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

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1
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2answers
65 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. ...
2
votes
2answers
35 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
82 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
53 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?
6
votes
6answers
450 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
57 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
28 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
708 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: $$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$ $$\log_{0.5}8 = -3$$ ...
47
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
36 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 ...
0
votes
3answers
49 views

Finding out how out how much is 100%

I know 14% is 41. So how much is 100%? I know it is simple math, but... you know how it is being out of school for a few years. If at all possible, I'd like to see the equation too.
2
votes
3answers
74 views

Show that for every integer $n ≥ 1$, $1 + \frac{1}{4} +\frac{1}{9} + · · · + \frac{1}{n^2} ≤ 2 − \frac{1}{n}$

I can just think of trying to prove $\frac{1}{4} +\frac{1}{9} + · · · + \frac{1}{n^2} ≤ \frac{1}{n}$, but remains stuck.
0
votes
3answers
47 views

Trying to subtract 2 fractional

I'm trying to solve $f(x)=0$ for $x$, but I can't figure it out as I have to get both the denominators to become for instance $8x$, and then only 1 numerator has $x$ in it. How can I solve this? $$ ...
13
votes
11answers
5k views

Dividing by 2 numbers at once, what is the answer?

Let's say i have 4/1/5. or 4 divided by 1 divided by 5. Are there any rules that i am allowed to use to stop any mistakes?, for example this has 2 solutions, 4/5 , and 20. Edit: Thanks for your ...
1
vote
4answers
46 views

Rewrite fraction to calculate limit

I am practising finding limits. However, I can't seem to figure out this one. $$f(x) = \frac{x^3 + 4x - 5}{x^2-1}\text{ as $x$ goes to $1$}$$ I understand I have to rewrite the fraction somehow for ...
3
votes
5answers
235 views

How do I solve this fraction addition problem?

$4\frac{2}{9} + -9\frac{1}{2}$ yeilds result of $-5\frac{13}{18}$ but WolframAlpha says the answer is $-5\frac{5}{18}$ fixed.
1
vote
1answer
42 views

Instead of mid-point, how do i find a third of the way up the line instead.

I'm working with the program Maya, and i need to script a way to find various fractions up a line for each 3D vector. For example, lets say I have the vector (0, 0, 0), and the vector (12, 12, 12). ...
0
votes
2answers
84 views

Simplifying fractional exponents

I am very confused about the following: whenever I put in into wolfram alpha the answer it gives me is "indeterminate", is it not possible to simplify fractional exponents or something? if the ...
2
votes
3answers
53 views

Multiplying fractions to answer story problems.

My daughter had a math question about finding how far someone walked using multiplication of fractions. The distance was $1 \frac 78$ and he walked $\frac 23$ of the way. The problem wanted to ...
2
votes
4answers
75 views

How do you add two fractions?

I have a fraction I am trying to solve. I know the answer already, as Wolfram says it is $\frac{143}{300}$. The fraction is: $$\frac{5}{12} + \frac{3}{50} = \space ?$$ Please explain why and how your ...
0
votes
2answers
67 views

How do you solve this fraction?

The problem is : $ \frac{2}{3} + \frac{-1}{16} $ The answer I got was $ \frac{1}{48} $ . I believe it to be incorrect. Three does not divide into $16,$ so I cross multiplied. What am I doing wrong? ...
0
votes
1answer
36 views

Regarding +/- fractions: what are some mental tests you can apply to uncommon fraction denominators?

When adding and subtracting fractions: what if there is no uncommon factor (for example 4=2,2 and 6=2,3). Does that always mean to use the LCM? What if the LCM is too big or time consuming to ...
0
votes
1answer
71 views

Why is (n+1)/2n = 1/2 + 1/n, and not 1/2 + 1/2n?

