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

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10
votes
2answers
355 views

Sum of series with binary parity in the numerator

I'm now stuck with this question, and I don't even know where to start: Find sum of series$$\sum_1^\infty \frac{f(n)}{n(n+1)}$$, where f(n) - number of ones in binary representation of n. I wish I ...
0
votes
1answer
37 views

$5.30$ converted to a fraction or mixed number in lowest terms

$5.30$ converted to a fraction or mixed number in lowest terms The correct answer they got on my worksheet is $5 \frac3{50}$, but I get $5 \frac{15}{50}$.
1
vote
3answers
75 views

How to put a fraction in simplest form, such as $140/255$?

Given the fraction $$\dfrac{140}{255}$$ How do I find a common factor so it can be easily simplified? I have already tried $2$, $3$ and $4$.
2
votes
4answers
62 views

Why does the least common denominator work?

Take for instance the following problem. You have two beakers of the same height. One has tick marks that break it into thirds. The other has tick marks that separate it into fourths. The water levels ...
4
votes
5answers
1k views

why is PI considered irrational if it can be expressed as ratio of circumference to diameter? [duplicate]

Pi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where both ...
-1
votes
2answers
31 views

Fraction confusion

I read in a set of memoranda that if $ \frac{b-x}{x}=\frac{b}{a}$, then $$x = \frac{ab}{a+b}$$ How is this true? I tried working it out but I could not understand. Please help.
0
votes
1answer
54 views

How can I integrate $\int{1\over 2x+2}$

$$\int{1\over 2x+2}$$ Method 1 $$\int{1\over 2x+2} = \frac 12\int{1\over x+1} = \frac 12 ln(x+1) + c $$ Method 2 $$\int{1\over 2x+2} = \frac 12\int{2\over 2x+2} = \frac 12 ln(2x+2) + c $$ Wolfram ...
4
votes
6answers
123 views

Find $\lim_{x \to \infty} \left(\frac{x^2+1}{x^2-1}\right)^{x^2}$

How to calculate the following limit? $$\lim\limits_{x \to \infty} \left(\frac{x^2+1}{x^2-1}\right)^{x^2}$$
7
votes
3answers
113 views

Show that $\frac{1}{a}+\frac{1}{b}\not=\frac{1}{a+b}$

Problem Assume that $a,b\in\mathbb{R}-\{0\}$ and that $a+b\not=0$. Prove that $\frac{1}{a}+\frac{1}{b}\not=\frac{1}{a+b}$. My Proof Let's assume that $\frac{1}{a}+\frac{1}{b}=\frac{1}{a+b}$, then ...
0
votes
3answers
69 views

Find the fractional representation $p/q$…

Been trying to get some sort of solution for this for hours now, with no avail. Find the fractional representation $p/q$, with $p \in \mathbb{N}$ and $q \in \mathbb{N}$, of the rational number whose ...
2
votes
3answers
68 views

If an object halves its speed every second (but never gets to 0), will it eventually get from point A to point B?

There is a ball that starts at point A on a line and moves toward point B. Every second, it moves half of the distance left, but never stops moving: Etc. Would the ball ever reach point B? In one ...
0
votes
4answers
41 views

infinity sum of the fractional

Can anyone explain how to simplify $ \frac{2}{3} + \frac{6}{9} + \frac{12}{27} + \frac{20}{81} + \frac{30}{243} + . . . $ I have no any idea since i dont have pattern i can't do it with integral or ...
5
votes
0answers
50 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
0
votes
1answer
45 views

Fraction in other bases

How to convert a base 10 fraction into fraction in other bases?. For example base 10 fraction 17/94, How we convert this 17/94 into base 2 fraction ?
1
vote
4answers
115 views

How come $\left(\frac{n+1}{n-1}\right)^n = \left(1+\frac{2}{n-1}\right)^n$

I'm looking at one of my professor's calculus slides and in one of his proofs he uses the identity: $\left(\frac{n+1}{n-1}\right)^n = \left(1+\frac{2}{n-1}\right)^n$ Except I don't see why that's ...
0
votes
2answers
24 views

Compare colon notation with fraction

I'm working on a job interview test and there is one answer which I just don't get. The test states that statement below is true. To me it just seems wrong. No box is provided to check. Then how do I ...
2
votes
2answers
53 views

simplifying and factoring a fraction

how i get $\frac{(a+b)^2+(a+c)^2+(b+c)^2}{2}$ from $\frac{a^4}{(a-b)(a-c)}+\frac{b^4}{(b-a)(b-c)}+\frac{c^4}{(c-a)(c-b)}$ assuming that $a\ne b\ne c\ne a$ i tried to make $$\begin{align} ...
1
vote
5answers
513 views

Un-Simplifying a fraction, i.e. computing partial fraction decomposition

$\frac{3x^2+17x}{x^3+3x^2+-6x-8}$ I need to find the value of C in the form of $\frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{x+4}$ which is based on the fraction give at the top. I can get so far to do ...
1
vote
1answer
11 views

Explaining the non-application of the multiplication law of logarithms, when logs are in the denominators.

