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

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2
votes
2answers
87 views

When is the class of a fraction, the set of multiples of the fraction?

Let $D$ be an integral domain and $F=\mathrm{Frac}(D)$ be the field of fractions of $D$. We will look at $F$ as a set of equivalence classes from the equivalence relation $\sim$ on $D\times ...
1
vote
1answer
105 views

change values by a percentage factor

I have a problem in that i have a camera with a zoom range values between 1,000 and 30,000. The problem is i am accessing these values through a slider bar with values starting 250 to 750. How can i ...
0
votes
3answers
86 views

Ratio problem to find the woman weekly salary

A woman spend $5/8$ of her weekly salary on rent, and $1/3$ of the remainder on food, leaving $40 available for other expenses. What is the woman's weekly salary ? I have tried , i am really confused ...
4
votes
1answer
182 views

Algebraic structure of a set of Egyptian fractions of a positive rational?

It is said that every positive rational number can be represented by infinitely many Egyptian fractions (defined as the sum of distinct unit fractions). I am struggling to understand in a formal way, ...
1
vote
2answers
256 views

Bar Notation Problem

everyone! I came across a problem in math that dealt with bar notation. Does anyone know how, for instance, 1.234(with a bar notation over the 34) is expressed as a fraction? I know already how ...
4
votes
2answers
88 views

Why does my intuition for “order of divergence” for algebraic fractions fail?

I come across this identity once in a while but I actually never grasped it: $$\frac{2}{1-x^2}=\frac{(1-x)+(1+x)}{(1+x)(1-x)}=\frac{1}{1+x}+\frac{1}{1-x}$$ I'm surprised by it because I would ...
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 ...
0
votes
1answer
157 views

Finding percentage which is less

The number that is 50% greater than $60$ is what percentage less than the number that is 20% less than $150$ ? My try : I considered a number is 50% of $130$ which is greater than the $60$ and 20% ...
38
votes
1answer
852 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
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 ...
22
votes
1answer
639 views

Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$

Simplify $$\frac{\displaystyle\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}{\displaystyle\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}$$ I don't have any good idea. I need your help.
3
votes
2answers
106 views

Question involving approximation, taylor series and proving

Question: Consider the approximation $$\ln(2)\approx 2\left ( \frac{1}{3}+\frac{1}{3\times 3^{3}}+\frac{1}{5\times 3^{5}} \right )$$ Prove that the error in this approximation is less than ...
1
vote
4answers
122 views

Simplifying compound fraction: $\frac{3}{\sqrt{5}/5}$

I'm trying to simplify the following: $$\frac{3}{\ \frac{\sqrt{5}}{5} \ }.$$ I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I ...
-2
votes
4answers
136 views

Canceling in fractions sometimes gives a wrong result

When the same factor appears in the numerator and denominator, it can be canceled out: $$\frac{4a}{4a} = 1$$ However, in this more complicated fraction this does not work: $$\frac{4ac-b^2}{4a} \neq ...
0
votes
2answers
2k views

Numbers that cannot be expressed as fractions

What are Numbers that cannot be expressed as Fractions called?
1
vote
2answers
169 views

How to solve a ratio question

Studying for the GRE. In the GRE guide, it says that If the ratio is $2x:5y$, and this equals the ratio $3:4$, what is the ratio of $x:y$? I tried cross multiplying but I don't get the answer. ...
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} = ...
0
votes
1answer
459 views

Ceiling to Floor Function Conversion Proof

I am working on a proof to convert a ceiling of a fraction to a floor of a fraction. I found this: \begin{aligned} q=\left\lceil \frac{n}{m} \right\rceil \;&\Leftrightarrow\; \frac{n}{m} \leq q ...
3
votes
2answers
88 views

How do we know $p/q$ can be expressed as a terminating fraction in base $B$ only if prime factors of $q$ are prime factors of $B$?

