Questions on fractions, numbers of the form $p/q$ where $p$ and $q$ are integers, and $q$ is not zero.

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2
votes
1answer
56 views

$\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $?

Is it always true that: $\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $ where $m,k \in \mathbb N$ ? I tried it with a few numbers and it seems to work every time.
0
votes
4answers
15 views

How to make sense of fractions concretely

I can solve fractions abstractly, for example, $\frac{5}{2}$ divided by $\frac{3}{2}$, you can flip $\frac{3}{2}$ so that $\frac{5}{2}$ multiplied by $\frac{2}{3}$. Specifically, math makes sense ...
0
votes
0answers
66 views

What is wrong with this, Calculator says OK, Homework doesn't

I'm sorry about this, this must seem like a stupid question, but I have homework showing me a "incorrectly answered" problem and asking me to find why it was Incorrect and Correctly solve the problem. ...
0
votes
1answer
37 views

fraction as index number?

given these inputs x = 4, S = [1 2 3 4 5 6 7 8 9 10], and n = 10 ...
6
votes
1answer
413 views

Does it make sense to multiply slopes?

Multiplying fractions is a regular occurance. If those fractions are considered slopes, does it make any sense? For example, if these fractions are slopes,$\frac{9}{8} \times \frac{49}{48},$ does the ...
1
vote
3answers
39 views

How have they done the algebra here?

Proof by induction Can someone explain these steps to me please? Did the $2^{k-1}$ change to $2^k$ by multiplying numerator by 2?? Even so, if you add them when they have the common denominator, ...
1
vote
1answer
37 views

Calculating the average of multiples and divisions

Imagine a number line that contains every value that is greater then (but not inclusive of) 0. The center of the line is 1. On the right side of the center(1), obviously, are all the whole and ...
-1
votes
1answer
15 views

What is the formula's are used to convert to meters/second?

What are the formula's to convert the following per hour intervals into meters per second (using meters/s from light speed): Kilometer Mile (US) Mile (Nautical) Feet The result should be decimal ...
0
votes
1answer
20 views

Addition on an Elliptic Curve and Modular Arithmetic involving fractions

I'm having a bit of an issue with addition on elliptic curves. For example, I've been given the curve $Y^2 = X^3 + 2X + 1$, working modulo 3. Now, say I want to add the point $(1,2)$ with itself. To ...
0
votes
2answers
30 views

Fractions with 3 diffferent variables

Calculate the values of $a$, $b$ and $c$ if: $$\frac{5}{13} = \frac{1}{a+\frac{1}{b+\frac{2}{c}}}$$ Can anyone give me a hint and not the answer? Thanks.
0
votes
0answers
12 views

Identity for fractional summation

I would like to know if there's an identity to represent the following summation $\sum_{i=0}^{n}\frac{x_i}{y_i}$ Where x and y are non integer values. The result of this is being calculated using ...
0
votes
2answers
27 views

Subtracting 2 fractions with variables in the denominator that have different exponents.

Sorry for the relatively elementary question, but I am having trouble remembering exactly how to do this type of problem. I am looking to simplify this: $$ \frac{3}{4t^{1/4}} - \frac{1}{2t^{3/4}} $$ ...
-1
votes
1answer
26 views

How do I use the partial fractions technique in this case?

How do I use the partial fractions technique in this case? $$\frac{(x - 1)}{(x^2 - x + 1)(x + 1)}$$
6
votes
1answer
108 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
3
votes
3answers
112 views

How do I integrate $\frac{1}{x^6+1}$

My technique so far was substitution with the intent of getting to a sum of three fractions with squares in their denominators. $t = x^2 \\ \frac{1}{x^6 + 1} = \frac{1}{t^3+1} = ...
1
vote
3answers
38 views

Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b,c > 0 $?

Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b> 0$ and $c\geq 0$ ? Is there a general proof for that relation ?
-3
votes
1answer
73 views

How to multiply, divide, add and subtract fractions

I've spent hours on this and I keep getting mixed answers. I need to know the rules for multipling, dividing, adding, subtracting equations involving fractions. I google search but the information is ...
0
votes
0answers
17 views

Calculating an average value based on separate subsamples from the same sample

I have a question coming from biological research. We routinely have to quantify on microscopic images certain values characteristic of a piece of tissue – for example the percentage of cells that are ...
3
votes
1answer
54 views

How do you calculate how many decimal places there are before the repeating digits, given a fraction that expands to a repeating decimal?