If I factor $(n+1)/2n$ by $n$, I get $n(1+1/n)/n(2)$. Simplifying, I end up with $(1 + 1/n) / 2$. This can be rewritten as: $1/2 + (1/n)/2$, which would give me $1/2 + (1/n) \times (1/2) = 1/2 + ...
1
vote
3answers
64 views

Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $

How can I prove that, for $a,b \in \mathbb{Z}$ we have $$ 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 \, ? $$ Here, $\left \lfloor\,\right ...
1
vote
2answers
92 views

Is there a faster way to add/subtract fractions then having to draw a factor tree each time?

Do you really have to draw a factor tree and work with primes every time you encounter adding or subtracting fractions? Not this way - LCM(8,15)... ...
0
votes
3answers
79 views

What is the correct way to divide a fraction?

This is a very basic question, but i'm struggling with it. Can someone explain the rules when dividing a fraction like this: $$\frac{\cos(\pi x)\sin(\pi x)}{\Large{\frac{\cos(\pi x)}{\sin(\pi x)}}}$$ ...
1
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2answers
47 views

Evaluating a limit as $x \to -\infty$ of a power of a rational function

Sorry for the weird title, I don't know how to put the equation on the title. $$\lim_{x\to-\infty}\left(\frac{1-x^3}{x^2+7x}\right)^5$$ Ok I divided inside the parenthesis by $x^2$, but now I am ...
0
votes
1answer
50 views

How to reduce this fraction?

I am trying to reduce the following fraction to its simplest form: $$\frac{2121212121210}{1121212121211}$$ Can someone please help me? Thanks.
0
votes
1answer
44 views

Calculate $X$ of a math problem

I am trying to learn some more math and I got stuck on this: $$\frac{0.2}{X} = 140$$ How do I calculate $X$? EDIT Sorry I meant to calculate $$\frac{28}{X} = 140$$ So that $X = 0.2$, but how do I ...
1
vote
1answer
94 views

Is there a simple closed form of $|\alpha(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor) + \beta(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor)|$?

Let $d_n(x)$ denote the $n$'th digit after the decimal point in $x$. Examples: $d_8(e) = 2,\;d_5(\pi) = 9$ $\alpha(x)$ and $\beta(x)$ are defined this way: $$d_n(\alpha(x)) = \left\{ ...
3
votes
4answers
95 views

Is there an example to demonstrate why $\frac{1}{(1/2)}$ equals $2$?

To explain why $\frac{1}{2}=\frac{2}{4}$ I use slices of pizza and show how eating one slice of a pizza cut in half is the same thing as eating two slices of a pizza cut in quarters. Is there a way ...
1
vote
1answer
46 views

The elementary question on sign of Rational numbers

The under picture show that $$+\dfrac{8}{3}=\dfrac{+8}{3}$$ Similarly we can show $-\dfrac{8}{3}=\dfrac{-8}{3}.$ Now How do can show that $$-\dfrac{8}{3}=\dfrac{8}{-3}?$$
3
votes
2answers
77 views

Simplifying a square root fraction

Simplify the following $$\frac{\sqrt{3}}{\sqrt{2}(\sqrt{6} - \sqrt{3})}$$ Apparently the answer is $\frac{1}{2} (2 + \sqrt{2})$ but can't for the life of me see how to get it. Any help is massively ...
0
votes
1answer
45 views

Sum of positive integers estimating sum of fractions

Given $m$ fractions adding up to an positive integer $n$ For example: $m=3\\n=10=\frac{30}{6}+\frac{20}{6}+\frac{10}{6}$ How can we find $m$ positive integers that sum to $n$ (a partition of $n$), ...
2
votes
3answers
113 views

What is the meaning of $dy=dx^2$?

When I read the mathematical analysis ,I think if the differential is $dy=Adx^2$ $A$ is a function about x, what will happen? Maybe, it is not proper defined ,but I think the "function" meet ...
0
votes
2answers
132 views

Convert a fraction to infinite repeating decimal?

We all know how to convert an infinite repeating decimal to fraction. It is simple. But now I have these fractions 10/23, 3/29, etc. I know these fractions can be written in infinite repeating ...
1
vote
1answer
56 views

Fractions and big $O$ notation.