I have an A' Levels student who had to solve the following problem: $ log_2 x + log_4 x = 2$ This was to be solved using the Change of base rule, and then substitution, as follows: $ \frac{1}{log_x ...
0
votes
1answer
25 views

Is there a value for $a$ other than a factor or a multiple of $c$ in $\frac{a}{b}=\frac{c}{d}$

Suppose $a,b,c,d$ to be whatever quantities whatsoever that satisfy the proportion $\frac{a}{b}=\frac{c}{d}$. Is there a value for $a$ other than a factor or a multiple of $c$. Or, is there a value ...
0
votes
0answers
20 views

GCD and fraction problem

If x/y = 1/a + 1/b + 1/c and GCD of a , b and c is 9 then find a) minimum of x and y which do not cause x/y repeating decimal b) the best of x and y that cause x/y nearly to 3/10 many ...
0
votes
2answers
44 views

Does there exist $a,b,c,d$ such that $\frac{a+b+c+d}{4}$ is an integer?

Let $a,b,c,d$ be defined as such: $$\{a,b,c,d\} \geq 1,\\ a\neq b\neq c\neq d,\\ a \not\in \{bx,cx,dx\},\\ b \not\in \{ax,cx,dx\},\\ c \not\in \{ax,bx,dx\},\\ d \not\in \{ax,bx,cx\},\\ \{a,b,c,d\} ...
3
votes
1answer
62 views

When is $(12x+5)/(12y+2)$ not in lowest terms?

I am struggling to solve this problem and would appreciate any help: When is $\frac{12x+5}{12y+2}$ NOT in lowest terms? (x,y are nonnegative integers) I have found that it is not in lowest terms for ...
4
votes
3answers
68 views

Proving $\lim _{x\to \infty }\left(\frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}}\right) = \frac35$

$$\lim _{x\to \infty }\left(\frac{\sqrt{x+1}-\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-3}}\right)$$ Can someone help me to solve it? result of online calculator: 3/5
4
votes
2answers
370 views

Equivalent of adding to a denominator?

Given the inequality $\frac{n}{m} \ge \frac{1}{2}$, I want to add $1$ to both $n$ and $m$: $$\frac{n+1}{m+1}.$$ What would be the equivalent operation on the RHS of the equation? Adding $1$ to $n$ ...
1
vote
2answers
102 views

How is $\frac{1-x}{x^2-1}=\frac{1}{x+1}$?

When integrating $\int \frac{1-x}{x^2-1} dx$ Maple rewrote it as $-\int\frac{1}{x+1}dx$ How is $\frac{1-x}{x^2-1}=\frac{1}{x+1}$?
0
votes
1answer
53 views

Shorten $\frac{n}{n^\frac{1}{2}}$?

I have a short question to solve my problem. Can I simplify $\frac{n}{n^\frac{1}{2}}$ ? Thanks already for answers.
0
votes
1answer
35 views

Rearranging equation

I'm reading a textbook in which an equation is rearranged and I'm failing to see how they've done it. I've tried writing it down step by step in my notebook but can't come up with the right answer. ...
0
votes
1answer
96 views

Fractions vs Decimal numbers

I want to know if there is any difference between Fractions and Decimal numbers, are Decimal numbers just Fractions that are written in a different way according to a predefined rule: using "a group ...
0
votes
1answer
45 views

Fractional Exponents and Fractions

When dealing with fractional exponents like in the question below, how do you combine them so the two "n's" in the first fraction become one? ((how do i combine $4/3$ with $1/3$)) The aim is to end ...
2
votes
5answers
47 views

need help on this fraction equation $2/5 = 2/3 - r/5$

$$\frac{2}{5} = \frac{2}{3} - \frac{r}{5}$$ I'm trying to find $r$. Can anyone give me a step by step?
1
vote
1answer
46 views

What is the smallest fraction produced by a sum of fractions with bounded denominator?

For $x$ a sum of fractions: $$ x = \sum_{i=1}^{N}\frac{a_i}{b_i} $$ for all $a_i, b_i \in \mathbb{Z}$ with $ 0 < b_i \leq D$ and $N$ are non-zero positive integers, I know that the denominator of ...
0
votes
1answer
33 views

Why do the denominators of two fractions with numerators $1$ add up to a third fraction that have the special things below?

I found out that the denominators of two fractions with numerators $1$ add up to a third fraction that has the sum in the numerator and the product in the denominator. For example, ${1\over ...
1
vote
2answers
27 views

How do I solve this equation?

