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

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0
votes
2answers
18 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
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
18 views

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

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
31 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 ...
1
vote
1answer
19 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 ...
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 $\frac 2 3$.
0
votes
2answers
52 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? ...
2
votes
4answers
75 views

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

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 $0.7$?
2
votes
4answers
68 views

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

To discover the intuition, I refer to concrete objects: define $a$ as apples and $c$ as children. Question 1. How and why is $\dfrac{a}{c} \qquad (3) \quad = \quad\dfrac{\color{red}{1}}{\dfrac{c}{a}} ...
0
votes
1answer
36 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
41 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
73 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 ...
0
votes
3answers
65 views

How to divide the fraction $1/1+1$?

This has to do with re-calculating the sigmoid function in ai. It isn't really important, but the simplest way to put it is I need a math guru to help my monkey brain do this: $$\frac{1}{1+e}$$ to ...
4
votes
2answers
64 views

If $x+y+z=0$, prove that $\frac{x^2}{2x^2+yz}+\frac{y^2}{2y^2+zx}+\frac{z^2}{2z^2+xy}=1$

A problem in my homework had asked me: When $x+y+z=0$, evaluate$$\frac{x^2}{2x^2+yz}+\frac{y^2}{2y^2+zx}+\frac{z^2}{2z^2+xy}$$ Without too much difficulty, one can see that the value should be ...
0
votes
1answer
31 views

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$. How come the answer is left as $\frac{5x}{(x+2)(x-3)} + \frac{4}{(x+2)^2}$. Why don't we go any further?
1
vote
2answers
57 views

Ratio Inequality

How can I prove that, $$\frac{a_{1}+a_{2}+\dots+a_{n}}{b_{1}+b_{2}+\dots+b_{n}} \le \max_i\left\{\frac{a_{i}}{b_{i}}\right\}$$ where $1 \le i \le n$, and $a_{i} \neq a_{j}$ and $b_{i} \neq b_{j}, ...
0
votes
1answer
19 views

Finding the number of chairs

In a school hall, $7/31$ of chairs is arranged in rows of 5, $11/31$ of chairs is arranged in rows of 13. The rest of chairs are not used. The total number of chairs does not exceed 4000. Find the ...
3
votes
4answers
91 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 ...
1
vote
2answers
22 views

Which rules are used to make function like one in Laplace Transformations table?

I have function like this: $$\frac{s^2+3s+3}{(2s^2+7s+7)} $$ It needs to be brought to the level of Laplace Transformations from table, like these two: $$\frac{a}{(s-b)^2 + a^2} $$ ...
1
vote
2answers
27 views

Fractions (Dividing… Maybe?)

The question... (With mixed fractions) (3)3/4 is bigger than (1)19/21 How many times bigger? I assume you divide the first fraction by the second but I cant seem to do it, could someone maybe answer ...
0
votes
1answer
36 views

Is there a better way of simplifying fractions?

The way I teach my students to simplify fractions is to first write the numerator and denominator as a product of prime numbers, and then cancel. For instance: $$\frac{15}{20} = \frac{3 \times 5}{2 ...
2
votes
0answers
37 views

Pull constant out of a summation of fractions

General problem $$ \sum_{i=1}^n \frac{a_i + x}{b_i + x} = 0 $$ Is it possible for solve for $x$? Some context I've hit a road block in my derivation... At this point, I need to pull the model ...
1
vote
1answer
18 views

(Straight line) Gradient for (-4, 0) (0, 2.5)

So I changed the question the textbook gave me to (-4,0) (0, 5/2). The question asked me what is the gradient for the X and Y.I was doing the question without a calculator and the answer I got was ...
0
votes
2answers
23 views

Cancellation fractions; why this is not equal to this?

I'm very bad with math, i will go right to the question: Why i can't make this cancellation? (or why this is False, I tested this in symbolab and gave false)... $$\frac{3^{-m}\cdot ...
0
votes
1answer
28 views

Square Root of Rational Number $\frac{A}{B}$

Here's the question: Let $x=\frac{A}{B}$ be a positive rational number in lowers terms (i.e., $A, B\in\mathbb{N}$ and $hcf(A,B)=1$). Prove that $\sqrt{x}$ is rational if and only if $A$ and $B$ are ...
0
votes
0answers
29 views

if $a/b$ and $c/d$ are consecutive fractions in a list, then prove $bc-ad=1$ .

Let n be a fixed positive integer, and suppose we list in increasing order all numbers $a/b$ where $1 ≤ a, b ≤ n$, and the fraction $a/b$ is in lowest terms. Show that if $a/b$ and $c/d$ are ...
1
vote
0answers
27 views

How to make continued fractions of any number?

I recently found an continued fraction representation of $\pi$, and I wondered how can I make an continued fraction that converges into a number? The MAIN question is: how do you make a continued ...
0
votes
1answer
22 views

Numerical analysis: what is the error term for the rule…?

The question goes: derive the error term for the rule $phi$ to approximate the third derivative of f(a). I have attached a screenshot I understand how to take the Taylor series in the hint, but the ...
1
vote
1answer
34 views

Is there a positive integer n such that the fraction $(9n+5)/(10n+3)$ is not in the lowest term?

Is there a positive integer n such that the fraction $(9n+5)/(10n+3)$ is not in the lowest term? Please explain. I found $n=2$ be a solution. Is it correct?
0
votes
2answers
28 views

Fractional Exponents - Is the sign discarded?

