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

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1
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3answers
43 views

Find two fractions such that their sum added to their product equals $1$

This is a very interesting word problem that I came across in an old textbook of mine. So I managed to make a formula redefining the question, but other than that, the textbook gave no hints really ...
5
votes
3answers
130 views

Interesting and unusual word problem with prime numbers and factors

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really and ...
0
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3answers
32 views

How does one find a rational number in fraction form, knowing the repeating decimal?

For example, I have 0.786786786... How do I find the fraction equivalent?
0
votes
0answers
22 views

Simplifying MATLAB fraction to make numerator equal to 1

I have a function which returns this fraction, which is not in the needed format. $$\frac{36893488147419103232*z^2}{36893488147419103232*z^2 - 672282507639892864*z + 6656262451880127}$$ I need it to ...
-1
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0answers
49 views

Using continued fraction find square root of even number [closed]

We can approximate the square root of 2 using the method of Continued_fraction_representation The square root of 2 is calculated using (1+r) where 1 is the lower ...
1
vote
0answers
12 views

unique ringhomomorphism from the Field of fractions to another field

$R$ is a ring, $L$ a field and $K$ the fraction field constructed from $R$. For any injective ring homomorphism $f=R \rightarrow L$, there is a unique ring homomorphism $\tilde{f}:K \rightarrow L$ ...
1
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1answer
25 views

Partial Fraction Decomposition - Multiple Answers-Question

Now I do understand how partial fraction decomposition works and why you can do it, but there is one case that I don´t understand. And that is, the following: $$\frac{A_{1}}{(x-x_{1})} + ...
1
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1answer
59 views

For which $a,b\in \mathbb{N},$ is $\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is a rational number. [closed]

I found the following problem on a Olympiad question paper: For which $a,b\in \mathbb{N},$ is $$\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$$ a rational number. I am unable to solve it. Any help ...
1
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1answer
19 views

Auto loan calculator website widget

I was going to try my luck on StackOverflow, but I have a feeling my issue here is on order of operations. I'm using the loan calculation found here to build a loan calculator for a clients website. ...
0
votes
1answer
27 views

Total Percentage Change - Why does this work?

Algebraically this works, but I'm looking to understand (1) why it works and (2) if there is a simpler formula. Problem: I want to get total % change over time. Facts: Example Data - Jan 2013 - ...
3
votes
0answers
28 views

Origin/history of mixed number notation with misleading hyphen, e.g. 1-1/2

So there is a system of writing mixed numbers (that is, a combination of whole number and fraction, used instead of an “improper” fraction) used in cases where typing vulgar fractions (e.g. ½) ...
5
votes
2answers
58 views

Can fractions be written as over 1?

I know that all whole numbers can be written as the whole number divided by one. I was wondering if fractions could be written the same way, for example.. Can $1\over2$ be written as $1/2\over1$ ...
0
votes
2answers
38 views

Partial fractions - alternative result

I have the following fraction: $$ \frac{z}{(z-1)(z-2)} $$ When I try to decompose it to partial fractions, I get: $$ -\frac{1}{z-1} + \frac{2}{z-2} $$ But the result in my book is: $$ ...
4
votes
7answers
2k views

When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$…

When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$. What is the denominator of the fraction? I tried, Let the numerator of the fraction be $x$ and the denominator ...
2
votes
1answer
19 views

Partial Fraction Decomposition with complex poles

I have a function which I'd like to perform partial fraction decomposition on, to allow easier inverse laplace transform. $$F(s) = \frac{1}{s(s^2+140s+10^4)}$$ I begin with finding the poles $$s = ...
1
vote
1answer
53 views

Is the answer key wrong? or it's me?

ok, so im reviewing for a math test and the following question is from the practice final exam. Rationalize the denominator in the example: $$\frac{\sqrt {2}}{\sqrt {x-3}}$$ after multiplying both ...
-4
votes
1answer
30 views

How would you solve this problem by multiplying by the reciprocal? [closed]

How would you solve 3/1 ÷ (1/2)/1 ? Please show all the steps clearly! I know you have to multiply by the reciprocal when dividing fractions, however you just get back to the original problem when ...
2
votes
4answers
97 views

Intuitively, why does $\dfrac{a}{c} = \dfrac{1}{\dfrac{c}{a}}$?