On cs.stackexchange I asked a math question: How to demonstrate only 4 numbers between two integers are multiples of .01 and also writable as binary. Yuval Filmus answered with a explanation ...
0
votes
2answers
561 views

Ratio - Basic Question

If ratio of A:B = 1:2 if it is doubled , should it be not 2:4 i see many problems where they are simply multiplying numerator by 2 please can some one explain
0
votes
4answers
95 views

breaking up fractions

I have these two fractions ${11 \over 31 }+{-11 \over 61}$ Adding them gives $330 \over 1891$ But how do I go back to the two fractions, once I've added them? I can get the denominators just by ...
1
vote
3answers
288 views

Reverse percentages

My mothers recently started doing a distance learning course. And is struggling with her mathematical questions. I'm trying to explain to my mother how to answer the following question. Despite my ...
1
vote
2answers
103 views

Fractional overlap of 1/2 and 1/3

Given a subset of the natural number sequence (positive integers starting from 1) we could say that $\frac12$ of the numbers in the set are divisible by 2. e.g if the set were ${[1,2,3,4,5,6,7]}$ we ...
18
votes
11answers
3k views

Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
0
votes
2answers
147 views

removing the remainder of a fraction

I would like to remove the remainder from a fraction if possible. I want a function $$f(x,y) = x/y - remainder$$ for example $$f(3,2) = 1$$ $$f(7,2) = 3$$ $$f(12,5) = 2$$ It seems so simple but ...
4
votes
3answers
391 views

Why are only fractions with denominator 2 and 5 non-repeating?

Given a rational number $\frac{n}{d}$, I understand that in the base $10$ number system, the number can be represented as a non-repeating decimal number if and only if $d$ has only prime factors of ...
1
vote
4answers
106 views

question about partial fractions why $\frac{6x^2+19x+15}{(x-1)(x-2)^2}=\frac{A}{(x-1)}+\frac{B}{(x-2)}+\frac{C}{(x-2)^2}$

can anyone tell me why $$\frac{6x^2+19x+15}{(x-1)(x-2)^2}=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{(x-2)^2}$$ I don't undestand the $\frac{C}{(x-2)^2}$ and also what is wrong according to basic math ...
4
votes
2answers
132 views

Proof for $\displaystyle\sum_{k=1}^n k^a$ equaling a sum of fractions

I know $\displaystyle\sum_{k=1}^n k^2$ equals $n/6+n^2/2+n^3/3$, but... why? And I also know that $\displaystyle\sum_{k=1}^n k^3$ equals $n^2/4+n^3/2+n^4/4$, but... is there a pattern so I can easily ...
2
votes
2answers
53 views

which parameters always make this rational equation evenly divisible?

Hi guys I have the following equation: $$x = \dfrac{a + b \times c - b}{c}$$ This is what I know about each variable: $$a \ge 64$$ $$b \ge 0$$ $$8 \le c \le a$$ My questions is there a concise way ...
-1
votes
1answer
35 views

$\frac{-4z^{-1}}{(1-\frac{1}{4}z^{-1})(1-4z^{-1})} = \frac{16}{15}\frac{1}{(1-\frac{1}{4}z^{-1})}-\frac{16}{15}\frac{1}{(1-4z^{-1})}$

Can anyone help me clarify how this rewriting is done? $$\frac{-4z^{-1}}{(1-\frac{1}{4}z^{-1})(1-4z^{-1})} = \frac{16}{15}\frac{1}{(1-\frac{1}{4}z^{-1})}-\frac{16}{15}\frac{1}{(1-4z^{-1})}$$
0
votes
1answer
37 views

How to show that $\frac{1}{(1-\frac{1}{4}z^{-1})(1-\frac{1}{4}z)} = \frac{-4z^{-1}}{(1-\frac{1}{4}z^-1)(1-4z^{-1})}$

Can anyone help me clarify what rule is used in this rewriting of this fraction? $$\frac{1}{\left(1-\dfrac{1}{4}z^{-1}\right)\left(1-\dfrac{1}{4}z\right)} = ...
2
votes
1answer
140 views

Tetration and its inverse to various exponents

I've recently seen in my studies tetration, or the next operation in the addition, multiplication, exponentation... series. I've also heard much discussion about how to extend this operation to ...
1
vote
2answers
181 views

Solving Simple Mixed Fraction problem?

How do you wrap your head around mixed fraction, does anyone knows how to figure out, can someone give me an example how it can be solved?
5
votes
1answer
374 views

Is it possible to rationalize a denominator containing two cube roots?