For example, 16^(3/4) Is the accepted as both -8 and 8 or just 8? I ask this because on an AS maths mark scheme it says to condone -8 Thanks
0
votes
2answers
29 views

How does this numerators and denominators relate with the fraction? [duplicate]

Suppose if we have two fractions $\frac{a}{b}$ and $\frac{c}{d}$ then how are their values related with the fraction $\frac{a+c}{b+d}$ ? I have observed this inequality: ...
0
votes
1answer
27 views

One tap fills a pool. The other one empties it. It's a word problem.

In a pool there are two taps, one for filling and one for emptying. Once, when the pool was empty they opened the filling tap for $4$ hours. Afterwards, they opened by mistake the emptying tap and ...
2
votes
3answers
29 views

Find $\frac{y}{x}$ from $3x + 3y = yt = xt + 2.5x$

I need to find the ratio of $$\frac{y}{x}$$ If given that $$3x + 3y = yt = xt + 2.5x$$ So what I tried is: $$t = \frac{3x + 3y}{y}$$ And then put it in the equation $$\frac{x(3x + 3y)}{y} + 2.5x ...
1
vote
2answers
49 views

Resolve $ \frac{120}{x+y} + \frac{60}{x-y} = 6;\,\frac{80}{x+y} + \frac{100}{x-y} = 7$

I want to resolve this system of equations: $$\begin{cases} \frac{120}{x+y} + \frac{60}{x-y} = 6 \\\frac{80}{x+y} + \frac{100}{x-y} = 7\end{cases}$$ I came to equations like $$x - \frac{10x}{x-y} + ...
3
votes
3answers
453 views

Find equation for mass in gravity

A satellite is moving in circular motion round a planet. From the physics we know that $$\Sigma F_r = ma_r = \frac{GMm}{r^2}$$ So I wanted to find the equation for $M$ knowing also that $$v = ...
0
votes
0answers
12 views

Find smallest counter and denominator of fraction to approach given real number

What is the smallest pair $(a,b)$ that serves the following condition: $$ \frac{\left| \frac{a}{b} -x \right|}{x} \leq p \\ a,b \in \mathbb{Z} \quad \text{and} \quad p,x \in \mathbb{R} $$ $x$ : the ...
4
votes
2answers
55 views

Simplifying Fractions involving negative numbers

I want to simplify $$\frac{\frac{7}{-10} \times \frac{-15}{6}}{\frac{7}{-19} + \frac{-17}{-8}}$$ I really don't understand how to do this, or even how to start? Negative numbers make it even harder ...
1
vote
4answers
51 views

What exactly goes raising a number to a fraction mean?

I apologise for asking something so fundamental, but what exactly does $$2^\frac{2}{5}$$ actually mean? I get raising a whole number to another whole number $$x^y$$ means you are multiplying x with ...
2
votes
3answers
51 views

Can someone explain why this happens? (Dividing variables with exponents)

Alright, so let's say I have $$\frac{x^{-6}}{-x^{-4}}$$ The answer is $\dfrac{1}{x^2}$, but why isn't it $\dfrac{1}{-x^2}$?
0
votes
3answers
35 views

Comparing two fractions

I saw this problem from an elementary textbook: Let $$ A = \frac{2014}{2015} + \frac{2015}{2016} $$ and $$ B = \frac{2014 + 2015}{2015 + 2016} $$ Compare $A$ and $B$. I know the answer is $A ...
0
votes
4answers
48 views

Prove a sum of fractions less than a value

I happened to see this problem from an elementary school textbook, but cannot solve it: $$ C = \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + ... + \frac{1}{15} + \frac{1}{16} + \frac{1}{17} $$ Prove $$C ...
1
vote
3answers
34 views

fraction division understanding

Want to visualize rule division of fraction. For example 1) 2 2 4 _ * _ = _ 2 3 6 in this case we "split" each piece of cake in numerator to the ...
1
vote
2answers
75 views

Is the ring $\mathbb{Q}$ isomorphic to $\operatorname{Frac}(\mathbb{Q}[x])$?

Is the field of rational numbers $\mathbb{Q}$ isomorphic to the fraction field $\operatorname{Frac}(\mathbb{Q}[x])$? Both are fields, I can't disprove it by some algebraic properties that hold for ...
2
votes
2answers
31 views

Square root fraction confusion

I was doing math for school and got to something that really confused me. With having the rule $\frac{2}{4} = \frac{4}{8}$ (or some simular fraction equation) in mind, I got to the following confusing ...
5
votes
2answers
83 views

If $\frac1x-\frac1y=\frac1z$, $d=\gcd(x,y,z)$ then $dxyz$ and $d(y-x)$ are squares

Let $x, y, z$ be three non negative integer such that $\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{1}{z}$. Denote by $d$ the greatest common divisor of $x, y, z$. Prove that $dxyz$ and $d(y-x)$ are ...
0
votes
3answers
67 views

Express $w$ and $1/w$ for $w=\frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3}$ in the simplest form with a rational denominator [closed]

If $w = \frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3} $ Express the following in the simplest form (with a rational denominator) i) $w$ ii) $\frac1w$ I'm confused about (ii) question :/ pls help me.
0
votes
0answers
44 views

Express $w$ and $\frac1w$ for $w=\frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3}$ in the simplest form with a rational denominator [duplicate]

If w = $\frac {\sqrt2+\sqrt3}{\sqrt5-\sqrt3} $ Express the following in the simplest form (with a rational denominator) i) w ii) $\frac {1}{w} $ It must be like this right ? w = $\frac ...
1
vote
1answer
29 views

When graphing both X, and Y are fractions

In my instructions, I am told to place the point on the coordinate system. My X, and Y value are $$(\frac 52, \frac 72)$$ at this point I am a little lost. Would I flip it, and multiply it like so? ...