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
62 views

Ring of fractions in $\mathbb{Z}/35\mathbb{Z}$

How can I determine $S^{-1}(\mathbb{Z}/35\mathbb{Z})$, where $S$ consists of of all elements of $\mathbb{Z}/35\mathbb{Z}$ except $0,5,10,15,20,25,$ and $30$?
1
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2answers
80 views

most efficient way to convert a number into a fraction

supposing I have a decimal like $$ 0.30000000000000027$$ What would be the best way to know the same number but in a fraction way like we know $\dfrac{1}{3} > 0.30 > \dfrac{1}{4}$ because ...
2
votes
1answer
66 views

Integer+fraction vs Top-heavy fraction

What is the name of a fraction like this: $$\frac{22}{7}$$ as opposed to one like this: $$3\frac{1}{7}$$ I've never actually had to describe this until today. Not only have I no idea how to ...
3
votes
2answers
59 views

Separating $\frac{1}{1-x^2}$ into multiple terms

I'm working through an example that contains the following steps: $$\int\frac{1}{1-x^2}dx$$ $$=\frac{1}{2}\int\frac{1}{1+x} - \frac{1}{1-x}dx$$ $$\ldots$$ $$=\frac{1}{2}\ln{\frac{1+x}{1-x}}$$ I ...
0
votes
2answers
70 views

system of equations with three equations.

We have to find all real solutions to this system of equations: $$x=\frac{4z^2}{1+4z^2},y=\frac{4x^2}{1+4x^2},z=\frac{4y^2}{1+4y^2}$$
1
vote
2answers
3k views

Why are fractions with a negative denominator valid?

Whenever in a fraction, there is $0$ in the denominator, the fraction becomes $\infty$ or indeterminate. But why do we consider those fractions valid that have some negative numbers in the denominator ...
0
votes
3answers
149 views

$3 \dfrac12$ divided by $\dfrac45\,$ ; why do I get 4.3?

This has been bothering me a lot, this is my thinking: $3\dfrac12 \implies \dfrac72 \implies \dfrac{35}{10}$ similarly $\dfrac45 \implies \dfrac{8}{10}$ So ...
3
votes
3answers
182 views

Lottery Odds as Multiples of Fractions

I run a Lottery syndicate for the UK lottery, and we play 30 lines per draw. The odds of winning £10 (3 matching numbers) is deemed to be 1 in 56.7 (or 1/57 for the purposes of this question). ...
0
votes
1answer
87 views

summation of fractions and inequalities

I am trying to prove that $\sum_{i=1}^{n}\frac{1}{a_i}\leqslant 2$, where all $a_i$ are less than 1000, and all $a_i$ have a lowest common multiple greater than 1000. This is what I have done so far: ...
0
votes
2answers
73 views

Simplifying $\frac{4}{2x-7}-\frac{3}{(2x-7)^2}$

A homework question asks me to "perform the addition or subtraction and simplify" $$ \begin{gather} \frac{4}{2x-7}-\frac{3}{(2x-7)^2}=4(2x-7)-3=8x-28-3=8x-31 \\ 8x=31 \\ x=\frac{31}{8} \end{gather} ...
0
votes
2answers
99 views

Aproximate calculation in decimals

I am trying to refresh on precision of calculations. If we have the decimal fractions: $.234673$, $.322135$, $.114342$, $.563217$ each known to be correct to six figures why are each of the decimals ...
6
votes
4answers
431 views

An interesting problem with fractions

These are a few examples of how "forbidden" procedures can lead us to the correct answer: $$\displaystyle\frac{1\not4^1}{2\not8_2} = \frac{11}{22}=\frac{1}{2}$$ ...
11
votes
8answers
3k views

Is 1 divided by 3 equal to 0.333…?

I have been taught that $\frac{1}{3}$ is 0.333.... However, I believe that this is not true, as 1/3 cannot actually be represented in base ten; even if you had ...
1
vote
3answers
81 views

Simplify a sum of fractions

I am stuck trying to get from: $$\frac{pZ(a)}{pZ(a) - (1-p)Z(b)} - \frac{p(pZ(a) - (1-p)Z(b))}{pZ(a) - (1-p)Z(b)} $$ to $$\frac{p(1-p)(Z(a) - Z(b))}{pZ(a) - (1-p)Z(b)} $$ Obviously my problem is ...
0
votes
2answers
5k views

How to enter subscript characters in WolframAlpha? [closed]

I'm trying to enter equations like this in WolframAlpha. How do I format this?
1
vote
2answers
71 views