For intuition, I reference objects. Imagine making a dessert with: $a$ as apples and $c$ as chestnuts. Question. How and why is $\dfrac{a}{c} \qquad (3) \quad = ...
2
votes
1answer
37 views

Division by rational (decimal) number meaning

When I say, that I exchanged 42 CZK into 1,5 euro. Why do I get the rate for one euro by dividing? 1) How do you explain this division in words. Like when you say when doing integer division, that ...
5
votes
4answers
76 views

Showing associativity of (x*y) = (xy)/(x+y+1)

In order to show something is associative one must show that $(x*y)*z$ = $x*(y*z)$. I want to show that $x * y = \frac{xy}{x+y+1}$ is associative. This is for self-study (I'm learning algebra over ...
1
vote
2answers
226 views

find a value of x in a fraction

Can anyone help me with this? I don't know where to start. I assume there is a trick. Fine the value of x if $$ \frac{1}{1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4 + \frac{1}{x}}} } } \ \ = \ \ ...
-1
votes
2answers
38 views

operations with fractions [closed]

Can anyone help me with this? I don't know where to start. If $$ 1 \ - \ \frac{1}{7 + \frac{1}{7 + \frac{1}{7} } } \ \ = \ \ \frac{1}{a + \frac{1}{b + \ \frac{1}{c + \frac{1}{d} } } } \ \ , $$ what ...
0
votes
1answer
24 views

Can fractions be written with a one under them and can problems be solved this way

How would you write a/.1 ? Are either of these ways acceptable? $$ \frac{a/1}{\frac{1/10}{1}} = \frac{a}{1} \times \frac{1}{1/10} $$ or $$ \frac{a/1}{1/10} = \frac{a}{1} \times \frac{10}{1} $$
5
votes
1answer
96 views

Transformation that preserves an increasing ratio between vectors

Consider two vectors $x=(x_1,x_2,\ldots,x_n)$, $y= (y_1,y_2,\ldots,y_n)$ such that all $x_i,y_i>0$ and \begin{align} \frac{y_1}{x_1}\le \frac{y_2}{x_2}\le\cdots\le \frac{y_n}{x_n} \end{align} Now ...
0
votes
1answer
28 views

Why is the coefficient of a fraction 1/denominator instead of simply 1?

Why is the coefficient of a fraction 1/denominator instead of simply 1? Wouldn't the result be the same?
0
votes
2answers
24 views

What are the steps to simplify the following equation (with simplified version included)

This is part of a question asking to find the differential of a polynomial fraction. I have already taken the derivative using quotient and chain rule, and this is where I am up to. Rather than go ...
0
votes
3answers
54 views

Where does this fraction come from in this integral?

So you have the integral: $$\int\frac{3v}{200 - 4v} dv$$ I tried to do $u$-substitution at first with $u = 200 - 4v$, but I could not get the correct answer which is: $$-\frac{3}{4}v - ...
1
vote
0answers
35 views

How much information is missing?

If we know the value of $\frac{(a-b)}{(c-d)}$, can we calculate the value of $\frac{(a-d)}{(c-b)}$ That is : Let $\frac{(a-b)}{(c-d)}=k$ , can we calculate $\frac{(a-d)}{(c-b)}$ in terms of $k$ And if ...
1
vote
4answers
110 views

Splitting fractions with a linear denominator: $\frac{2x-1}{x+2}$

How can $$\frac{2x-1}{x+2}$$ be split to give $$A-\frac{B}{x+2}$$ where $A$ and $B$ are integers? The solution is $$2-\frac{5}{x+2}.$$
5
votes
5answers
541 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
1answer
21 views

Expressing fractions in different bases. [closed]

In particular, a fraction that, in a certain base, will be recurring - such as $\frac12$ in base 3. (I used a base changing calculator to find it out) I have tried using repeated multiplation as ...
0
votes
3answers
22 views

Need to overcome erroneous result when differentiating natural log of a fraction

I am trying to differentiate the following: $$ln(3x-8/6x+2)$$ my (incorrect) method is: let $$ln(x) = ln(u)$$ therefore when differentiating u.. $$ln(u) = 1/u$$ and diff of$$$$(3x-8/6x+2) = 3/6 = ...
0
votes
0answers
24 views

Let $r$ be a rational no. with $0<r<1$. Then $r$ can be expressed as a sum of reciprocals of finitely many distinct positive integers [duplicate]

Let $r$ be a rational no. with $0<r<1$. Then $r$ can be expressed as a sum of reciprocals of finitely many distinct positive integers . Show that this is true even for $r>1$. If there is a ...
1
vote
1answer
31 views

How to do 24.89 / 26.15 with long division?