If you have a fraction such as $$\frac{7}{26}=0.269230\overline{769230}$$ where there are a number of digits prior to the repeating section, how can you tell how many digits there will be given just ...
0
votes
2answers
24 views

Is there a formula for this?

Take a test score represented by the fraction ${a\over b}$. This test score could be curved by removing a wrong answer to get ${a\over b-1}$ or adding a correct answer to get ${a+1\over b+1}$. ...
0
votes
1answer
26 views

Find pdf of $f(x)$ such that $g(x)/f(x)$ is approximately a constant

My friend asked me a question that asks to find a pdf function $f(x)$ such that $f(x)/g(x)$ is approximately a constant, where $g(x)=\sqrt{e^{x^2}+e^x}$, and $f(x) \neq g(x)$. And the range of x is ...
0
votes
0answers
23 views

How can I simplify the following expression with exponents.

$$\frac{(t+1)^{\frac{1}{3}}-\frac{1}{3}t(t^2+1)^{-\frac{2}{3}}}{(t^2+1)^{-\frac{2}{3}}}$$ I found this problem from a book and its answer is $\frac{2t+3}{3(t+1)^{\frac{4}{3}}}$(as in the book's ...
0
votes
2answers
32 views

Simple ratio problem that I can't solve.

One fifth of criminals are hard-core criminals. The hard-core criminals commit two-thirds of the criminal acts. What is the ratio of the number of criminal acts committed by the average hard-core ...
2
votes
1answer
68 views

Simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and a natural ...
0
votes
2answers
48 views

Rationalising the Surds

Please help me rationalise and simplify: $$ \frac{1}{\sqrt[3]{2} - 1} \ - \ \frac{2}{\sqrt{3} - 2} \ . $$ I have tried using the cube of the denominator and the square of the denominator on the ...
0
votes
4answers
69 views

How to simpify this?

How to simplify following fraction? I have tried everything, but nothing seems to work... $$-a^3 (c^2 - b^2) + b^3 (c^2 - a^2) - c^3 (b^2 - a^2)\over (c-b)(c-a)(b-a)$$
-2
votes
4answers
54 views

how do you solve this? [closed]

a baseball team has won 15 games and lost 9. If these 24 games represent 1/6 of the games played during the entire season, how many more games must the team win in order to win 3/4 of their games for ...
8
votes
2answers
77 views

Integrating $\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$

I am a little bit lost with integral: $$\int{\frac{\sqrt{1-x^2}}{(x+\sqrt{1-x^2})^2} dx}$$ I have already worked on in and done substitution $x = \sin(t)$: This brings me to: ...
1
vote
2answers
80 views

Dividing amount unequally in tournament winners?

Hi I am trying to write the algorithm to calculate the winning amount of the winners. My problem is as below, 1) I have open ended tournament in which participants can be in ratio of 2 (i.e 2,4,8,16 ...
0
votes
4answers
98 views

Solution of $\dfrac{a}{b}=\dfrac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$

Let $\dfrac{a}{b}=\dfrac{a'}{b'}$ , $a,b,a',b' \in \mathbb{N}$ s.t. $a$ and $b$ have no common factors and it is presumed that $a'>a$ and $b'>b$. How can we show that the only solution to this ...
0
votes
1answer
56 views

Integration involving $\log_2(x)$

Having a hard time going about this problem: $$\int{\frac{\ln(2)\log_2(x)}{x}}$$ I believe $\ln(2)$ would be considered a constant, so than the equation would then changed to: ...
1
vote
0answers
58 views

Prove that there exists a subset with sum >=1 such that the remaining integer sum reduces by 1

let $ n \in \mathbb{N} $ and $ \frac{1}{w_1},\ldots, \frac{1}{w_n} $ for some (not necessarily distinct) $ w_1,\ldots,w_n \in \mathbb{N} $ and $ w_1,\ldots,w_n \ge 2 $ be given. Assume that $ ...
0
votes
0answers
21 views

How to find maximum and minimum of (x+y+z)/(ax+by+cz) where 0≤x≤y≤z≤1 for given positive real numbers a,b,c

How to find maximum and minimum of $$\frac{x+y+z}{ax+by+cz}$$ where 0≤x≤y≤z≤1 for given positive real numbers a,b,c? I guess those are one of $\frac{3}{a+b+c}$ or $\frac{2}{b+c}$ or $\frac{1}{c}$, ...
0
votes
1answer
36 views