Monotonicity of a fraction

If I have a fraction $f(x) = \dfrac{n(x)}{d(x)}$, where $n(x)$ increases monotonically and $d(x)$ decreases monotonically; as functions of $x$. Can I be sure that $f(x)$ increases monotonically as a ...
5
votes
1answer
159 views

finite field to rational fraction

Suppose I have a number $n\in\mathbb F_p$, i.e. an element of the finite field obtained by arithmetic modulo some (odd) prime $p$. I'm looking for a way to find a simple description of $n$ as a ...
3
votes
1answer
77 views

Given two ratios $\frac{p_i}{q_i}$, what is $\frac{p_1+p_2}{q_1+q_2}$ in their terms

I am ashamed to say that I cannot figure this one out: I am given two ratios $\dfrac{p_i}{q_i}$ where $i=1$, $2$. (We just know the ratios and not the numbers $p_i, q_i$. What I mean by this is ...
0
votes
1answer
124 views

Inventory of Clocks and Frequency of Chimes

How do you determine the hours for which the clocks chime?
6
votes
9answers
2k views

Are all integers fractions?

In a college class I was asked this question on a quiz in regards to sets: All integers are fractions. T/F. I answered False because if an integer is written in fraction notation it is then ...
2
votes
1answer
72 views

Which numbers will remain if I keep removing the second third of the remaining interval?

Inspired by this Google Code Jam problem - Vanishing Numbers There is a pool of numbers which are arbitrary decimal fractions from the interval (0, 1). In the first round of the game the middle ...
1
vote
3answers
10k views

Most efficient method for converting flat rate interest to APR.

A while ago, a rather sneaky car salesman tried to sell me a car financing deal, advertising an 'incredibly low' annual interest rate of 1.5%. What he later revealed that this was the 'flat rate' ...
4
votes
3answers
147 views

How does $({{n/e})^n} / ({({n/{2e}})^n})$ simplify to $2^n$ (MIT OpenCourseware 6.006)

As stated in the title, how is the following simplification performed? $$\frac{\left(\frac{n}{e}\right)^n}{\left(\frac{n}{2e}\right)^n}=2^n$$ This was shown by a student in this Recitation video ...
9
votes
4answers
778 views

How do I rewrite -100+1/2 as the mixed number -99 1/2?

This has been bugging me for some time now, so I ask you to try to help me realize what is going on here. I just can't get my brain around this. I have a proper fraction and a negative integer. The ...
1
vote
1answer
73 views

Problems with basic algebra

I'm studying for an exam in a digital communications course I'm taking, and the solution to one question has me totally lost. While finding the Inverse Fourier Transform of a function, there's one ...
2
votes
3answers
2k views

Simplification of fraction with factorials

I'm stuck on a simplification, used to prove $C(n - 1, r - 1) + C(n - 1, r) = C(n, r)$ Could somebody clarify the step(s) from: $\frac{(n - 1)!}{(r - 1)!(n - r)!} + \frac{(n - 1)!}{r!(n - r - 1)!}$ ...
0
votes
1answer
55 views

How is this complex fraction getting cleared using this method?

Can someone please explain how multiplying these two fractions together is clearing the complex fraction? I know that by multiplying these two fractions clears the complex fraction, I just can't ...
1
vote
2answers
55 views

Simplifying $\frac{1}{x} + \frac{5+x}{x+1} - \frac{7x^2 + 3}{(x+2)^2}$

I'm having trouble simplifying this expression: $$\frac{1}{x} + \frac{5+x}{(x+1)} - \frac{7x^2 + 3}{(x+2)^2}$$ Would you first do the addition or subtraction? What's the steps to solve this? The ...
3
votes
2answers
859 views

How to solve this? (adding polynomial fractions)

I'm having trouble solving this expression: $$\frac{(x - 1)(7x + 6)}{(x - 1)(x + 1)^2 }-\frac{ 7}{ (x + 1)}$$ What's the steps to solve this? I know you expand $(x + 1)^2$ to $(x + 1)(x + 1)$, ...
1
vote
1answer
61 views

General solution for x of C = 100/(1+aX) + 100/(1+bX) … + 100/(1+zX)

Please can someone help me find a general solution for X $C = \frac{100}{(1+aX)} + \frac{100}{(1+bX)} ... + \frac{100}{(1+zX)}$ UPDATE Its not ideal but if we make C = 350 would this help?
-4
votes
2answers
136 views

how to add/subtract than multiply fractions?

Q). 1 + 9/2 x -5/7 Q). 1 - y^2 x 9/4 just guide me how to solve this question Edit: From the title of the question I would infer that he/she meant $ 1 + \frac92 \times \frac{-5}{7}$ $ 1 - y ^2 ...
2
votes
1answer
64 views

Problem with slopes.

I currently have a slope that looks like this: $\frac{-5}{10}$ However, I need to bring it down to it's lowest terms, so I divided the numerator and denominator by -5 and I got: $\frac{1}{-2}$ ...
2
votes
3answers
8k views

How can I convert this negative fraction to a positive one?

