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

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0
votes
3answers
53 views

Given that $yz:zx:xy = 1:2:3$ and $\tfrac{x}{yz}: \tfrac{y}{zx} = 1:k$, find $k$

Given that $yz:zx:xy = 1:2:3$ and $\dfrac{x}{yz}: \dfrac{y}{zx} = 1:k.$ Find $k$. I understand that $ k = \frac{y^2}{x^2}, y = 1,$ and $x = 2$. Therefore $k = \frac{1}{4}$. This also brings me ...
59
votes
14answers
3k views

Why rationalize the denominator?

In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "there ...
-1
votes
3answers
63 views

How to show that show that $\frac{v+u}{1+ uv/c^2}=c$ when $u=c$?

I am trying to show that $\dfrac{v+u}{1+\dfrac{uv}{c^2}}=c$ when $u=c$. Context It's needed for a physics proof that I'm working on. This is the formula for relative velocity, $u$ represents the ...
13
votes
2answers
183 views

Why do I get $0.098765432098765432…$ when I divide $8$ by $81$?

I got this remarkable thing when I divided $16$ by $162$, or, in a simplified version, $8$ by $81$. It's $0.098765432098765432\cdots$, or more commonly known as $0.\overline{098765432}$, with all the ...
5
votes
6answers
172 views

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$?

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$? I got answer $\frac{1}{\sqrt{2}}$, but the real answer is $\frac{\sqrt{2}}{2}$. Anyway, calculator for both answers return same numbers. ...
0
votes
2answers
87 views

How to simplify this expression with radicals? $3\sqrt2 - \sqrt{32} + \sqrt{\frac{80}{16}}$

I don't understand how I could calculate this: $3\sqrt2 - \sqrt{32} + \sqrt{\dfrac{80}{16}}$ My answer is $-\sqrt2 + \sqrt5$, but the real answer should be $\dfrac{9-4\sqrt2}{4}$.
0
votes
1answer
28 views

How to differentiate for critical points with variable in denominator

sorry for posting a particular problem, this is maybe more of an algebra problem than a calculus problem, but it does involve differentiating so I thought I would state the problem as one. I am ...
1
vote
3answers
129 views

How to solve following limit

I've been struggeling a bit with the following limit: $\lim\limits_{x \to 0} \frac{a- \sqrt{a^2 - x^2}}{x^2}$ The solution is: If a < 0 then -$\infty$ . If a > 0 then $\frac{1}{2a}$ But I don't ...
1
vote
1answer
45 views

Are some infinite fractions in one counting system non-infinite in another?

I'm curious whether some infinite periodic fractions in one counting system (e.g. decimal - 10/3 = 3.33333...) turn out to be non-infinite in another system and vice - versa. Please excuse me if my ...
0
votes
3answers
38 views

Split a whole number into fractions, find which of thoses fractions another belongs in

I want to split a number into $n$ parts and then take another number (which is less than or equal to the first number) and see which of the fractions the other number belongs to. For example, split ...
2
votes
2answers
253 views

Determine if $\frac{k-1}{k}+\frac{1}{k(k+1)}=\frac{k}{k+1}$ holds

How to prove if the following equality holds? $$\frac{k-1}{k}+\frac{1}{k(k+1)}=\frac{k}{k+1}$$ Maybe finding a common denominator would work, but I have no idea how to do it in this example. I see ...
1
vote
1answer
43 views

What is the chances of a duplicate in this equation

I'm not very good at math; However I have a scenario where I'm trying to find the chance of duplicate for randomly generated data. In a nuttshell I have a "bag" with 62 different items, lets say a ...
0
votes
3answers
34 views

How to find the value of $a$ and $b$ from this limit problem with or without L'Hopital's formula?

Consider the limit: $$\lim_{x\to4} \frac{x^2+ax+b}{x-4} =14$$ Question: How can I find the values of $a$ and $b$? Attempt: My first thought is, we need to use L'Hopital's rule to make sure ...
5
votes
4answers
318 views

Find the fraction where the decimal expansion is infinite?

Find the fraction with integers for the numerator and denominator, where the decimal expansion is $0.11235.....$ The numerator and denominator must be less than $100$. Find the fraction. I ...
15
votes
1answer
245 views

Closed-form of infinite continued fraction involving factorials

Is there a closed form of this: $$ 1!+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}} $$
1
vote
1answer
48 views

Can two integer divisions be unified above one whole fraction line?

Is there a way to combine two integer divisions (i.e. division with the result rounded down to the nearest integer) into a single division operation? What I mean is that, when working with with real ...
12
votes
4answers
1k views

Is Cantor's diagonal argument dependent on the base used?

Applying Cantor's diagonal argument to irrational numbers represented in binary, one and only one irrational number can be generated that is not on the list. Wikipedia image: But if you change ...
0
votes
1answer
19 views

Help clarifying fraction and percentage question.

