac{x^2}{x^2 + x + 2}$
I am having a hard time long dividing:
$$\frac{x^2}{x^2 + x + 2}.$$
Could someone please show a step by step way to divide this, as I can only get it down to : $1 + \frac{x^2}{x + 2}$.
Thank you ...
3
votes
3answers
159 views
Constructing Farey sequences inductively
Objective: I'd like to prove that $F_{n+1}$ (the Farey sequence of order $n+1$) is obtained form the Farey sequence $F_n$ of order $n$ by adding all fractions of the form $\frac{a+c}{b+d}$ when ...
3
votes
4answers
192 views
Negative fractions - what's the difference?
What's the difference between the following fractions:
$ \frac{-4}{-5}$
$ \frac{4}{-5}$
$ \frac{-4}{5}$
$ - \frac{4}{5}$
3
votes
1answer
83 views
How fast is a low denominator encountered, when using only mediants?
This question is (remotely) related to How to find a "simple" fraction between two other fractions?, but is not answered in that older post.
Let $f_1=\frac{a}{b}$ and $f_2=\frac{c}{d}$ be ...
3
votes
1answer
87 views
Is there a direct proof of this inequality between quotients of integers?
Let $\frac{a}{b}$ and $\frac{c}{d}$ be two reduced fractions with
$bc-ad > 1$ (and hence $\frac{a}{b} \lt \frac{c}{d}$) and
$a,b,c,d$ positive. It is well known that there are integers $u,v$
...
3
votes
3answers
301 views
How to find a “simple” fraction between two other fractions?
If we have two fractions $a = { a_1 \over a_2} $ and $c = {c_1 \over c_2}$ with $a<c$,
how to find the fraction
$b = { b_1 \over b_2 }$ , $a < b < c$
for which some measure of ...
3
votes
2answers
107 views
Integers and fractions
How would I write this as an integer or a fraction in lowest terms?
$(1-\frac12)(1+\frac 12)(1-\frac13)(1+\frac13)(1-\frac14)(1+\frac14).....(1-\frac1{99})(1+\frac1{99})$
I really need to understand ...
1
vote
1answer
77 views
Looking for hints of this inequality
I think the following two inequalities are true. However, the proof may not be easy. Does anyone have any hints? Thank you very much!
Fix $a>1$. there exists two constants $K_1$ and $K_2$, such ...
3
votes
3answers
197 views
Is it possible to have a fraction wherein the numerator and denominator are also fractions?
For example:
$$
\frac{\ \dfrac{3}{5}\ }{\dfrac{7}{8}}
$$
I was wondering if such a situation had a name, wherein both the numerator and the denominator of a fraction consist of fractions ...
1
vote
0answers
236 views
Contribution (weighted average) of change in rate over time
I'm trying to determine the weighted average impact of one customer's change in rate on the total change in effective rate.
Let's say I have two customers and two time periods:
...
0
votes
1answer
102 views
Help with trinomial fractions [closed]
The denominator is $x-6$, and the numerator is a trinomial. I figured the trinomial out - $(x-3)(x-2)$; I'm not sure what to do after this step though.
$$\frac{x^2}{x-6} - \frac{5x+6}{x-6}$$
5
votes
2answers
253 views
Evaluate fraction of sum
So i have to evaluate this sum:
$\displaystyle \frac{1-2^{-2}+4^{-2}-5^{-2}+7^{-2}-8^{-2}+10^{-2}-11^{-2}+\cdots}{1+2^{-2}-4^{-2}-5^{-2}+7^{-2}+8^{-2}-10^{-2}-11^{-2}+\cdots}$
it has the form :
...
1
vote
1answer
93 views
Simplifying a Multi-Variate Fraction
I am an eighth grader in need of a little assistence. I was given a multi-variate fraction, and was told to simplfy it to lowest terms. On of my fellow classmates that is ahead of me in math, tried ...
2
votes
2answers
495 views
Simplifying Square Roots with Fractions?
I know this is a very basic question, but could someone please mathematically explain, why this is true:
$\sqrt{x} \cdot \frac{1}{x} = \frac{1}{\sqrt{x}}$
Wolfram|Alpha can confirm this.
8
votes
1answer
98 views
cyclic permutations of periods of recurring fractions
In base 10, the recurring bits of the fractions $\frac{1}7,\ldots,\frac{6}7$ are cyclic permutations of each other.
e.g. $$\frac{1}{7}=0.(142857)$$
$$\frac{2}{7}=0.(285714)$$
...
0
votes
0answers
94 views
General advice for dividing bigger fraction into partial fractions
There's a lot of cases where dividing a larger fraction into smaller ones helps a lot in making calculations a lot simpler. For example, the first step in solving this integral below is to divide ...
5
votes
1answer
176 views
When is the $lcm$ of a fraction sum the actual denominator.
Consider a sum
$$\frac{a}{b}+\frac{c}{d} = \frac{x}{y}$$
where each fraction is reduced. Alternatively using the familiar process of lowest common denominators, we have
$$\frac{a}{b}+\frac{c}{d} = ...
2
votes
2answers
136 views
How much can a fraction reduce?
Assume $x/a$ and $y/b$ are positive fractions in it's reduced form.
If $x/a+y/b=z/c$, where $z/c$ is also reduced. What can we say about $c$?
Does $\frac{ab}{\gcd(a,b)^2}|c$?
If it's not true. Is ...
2
votes
1answer
169 views
How to calculate percentage of comment lines in a code?
I have a file within which I have 6 lines of code and 8 lines of comment.
What's the formula to calculate how much percent of the whole file comments have?
4
votes
2answers
237 views
Bound on lcm of denominators of rational numbers that sum to 1.
This is related to the question If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators?
Suppose $1 = ...
3
votes
1answer
188 views
If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators?
I was experimenting with an algorithm for generating random numbers from a discrete distribution and came across an interesting observation. Suppose that you have any finite set of rational numbers ...
1
vote
0answers
350 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 ...
2
votes
1answer
160 views
Adding a different constant to numerator and denominator
Suppose that $a$ is less than $b$ , $c$ is less than $d$.
What is the relation between $\dfrac{a}{b}$ and $\dfrac{a+c}{b+d}$? Is $\dfrac{a}{b}$ less than, greater than or equal to ...
11
votes
1answer
606 views
Interesting pattern in the decimal expansion of $\frac1{243}$
There appears to be an interesting pattern in the decimal expansion of $\dfrac1{243}$:
$$\frac1{243}=0.\overline{004115226337448559670781893}$$
I was wondering if anyone could clarify how this ...