This question may be very simple, but I get confused on things like it. If I have a fraction like this: $-\frac{x}{-2}$ How can I convert this negative fraction to a positive one? It does not ...
1
vote
2answers
2k views

Problem with getting variable by itself in fraction.

I have a problem that looks something like this: The difference of the quotient of a number and $-2$ from $12$ is $15$. So I started off like this: $12-\displaystyle\frac{x}{-2}=15$ Then I ...
1
vote
1answer
89 views

What fraction of smarties are brown?

If I have 26 smarties and 2 of them are brown, what fraction is that?
2
votes
2answers
2k views

What's the largest possible repeating decimal that can be created from a fraction (if n <= 9,999 & d <= 9,999)

What's the largest possible repeating decimal that can be created from a fraction if: The numerator has to be less than or equal to 9,999 The denominator has be less than or equal to 9,999? I know ...
3
votes
2answers
2k views

What are some effective ways in teaching fractions to 5th graders who are behind (special needs)

I am teaching a group of 8 kids on fractions and I did not realize how difficult this can be. These kids were selected by their teachers for needing additional outside help. I really need some advice, ...
6
votes
6answers
346 views

How to show that $\frac{x^2}{x-1}$ simplifies to $x + \frac{1}{x-1} +1$

How does $\frac{x^2}{(x-1)}$ simplify to $x + \frac{1}{x-1} +1$? The second expression would be much easier to work with, but I cant figure out how to get there. Thanks
6
votes
1answer
571 views

Check my proof of an algebraic statement about fractions

I tried to prove the part c) of "Problem 42" from the book "Algebra" by Gelfand. Fractions $\frac{a}{b}$ and $\frac{c}{d}$ are called neighbor fractions if their difference $\frac{cb-ad}{db}$ has ...
1
vote
1answer
84 views

How to chose a rational with a non-repeating fractional part in an arbitrary base?

How can I choose an $x\in[a,b)\subseteq[0,1)$, where $a,b\in\mathbb{Q}$, such that $x$ has a non-repeating fractional part in some chosen base? For example, say I'm looking at ...
4
votes
2answers
359 views

How to prove that construction of Farey sequence by mediant is coverage?

Farey sequence of order $n+1$ ($F_{n+1}$) can be construct by adding mediant value (${a+c \over b+d}$) into $F_{n}$, where ${a \over b}$ and ${c \over d}$ are consecutive term in $F_{n}$, and $b+d = ...
1
vote
3answers
105 views

In a fraction between integers, what denominators produce a periodic result?

I'm trying to remember something that a math teacher told me many years ago about fractions. If I remember correctly he said that, in a fraction between integers, when the denominator is a multiple of ...
1
vote
2answers
104 views

Is the index in an nth root allowed to be fractional?

We have nth roots that can be rewritten as fractional powers: $$\sqrt[n] x = x^\frac 1 n$$ I was looking around on Wikipedia and some other online material, but I couldn't find any definitive set of ...
3
votes
3answers
188 views

Finding the minimum of $\frac pq + \frac rs$ for distinct integers $p, q, r, s$ from $\{1,2,3,4,5,\ldots,16,17\}$

Here is the question: Four distinct integers $p$, $q$, $r$ and $s$ are chosen from the set $\{1, 2, 3, 4, 5, \ldots, 16, 17\}$. The minimum possible value of $\frac pq + \frac rs$ can be written ...
6
votes
6answers
1k views

How to simplify $\frac{4 + 2\sqrt6}{\sqrt{5 + 2\sqrt{6}}}$?

I was tackling through an olympiad practice book when I saw one of these problems: If $x = 5 + 2\sqrt6$, evaluate $\Large{x \ - \ 1 \over\sqrt{x}}$? The answer written is $2\sqrt2$, but I ...
8
votes
4answers
196 views

What is the 'physical' explanation of a division by a fraction?

For example, dividing by 2, means we cut something in two. But dividing by 0.5, can only be explained with multiplying something by 2. So, is there a "physical" explanation of dividing by 0.5? Is it ...
2
votes
1answer
57 views

Need an explanation of a particular expression transformation

Please, I need an explanation of the one transformation. I have the equation set and its solution. $$ \begin{cases} \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 3\\\\ \frac{y}{x} + \frac{z}{y} + ...
3
votes
2answers
914 views

How do I get the integer part of a number by using basic arithmetic?

While it is trivial to simply remove the fractional part of an irrational or rational number, and in programming I could just use the floor() or ...
12
votes
2answers
451 views

proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$

i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows: $$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
1
vote
1answer
20 views

Laplace Transform Help

The Laplace Transform of $\frac{3}{(2s+5)^3}$ is given as $\frac{3 t^2}{16}e^{-\frac{5}{2}t}$ Can someone please walk through how this was obtained? Especially the $\frac{3}{16}$?