I have the following problem: Convert the mixed number $18 \frac 25 \%$ to an improper fraction, then, use the definition of percent to convert to fractional notation. If I follow the steps of ...
3
votes
7answers
180 views

Why Is $y^{-1}$ = $\frac{1}{y^1}$?

Basically, I'm asking 'Is there any place where I can access a compendium of formal mathematical proofs'? I need to know what processes mathematicians went through to declare $(-1)(-1)=1$ and so on. I ...
0
votes
1answer
30 views

Relating an expression to two similar ones

Is it possible to express $C$ solely in terms of $A$ and $B$, where $$A = \dfrac{m}{x+z}, B = \dfrac{n}{y+z}, C=\dfrac{m+n}{x+y+z}$$ and $m,n,x,y,z>0 \ ?$ If not, how close can I get?
1
vote
2answers
60 views

Hard problem with fractions

I can't solve the following problem. A person is $x$ years old. Find his age if the following is true. In a group of $x$ people each one started taking pictures of each of the others. At some point ...
0
votes
1answer
28 views

Basic solving for fractions

Can someone help me with this? I am a beginner to physics and in a question i need no isolate a constant to solve for another one. A/B=C/D solve for D step by step please.
0
votes
2answers
42 views

How to re-write one fraction as two others.

I have the two following fractions. $$ \dfrac{A}{Bx^{\alpha+1}}$$ and $$ \dfrac{C}{Dx^{\alpha+\beta}}$$ The form i want $$ \dfrac{E}{Fx^{\alpha+\beta+1}}$$ I was thinking to do partial fractions or ...
2
votes
4answers
81 views

How to extract fraction from a floating point number

I'm making some tests with float type (floating point number) with programming and in some of my tests I need to extract the fraction that originates the float value. Let $ x $ be a floating point ...
-1
votes
1answer
40 views

How is it possible that when you divide 1 by 9,899, you get two-digit Fibonacci numbers also being carried, etc.? [duplicate]

When I divided $1$ by $9,899$, I got two-digit Fibonacci numbers also being carried: $0.0001010203050813213455904636\dots$ When I divided $1$ by $89$, I got one-digit Fibonacci numbers at the ...
0
votes
0answers
108 views

How is it possible that you get two digit numbers multiplied by three at the beginning if you divide 1 by 97, etc.? [duplicate]

I divided $1$ by $97$ and I got this: $0.0103092783\dots$; I got two digit numbers being multiplied by three at the beginning. I divided $1$ by $997$ and I got a similar thing: ...
1
vote
3answers
111 views

What is $\frac{9}{3} - \frac{1}{2}$?

I need to compute $\frac{9}{3} - \frac{1}{2}$. I got an answer of $\frac{8}{6}$ but that is incorrect. $\frac{5}{2}$ is the correct answer. How is this possible?
0
votes
0answers
12 views

What can we say about the function $f(x)$ in this case?

Alright, I'm little bit confused about what's happening here to the function $f(x)$, i thought that the formula of $f(x)$, have nothing to do with its behavior or domain. there are two or many ...
0
votes
2answers
17 views

Determine rational expression

Can somebody help me with the following word problems: Problem #1 Russel's combine can clear a field in 24 tractor hours. Jerome's combine can clear the same field in 30 hours. If they work ...
2
votes
1answer
56 views

Rewriting to quintic equation

Consider the force $F_\Omega$: $$ F_\Omega= \Omega^2\left(x -\frac{\beta\left(x+\alpha R\right)R^3}{\left(\left(x+\alpha R\right)^2\right)^{3/2}} -\frac{\alpha\left(x-\beta ...
1
vote
1answer
47 views

$(-3)^{3/2} \neq (-3)^{6/4}$

$(-3)^{\frac{3}{2}}=-3\sqrt{3}i$ $(-3)^{\frac{6}{4}}=\sqrt{27}$ (not the same thing). What's the deal? It's interesting because people work with fractional exponents all the time and I've never ...
0
votes
1answer
27 views

Question on three algebraic expressions

Assume that $a < b$ , and both are natural numbers (like 2 or 4). Is $a/b$ * $a/b$ more, less or equal to $a/b$ ? Is $a/b$ / $a/b$ more, less or equal to $a/b$ ? Is $ab/a$ - $ab/b$ more, less, ...
0
votes
1answer
20 views

Algebra - fraction problem

"The cooler in a car contains $8$ litres. The coolant fluid contains $\dfrac3{10}$ of glycol and rest is water. To increase the glycol content to $\dfrac35$ you drop some of the coolant fluid and fill ...
1
vote
3answers
73 views

easier way to decompose fraction into partial fraction

It is a question in a test, and I couldn't manage to complete it. Given a complex fraction $\frac{1}{(z-1)^3(z+1)^3}$, we know that it can be decompose into ...
1
vote
2answers
36 views