I have an equation, where I need to find n, that I need help solving. I already cheated a little bit by using a CAS (Maple) to solve the equation, so i know what the result should be, but I need to ...
24
votes
8answers
2k views

How to make sense of fractions?

Can anybody explain what a fraction is in a way that makes sense. I will tell you what I find so confusing: A fraction is just a number, but this number is written as a division problem between two ...
0
votes
2answers
53 views

variables to the power of a fraction

I have this question for advanced math, I can't seem to get my head around. $$\frac{x^{5/2}}{(x^{1/3})^4}$$
6
votes
1answer
323 views

Will 0.99999999 eventually become equal to 1?

I am currently learning about fractions, and there is something that I am finding it hard to make sense of. When a fraction it added to the right of the decimal point, the number becomes slightly ...
0
votes
2answers
38 views

what is the default order and direction of operation?

I have a division like this 16/8/4/2 what is the default way to do calculations when the bracket is not specified . Method 1 : Is it correct to go from right to left like [16/ (8 / { 4 / ...
5
votes
2answers
147 views

How do I simplify this expression about factorization?

I am trying to simplify this $$\frac{9x^2 - x^4} {x^2 - 6x +9}$$ The solution is $$\frac{-x^2(x +3)}{x-3} = \frac{-x^3 - 3x^2}{x-3} $$ I have done $$\frac{x^2(9-x^2)}{(x-3)(x-3)} = ...
1
vote
0answers
63 views

exponential integration with fractional powers

I am trying to solve the following integral $$\int_{-\infty}^a \frac{\beta_1 \beta_2}{y^2(c-y)^2} e^{-\beta_1/(c-y)} e^{-\beta_2/y} \, dy$$ where $a<0$, $c>0$, $\beta_1>0$, $\beta_2>0$ I ...
2
votes
3answers
68 views

Why does $\sqrt{x} / y =\sqrt{x/y/y}$?

Sorry for the awkward title, hard to to sum a mathematical problem with words alone. Having said that, I recently learned that the root of any value, $x$, and then that over value $y$, is identical ...
6
votes
3answers
246 views

Why is $\frac{\sqrt{x+1}-1}{x}$ equal to $\frac{1}{\sqrt{x+1}+1}$?

I'm working with the expression $$\frac{\sqrt{x+1} - 1}{x}.$$ According to Wolfram Alpha "alternate form" section (http://www.wolframalpha.com/input/?i=%28%28x%2B1%29%5E1%2F2-1%29%2Fx) it is equal to ...
1
vote
1answer
25 views

Function that maps a rational number to its numerator and denominator

Question: Is there a simple way to represent a function $f:\mathbb Q\to \mathbb Z^2$ that maps a rational number in lowest terms $r=\frac ab$ to the ordered pair of its numerator and denominator ...
0
votes
4answers
188 views

What does Pi equal to [duplicate]

What is the approximation of pi in a fraction form. I am very curious to know what it is. I have been seeing pi almost everywhere.
0
votes
1answer
30 views

Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$

Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$ So need to show $\frac c {1+c} \le \frac {a+b+2ab} {1+a+b+ab}$ We have $\frac c {1+c} \le \frac {a+b} ...
93
votes
12answers
11k views

Can you be 1/12th Cherokee?

I was watching an old Daily Show clip and someone self-identified as "one twelfth Cherokee". It sounded peculiar, as people usually state they're "1/16th", or generally $1/2^n, n \in \mathbb{N}$. ...
1
vote
3answers
82 views

How do I solve this, first I have to factor $ 2x\over x-1$ + $ 3x +1\over x-1$ - $ 1 + 9x + 2x^2\over x^2-1$?

I am doing calculus exercises but I'm in trouble with this $$\frac{ 2x}{x-1} + \frac{3x +1}{ x-1} - \frac{1 + 9x + 2x^2}{x^2-1}$$ the solution is The only advance that I have done is factor $ ...
1
vote
1answer
57 views

Integral exponential and fraction of powers

I am trying to solve the following integral $$ \int_0^y \frac{x^{m-1}}{(1+x)^{m+k}}\, \exp\left(-\frac{m}{\gamma} x \right) \,dx. $$ I tried to look into different books such as Gradshteyn and ...
0
votes
1answer
42 views

power of a fraction

I am having trouble understanding where the numbers are coming from in this question. John and Melissa wonder about the potential increase in the value of their house. Assuming a 6% appreciation per ...
4
votes
5answers
101 views

$-\frac76=-\frac {4n}{3}$ struggling on solving this equation

I was wondering how to do this equation step by step. I forgot how to but it would be awesome if someone could help me out $$-\frac76=-\frac {4n}{3}$$