I need to find the principal part of the holomorphic functions: $$ \frac{1}{1-\cos z},\,\,\text{in $z_0=0$, and}\,\, \tan z\,\,\text{in $z=\pi/2$} $$ So using the Taylor series of $\sin z$ and $\cos ...
0
votes
1answer
226 views

Show the minimum value for v

Really struggling with this question... If $$v=\frac{(L\cdot V1-V1\cdot x+V2\cdot x)\cdot R}{2Lrx-2rx^2+LR}$$ Prove that the minimum values ($x>0$) for $v$ will occur at: ...
1
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3answers
34 views

Why is $1 \times 3 \times 5 \times \cdots \times (2k-3) = \frac{(2k-2)!}{2^{(k-1)}(k-1)!}$

In order to find out the Catalan numbers from their generating function you have to evaluate the product above. Here is what I thought: \begin{align*} 1 \times 3 \times 5 \times...\times (2k-3) ...
1
vote
3answers
84 views

How would I find the decimal expansion of $1/99^2$

I want to find the repeating decimal expansion of $1/99^2$. All I know is that $1/99 = 0.010101\cdots$. How would I continue?
1
vote
4answers
73 views

$\text{If } |z_1| = |z_2|, \text{ show that } \frac{z_1 + z_2}{z_1-z_2} \text{is imaginary.} $

$\text{If } |z_1| = |z_2|, \text{ show that } \frac{z_1 + z_2}{z_1-z_2} \text{is imaginary.} $ The first thing I tried to do was to multiply both top and bottom by the conjugate of the denominator... ...
2
votes
1answer
231 views

Equation simplification, can't get it right

$$\frac{1}{\frac{x-1}{x+2}}-\frac{2}{x^2-1}$$ should be simplified into $$\frac{x^2+3x}{x^2-1} \quad .$$ However, when I try to do it (tried several times), I fail to get it done right: ...
4
votes
3answers
86 views

Prove the following fraction is irreducible

Prove $\frac{21n + 4}{14n + 3}$ is irreducible for every natural number $n$. I was thinking of taking a number-theory based approach. Can you suggest the following method Calculus/Number theory ...
0
votes
1answer
28 views

How to label a tenth of a second properly in a graph?

I'm making a graph for a science class, and the x-axis represents every tenth of a second. What's the best way of labeling that axis other than "time (one tenth of a second)", or is that the best way? ...
2
votes
1answer
30 views

Finding the limit by factoring the denominator and canceling

I have the problem $$\lim_{x\to10} \frac{x-3}{x^2+7x-30}$$ If I factor it to $\dfrac{x-3}{(x+10)(x-3)}$ then $x-3$ cancels and I'm left with $0$. I know the real answer is $1/20$, but why is zero ...
1
vote
1answer
40 views

Simplifying an expression written as the sum of three fractions

Specifically, I don't know what to do first given the following expression: $$ \frac{4x - 2}{6} - \frac{2 - x}{4} + \frac{x + 3}{3} $$ So I think of it as $\frac 16(4x-2) - \frac 14(2-x) + \frac ...
3
votes
2answers
64 views

Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$

If $x.y.z=1$ show that $\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}=1$ My attempt - L.H.S$=\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}$ $=\dfrac y{y+xy+1}+\dfrac ...
1
vote
3answers
44 views

Show that $\frac{(b+c)^2} {3bc}+\frac{(c+a)^2}{3ac}+\frac{(a+b)^2}{3ab}=1$

If $a^3+b^3+c^3=3abc$ and $a+b+c=0$ show that $\frac{(b+c)^2} {3bc}+\frac{(c+a)^2}{3ac}+\frac{(a+b)^2}{3ab}=1$
0
votes
2answers
99 views

Parametric solution of the Diophantine equation $\frac{1}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z} ,x,y,z∈Z^+.$

I have prove that, for any given positive integer $p,$ parametric solution of the Diophantine equation $$\frac{1}{p}=\frac{1}{x}+\frac{1}{y}$$ can be written in the form $x=ac(a+b),y=bc(a+b),$ where ...