How to do 24.89 / 26.15 with long division? Question is to find applied tax if 24.89 is pre-tax and 26.15 is post-tax. I'm aware of how to solve this with a calculator (1-(24.89/26.15))= ~.0482 = ~ ...
37
votes
4answers
12k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
0
votes
2answers
19 views

Integrate algebraic fraction with constant on top?

I understand that if you have $\int \frac{1}{x + 1} dx$ you simply do $\ln(x + 1) + C$. Now I'm slight confused because in my text book, $\int \frac{31}{x - 4} dx$ evaluates to $31\ln(x - 4)$ but ...
1
vote
1answer
21 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (iii) and (iv) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
1
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1answer
26 views

Is this fraction non-terminating?

I recently stumbled upon an observation: the fraction $\frac{x}{y}$ terminates if and only if $y$ only has prime factors $2$ and $5$. For example: $$\frac{1}{20} = \frac{1}{2\cdot2\cdot5} = 0.05$$ ...
-1
votes
1answer
21 views

Partial fractions (How do I get from x to y) [closed]

how do I get from $$\gamma * \left( \frac{\frac{\lambda_0 w}{(1+r)^t \beta^t \alpha}}{\frac{\lambda_0 w}{(1+r)^t \beta^t \alpha}-\frac{\lambda_2}{\beta^t \alpha}} \right)$$ to $$\frac{\gamma r w ...
2
votes
1answer
36 views

Fraction simplification Rules

I am studying for GRE and One of the practice questions is a division. After converting my Mixed numeral I get 90/72 now I just have to simplify. What I understood is that you divide by Least common ...
0
votes
2answers
55 views

Triple fractions (and more complex fractions)

Usually $$\frac{a}{\frac{b}{c}} = \frac{ac}{b}$$ i.e. $b/c$ is seen as the denominator, and $a$ is the numerator. If you have $a/b/c/d$, what do you choose to take as the denominator? ...
0
votes
0answers
15 views

Term for “Remainder in the Whole”

If I have a proper fraction I want to know what the name is for the amount remaining in the whole. So given $\frac1 3$ I want the name of the term $\frac 2 3$.
2
votes
4answers
79 views

Is $1.0000…$ ( $1$ with infinite zeros) greater than $1.0$? [closed]

Given that $0.3333...$ is greater than $0.3$ and similarly $0.777...$ is greater than $0.7$, does it follow that the sum of $0.33...$ and $0.77...$ is greater than sum of $0.3$ and $0.7$?
3
votes
4answers
93 views

Easier way to divide a fraction by a fraction.

Say this is the problem: $$\frac{3/8}{4/5}$$ As of right now, I would multiply both fractions by $40$ then simplify to get $\frac{15}{32}$ Or I would multiply $\frac{3}{8}$ and $\frac{5}{4}$ to ...
21
votes
12answers
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
1answer
39 views

Combined Probability

$$P(A)\frac{2}{3}, P(A | B)= \frac{1}{3}$$ and $$P(A ∪ B)= \frac{4}{5}.$$ Find P(B). I honestly have no idea how to even approach this problem, as I cannot find any helpful online notes on Combined ...
1
vote
0answers
42 views

Reverse engineer numerical results to fractions of remarkable numbers?

Numerical methods output decimal numbers that oftentimes result from the division of two (or more) numbers: $1.5708... = \frac{\pi}{2}$ $0.3679... = \frac{1}{e}$ $0.7071... = \frac{\sqrt2}{2}$ ...
0
votes
0answers
40 views

How to find the common denominator with multiple variables

Find $\frac{zf^{'}(z)}{f(z)}$, where $-1 \leq \alpha \leq 0 $ and $0< v < 1$ Given: $f(z)= \frac{1}{\pi}(-\log (1-vz)+ \alpha \log(1-vz^{-1}))$ and $f^{'}(z)= \frac{1}{\pi}\left(\frac{v}{1-vz} ...
0
votes
0answers
43 views

Show that if $\frac{a}{b} < \frac{c}{d}$, then $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$ [duplicate]

This problem was asked in a 9th-grade textbook: Show that if $$\frac{a}{b} < \frac{c}{d}$$ then $$\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$$ However, I haven't been able to find a ...
-2
votes
1answer
75 views

Does this equation my professor wrote actually work? [closed]

Does this make sense to you? Peter's classroom example $\frac{1}{2^{(\text{any power})}} = \frac{1^{(\text{any power})}}{2^{(\text{any power})}} = (\frac{1}{2})^{(\text{any power})}$ If that's the ...