The fraction in question is $$-\frac{12}{\sqrt[3]{12\sqrt{849} + 108} - \sqrt[3]{12\sqrt{849} - 108}}$$ And was reached in calculating the solution to $x^4 - x - 1 = 0$. I've tried all the standard ...
1
vote
1answer
111 views

Am I right or is Wolfram right?

Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies $$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$ Determine a recurrence relation that ${a_n}$ satisfies. I ...
1
vote
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
vote
1answer
61 views

Series of Fractional Sums

What is the sum of $$\frac{1}{2\times 5} + \frac{1}{5\times 8}+ \frac{1}{8\times 11}+...+ \frac{1}{2009\times 2012}?$$ What is the easiest way to solve this kind of problem?
3
votes
6answers
572 views

Simplify with fractional exponents and negative exponents

I am trying to simplify $$ \left(\frac{3x ^{3/2}y^3}{x^2 y^{-1/2}}\right)^{-2} $$ It seems pretty simple at first. I know that a negative exponent means you flip a fraction. So I flip it. $$ ...
3
votes
2answers
130 views

Using the hypothesis $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ to prove something else

Assuming that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$$ Is it possible to use this fact to prove something like: ...
1
vote
3answers
91 views

Simplifying fractions - Ending up with wrong sign

I've been trying to simplify this $$ 1-\frac{1}{n+2}+\frac{1}{(n+2) (n+3)} $$ to get it to that $$ 1-\frac{(n+3)-1}{(n+2)(n+3)} $$ but I always end up with this $$ 1-\frac{(n+3)+1}{(n+2)(n+3)} $$ Any ...
8
votes
4answers
1k views

Remainder when $20^{15} + 16^{18}$ is divided by 17

What is the reminder, when $20^{15} + 16^{18}$ is divided by 17. I'm asking the similar question because I have little confusions in MOD. If you use mod then please elaborate that for beginner. ...
4
votes
3answers
204 views

Why does the Denominator of the Denominator go to the Numerator?

Quite blindly I've learnt a basic rule about fractions: The Denominator of the Denominator goes to the numerator. I'm confused about it and I'll give an example as to why. Imagine the following: ...
1
vote
2answers
63 views

Proving that $\frac{\sigma_{n-1}}{\omega_n} = n$ in $\mathbb{R}^n$

If $\sigma_{n-1}$ was the surface area of the unit sphere in $\mathbb{R}^n$ and $w_{n}$ was the area of the unit ball in $\mathbb{R}^n$, my lecture notes prove that $$\frac{\sigma_{n-1}}{\omega_n} = ...
1
vote
2answers
99 views

when the numerator is less than the denominator

when the numerator is less than the denominator the result is always between 0 and 1? for example if I have a number like x/y where x<y then the result will be ...
2
votes
2answers
40 views

Fraction question

Alvin and Bob had a total of 60 marbles. Alvin gave 1/4 of his marbles to Bob. Bob then gave 1/3 of the total number of marbles he had to Alvin. In the end, each of them had the same number of ...
2
votes
2answers
45 views

Simple fractions question

Mary and Henry shared a collection of stamps. Mary had 7/10 of the total number of stamps. If Mary gave 38 stamps to Henry, she will have thrice the number of stamps Henry have. How many stamps did ...
3
votes
4answers
90 views

Dividing and multiplying surds - Rule

What rule/process allows me to take this equation: $$6x^{2} \cdot \sqrt{\frac{y}{x}}$$ And simplify it to become: $$6x \cdot \sqrt{xy}$$
1
vote
2answers
92 views

Recurring decimals to fraction

$.\overline{36}=\frac{36}{99}$ $2.1\overline{36}=2\frac3{22}$ The part I do not understand however, is "you could used 1) to speed up the working of 2)" which is written in the book. How would ...
2
votes
2answers
273 views

How to prove the fraction identity without using calculator

How to prove without calculator that $$ \frac{1}{1001} + \frac{1}{3001} > \frac{1}{1000}$$
3
votes
2answers
53 views

To split in to partial fractions, the expression $\frac{1}{x^2(x+a)^2}$

To split in to partial fractions, the expression $\frac{1}{x^2(x+a)^2}$ $\frac{1}{x^2(x+a)^2}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{(x+a)}+\frac{D}{(x+a)^2}$ One method of finding the values of the ...