Algebra problem with fractions

"In a musical class, the students either played piano or violin as head instrument. By a concert, the students got to choose whether they would do a solo or pair-performance. A piano player can only ...
5
votes
5answers
108 views

Calculating the value of $\frac{a-d}{b-c}$

If $\frac{a-b}{c-d}=2$ and $\frac{a-c}{b-d} = 3$ then determine the value of: $$\frac{a-d}{b-c}$$ Where $a,b,c,d$ are real numbers. Can someone please help me with this and give me a hint? I tried ...
0
votes
1answer
34 views

Calculate discount needed in order to achieve required profit margin. (algebra with fractions)

This problem seems to require algebra with fractions. This is basic high school (or even middle school?) stuff, but embarrassingly, I seem to have forgot it. Background I own a small business and ...
0
votes
2answers
37 views

Writing the sum of two rational functions as a single rational function.

Write as a single fraction: $$\frac{2x}{x-1} - \frac{x}{x+1}$$ Please can somebody talk me through this question as I don't understand how to get a common denominator. Thank you.
1
vote
2answers
54 views

limits question with radicals, rationalizing

Find the limit value Here's what I did (Above) I think I can rationalize the numerator to solve it, but I'm having trouble rationalizing numerator, when I'm usually rationalizing the denominator. ...
0
votes
0answers
23 views

Freshman sum related question.

I want to prove the following: If $\frac{u_1}{d_1} > \frac{u_2}{d_2} > \frac{u_3}{d_3} > \frac{u_4}{d_4}$ and $d_1 + d_3 > d_2 + d_4$ and $u_1 > u_2$ then $u_1 + u_3 > u_2 + ...
2
votes
2answers
33 views

Can ratios really be manipulated as fractions?

In high-school Maths, we were taught that it was possible to manipulate ratios as fractions. For example, $$ 1 : 7 = 3 : x \\ \frac{1}{7} = \frac{3}{x} \\ \frac{x}{7} = 3\\ x = 3 \times 7\\ x = 21\\ ...
1
vote
2answers
62 views

Is there an operator for adding the numerator and denominator of a fraction separately?

Numbers in the Farey sequence are expressed as fractions e.g $F_5$: $0\over1$ $1\over5$ $1\over4$ $1\over3$ $2\over5$ $1\over2$ $3\over5$ $2\over3$ $3\over4$ $4\over5$ $1\over1$ All of the $n\over5$ ...
0
votes
2answers
52 views

What is the property that allow the transformation $\frac{16a^3}{8ac}=\frac{16}8\cdot\frac{a^3}a\cdot\frac1c$?

In a monomial division like this: $$\frac{16a^3}{8ac}=\frac{16}8\cdot\frac{a^3}a\cdot\frac1c$$ Why I can do this $\dfrac1c$? Where this 1 come from?
5
votes
6answers
344 views

The steps of simplifying a fraction?

So I'm in an Adult Education class for my GED and I'm trying hard to study on my Math which is the only subject I have trouble with. I only have "barely" a 6th grade education to Math so I'm having a ...
0
votes
3answers
45 views

Basic algebra/fractions: derivation

How do I do this derivation step? I don't understand why there is equality. The derivation is from my textbook. $$mg-m\left(\frac{g}{1+\frac{M}{2m}}\right)=\frac{mg}{1+\frac{2m}{M}}$$
0
votes
1answer
26 views

Trying to figure out a Mathematical pattern

I am given a start point, a, an end point b, and a number of values x. With that I am supposed to come up with the points between the start and end point. Below is an example ...
6
votes
5answers
612 views

How do you solve a logarithm with a non-integer base?

How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example: $$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$ $$\log_{0.5}8 = -3$$ ...
45
votes
3answers
5k views

Why do we miss 8 in 0.012345679…, 98 in 0.0001020304050607080910111213…, and so on in fractions like 1/81, 1/9801, and so on?

I've seen this happen that when you divide by a fraction using the square of any set of nines in the denominator depending on how many there are like ${1\over 99^2}={1\over 9,801}$, you get ...
1
vote
1answer
54 views

Find the value of x

$$\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\ldots\left(1-\frac{1}{2011^2}\right)=\frac{x}{2\cdot 2012}$$ Help me solve this, obviously there is a shortcut ...
0
votes
2answers
34 views

How to find the value of $\sum_{k=1}^\infty (\frac{1}{9})^k$ using partial sums?

So I was trying to prove an infinite sum by looking at the partial sum, when I ran into a problem. Consider: $$\sum_{k=1}^n \left(\frac{1}{9}\right)^k = \frac{1}{8} 9^{-n} (9^n-1)$$ but as there are ...