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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1answer
431 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 ...
18
votes
5answers
497 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, ...
1
vote
1answer
818 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
49 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
117 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
115 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
91 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
737 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 ...
2
votes
2answers
1k 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
128 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
68 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
119 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
712 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
170 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: ...
2
votes
2answers
297 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: ...
8
votes
2answers
268 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
105 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
210 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
360 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
124 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 ...
4
votes
3answers
316 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 ...
4
votes
4answers
338 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
95 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
100 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
563 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
131 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
80 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
249 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
620 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: ...
5
votes
2answers
328 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
352 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
575 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
143 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. $$\frac{1}{7}=0.(142857)$$ $$\frac{2}{7}=0.(285714)$$ ...
5
votes
1answer
212 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
154 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
299 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
263 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
279 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 ...
2
votes
0answers
541 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 ...
3
votes
2answers
331 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
736 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 ...
0
votes
0answers
89 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: ...
3
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1answer
134 views

Ways to teach fractions

I'm tutoring elementary-level kids on equivalent fractions and am not doing a very good job of explaining it. I've tried using the example of a pizza or a pie and have shown them how they can come up ...
2
votes
2answers
87 views

How do you convert $(12.0251)_6$ into fractions?

How do you convert $(12.0251)_6$ (in base 6) into fractions? I know how to convert a fraction into base $x$ by constantly multiplying the fraction by $x$ and simplifying, but I'm not sure how to go ...
2
votes
2answers
95 views

Simplifying fractions

How to transform $$\left({1+\sqrt{1-4ab}\over2a}\right)^n-\left({1-\sqrt{1-4ab}\over2a}\right)^n\over \left({1+\sqrt{1-4ab}\over2a}\right)^m-\left({1-\sqrt{1-4ab}\over2a}\right)^m$$ into ...
8
votes
1answer
703 views

Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$

Let $x\in\mathbb{Q}$ with $x>0$. Prove that we can find $n\in\mathbb{N}^*$ and distinct $a_1,...,a_n \in \mathbb{N}^*$ such that $$x=\sum_{k=1}^n{\frac{1}{a_k}}$$
1
vote
6answers
556 views

Proof of dividing fractional expressions

For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot ...
1
vote
3answers
2k views

Distributive Property on Fractions: Swapping Denominators

I'm learning Algebra and am curious about some methodological fundamentals here. One, in particular is why the following equation: 6(2x + 1 / 3) = 6(x + 4 / 2) ...
4
votes
2answers
3k views

How to convert a fraction into ternary?

In a ternary system how is $\dfrac{1}{2}=0.\bar1$ ,$\dfrac{1}{3}=0.1=0.0\bar2$??, etc. In general, how does one write the ternary expression for a given fraction?
0
votes
1answer
247 views

Simplify a equation containing factorial, summation, and fraction

I really need some help on simplifying this math equation. Please help me reduce it as simple as possible! Thanks in advance! $$\large P_0=\frac{1}{\left[\sum_{i=0}^{M-1} ...