How to solve a inequality with fractions and roots in denominator and numerator

The inequality is like that: $$ \sqrt{\frac{3x+1}{2}}>1 $$ I have no idea how should i begin with it.
0
votes
2answers
46 views

Counting four-digit numbers with repeating digits

Of all the four-digit positive integers containing only digits from the set $\{2,4,6,8\}$, what fraction of them have at least one of their digits repeating? Express your answer as a fraction. ...
6
votes
1answer
96 views

$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$

I have a pretty simple straightforward question. Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$ Instinctively, I do the quickest thing I know how to ...
0
votes
2answers
80 views

Solve algebraically: $\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$

$$\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$$ The professor says we can't use l'hopital's rule and must solve algebraically.
7
votes
2answers
63 views

Why does partial fraction decomposition always work?

Say you have a function $p(x)/q(x)$ for some polynomials $p(x)$ and $q(x)$ and $p$ has a lower degree than $q$. Say $q$ has degree three and $p$ has degree two. If you partially decompose it, you'll ...
2
votes
2answers
54 views

Proper decimal fraction for $\frac{4n+1}{n(2n-1)}$

Assume I have a function $f(n) = \frac{4n+1}{n(2n-1)}$ with $n \in \mathbb{N} \setminus \left\{ 0 \right\}$. The objective is to find all $n$ for which $f(n)$ has a proper decimal fraction. I know ...
0
votes
1answer
48 views

Is there an abstract algebraic way to analyze the rational numbers between 0 and 1 (inclusive)? [closed]

I'm wondering if the rationals between $0$ and $1$, have been studied in a systematic manner using abstract algebra. Is there any interesting theory behind this set?
1
vote
3answers
136 views

If the sum of two irreducible fractions is an integer, then the denominators are equal

I have to show the following:"If the sum of two irreducible fractions with positive denominators is an integer, then the denominators are equal." $$\frac{a}{b}+\frac{c}{d}=k, \text{ where k an integer ...
8
votes
4answers
123 views

Primary/Elementary Pedagogy: What is the rationale for the absent '+' in mixed fractions?

Why are elementary students taught to represent one and a half as 1 1/2 rather than 1 + 1/2? This mode of expression seems standard throughout at least North America. I think it is bad pedagogy for a ...
1
vote
1answer
34 views

Simplify this expression $w\left(\cfrac Q {1+\frac w r}\right)^2 + r\left(\cfrac Q {1 + \frac r w}\right)^2$

I can't figure out how to do the algebra to simplify this expression! $$w\left(\cfrac Q {1+\frac w r}\right)^2 + r\left(\cfrac Q {1 + \frac r w}\right)^2$$ It's supposed to turn out to $\cfrac {Q^2} ...
-2
votes
3answers
76 views

Explanation of sin(x) and cos(x) [closed]

Can anyone explain me what is this equation telling us? I need to implement it in my computer program :P AND
1
vote
1answer
15 views

Help with load balancing math based on fractional capacity

I'm looking to create an algorithm that allows me to select a number(index) from a list based on it's fractional weight component. It's for load balancing, I'll give an example below of what I mean. ...
7
votes
2answers
151 views

Rationalizing the denominator 3

It is a very difficult question. How can we Rationalizing the denominator? $$\frac{2^{1/2}}{5+3*(4^{1/3})-7*(2^{1/3})}$$
3
votes
2answers
102 views

What is $\lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$?

These are two little questions that came to mind while I was looking at this problem. What is $\displaystyle \lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$? I am fairly certain that the ...
1
vote
3answers
79 views

Finding the sum of fractions with increasing denominator and decrease numerator for n iterations?

Considering something like this: $ \frac{10}{10} + \frac{9}{11} + \frac{8}{12} + ...$ Where denominator increases each iteration while the numerator decreases. Is there a simple way to find the ...
0
votes
2answers
26 views

Fractional Power Interpretation

I have a following query in my mind. It has been in my mind since i was a kid. I know that 2^3 means that multiply 2 three times,3^-2 means multiply (1/3) two times.What does 2^(0.22) means. multiply ...
1
vote
2answers
30 views

Simplified way of writing summation of $1$ to $n$ in fractions

I have the following expression that I'm trying to simplify: $$\frac{1}{3} + \cdots+\frac{1}{3^{n-1}} +\frac{1}{3^n}.$$ This looks like a summation of $1$ to $n$ but in different terms. Can someone ...