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

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0
votes
4answers
55 views

At what $a$ is $\lim_{x\to0}\frac{e^{ax}-e^x-x}{x^2}$ finite? What is that limit?

Here is a function: $$\frac{e^{ax}-e^x-x}{x^2}$$ In which $a$ is a coefficient. The problem whats us the value for $a$ which gives a finite value for the limit, $$\lim_{x\to0}\frac{e^{ax}-e^x-x}{x^...
0
votes
1answer
20 views

What do you think it means when I ask for a 0.5% transaction fee?

What comes to mind when I tell you that im charging a fee of 0.5% when you buy flowers from me? You can either think of it as If I want to purchase 1000 dollars worth of flowers I need to pay 1005. ...
2
votes
1answer
47 views

Number theory problem, fractions and gcd, please help!!!

The problem says "if a,b are positive integers such that $\frac{a+1}{b}+\frac{b+1}{a}$ is an integer then show that $\sqrt{a+b}\ge$ gcd(a,b)" Adding $\frac{2ab}{ab}$ to $\frac{a+1}{b}+\frac{b+1}{a}$ ...
5
votes
4answers
218 views

Converting repeating decimal in base b to a fraction the same base

The repeating decimal .36666... in base 8 can be written in a fraction in base 8. I understand simple patterns such as 1/9 in base 10 is .1111.... so 1/7 in base 8 is .1111. But I'm not too sure how ...
0
votes
2answers
27 views

Fraction and Decimal: Reciprocal of x's non-integer

The reciprocal part of $x$'s non-integer decimal part equals $x+1$, and $x>0$. What is $x$? Solution: I tried this way- Let's $n$= integer part of $x$ $1/x-n = x+1$ or, $1=(x-n)(x+1)$ or, $1= ...
-7
votes
1answer
21 views

A word problem involving fractions [on hold]

A roofer and an apprentice are roofing a newly constructed house. In one day, the roofer completes 2/5 of the job and the apprentice completes 1/4 of the job. How much of the job remains to be done?
0
votes
4answers
53 views

If $x$ is positive, then why does $\frac{1}{\sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} - \sqrt{x}$?

Given that $x$ is positive, $\frac{1}{\sqrt{x+1} + \sqrt{x}} = \sqrt{x+1} - \sqrt{x}$ I've been trying to convert the left side of the equation to the right side: $$ \frac{1}{\sqrt{x+1} + \sqrt{x}}$...
0
votes
2answers
70 views

If $\frac{a}{b}=\frac{c+d}{e+f}$ so $a$ is equal to $c+d$?

I have these question, a is always c+d and b is e+f ?. Thanks.
8
votes
5answers
255 views

What is the 'physical' explanation of a division by a fraction?

For example, dividing by 2, means we cut something in two. But dividing by 0.5, can only be explained with multiplying something by 2. So, is there a "physical" explanation of dividing by 0.5? Is it ...
2
votes
2answers
66 views

Pairs of irreducible fractions that add up to a given irreducible fraction

Given the irreducible fraction $\frac a b$, with $a, b \in \mathbb N$, what is the expression that enumerates all the irreducible fractions of integers that add up to $\frac a b$? Namely, an ...
0
votes
1answer
98 views

why $1+\frac{1}{1+\frac{1}{\frac{1}{1+\dots}}}$ cannot be equal to $\frac{1-\sqrt{5}}{2}$

why $1+\frac{1}{1+\frac{1}{\frac{1}{1+\dots}}}$ cannot be equal to $\frac{1-\sqrt{5}}{2}$ I know that some of you will answer because it is positive.I know it but I don't know how to prove it.Maybe ...
-1
votes
1answer
6 views

Why is this true? $p_x$ $x$ $(1 + {(1 - a) \over a})$ = $p_x w_x + p_y w_y$ $\Rightarrow$ $x$ = $a$ ${p_x w_x + p_y w_y } \over p_x$

I don't see why the following equation is true - although wolfram-alpha gives me the same result, I can't figure out the steps that were made. Sure, we can simply divide the equation by $p_x$, but ...
2
votes
1answer
70 views

Is there an equivalent word for '3/4?'

It's already known that the most of the quarter fractions have a single word equivalent that correspond with its numerical counterpart, such as '1/4' is a quarter, '1/2' is half, and '4/4' is the same ...
1
vote
1answer
53 views

Simplify simple fractions after Hospital's rule

I'm having trouble simplifying the result of complicated limits which contain mutliple fractions. I understand this is basic math level, but i find it difficult to find exercices to practice that ...
-2
votes
1answer
47 views

Why won't my calculator display this fraction? [closed]

I am using my calculator to convert 2.6(457) 457 is recurring. This should be 2 6451/9990, but my calculator only displays the decimal 2.645745764. How can I fix this, incase I need to do this in an ...
8
votes
1answer
89 views

Numbers whose reciprocals sum to $1$

What are all the numbers that can be written as $a_1+a_2+\dots+a_n$, where $a_1,\dots,a_n$ are positive integers such that $\frac{1}{a_1}+\dots+\frac{1}{a_n}=1$? For instance, such numbers include $4=...
1
vote
1answer
37 views

Partial Fractions Question: $\frac1{s(n-s)}$

How do you decompose the fraction $$\frac1{s(n-s)} $$ to two fractions? ($s$ and $n$ are variables).
3
votes
1answer
46 views

Laplace Transform for Solving Differential Equation

I solved the following task, but since I am new in this field I need to check if it is correct or if there is anything I am missing or doing wrong. Task : Solve differential equation using Laplace ...
0
votes
1answer
400 views

Quotient field of gaussian integers

Let $D$ be the set of all gaussian integers of the form $m+ni$ where $m,n \in Z$. Carry out the construction of the quotient field $Q$ for this integral domain. Show that this quotient field is ...
4
votes
2answers
21 views

Adding fractions of Groups of People

I understand the rules of adding fractions perfectly well. I know how to find common denominators, and understand why adding fractions without common denominators doesn't make sense. But, today ...
0
votes
4answers
64 views

If you take the reciprocal in an inequality, would it change the $>/< $ signs?

Example:$$-16<\frac{1}{x}-\frac{1}{4}<16$$ In the example above, if you take the reciprocal of $$\frac{1}{x}-\frac{1}{4} = \frac{x}{1}-\frac{4}{1}$$ would that flip the $<$ to $>$ or ...
1
vote
5answers
45 views

Simplify the following expression

$$\left(\frac{4}{27}\right)^{3/2}$$ I am trying to figure out how to solve this problem and my teacher was not explaining it well enough for me to grasp it. Can anybody maybe help me with step-by-...
0
votes
2answers
20 views

Finding LCM of an expression

Image link I am currently trying to figure out how to solve this problem. I know how they got the denominator but am unsure how they got 6x in the numerator? Can somebody explain this to me?
5
votes
7answers
177 views

What happens when you add $x$ to $\frac{1}{3}x$?

I am dealing with an equation that requires me to add $x$ to $\frac{1}{3}x$: $x + \frac{1}{3}x$ = ?? I know this might be simple to any of you on this site, because you are all asking questions with ...
-1
votes
1answer
27 views

Bob and Jim's Coins [closed]

Bob had 52 more coins that Jim after Jim gave 1/5 of his coins to him. If they had 260 coins total, how many coins did Bob have at first?
1
vote
5answers
85 views

When is $\sqrt{x/y^2}$ equal to $\sqrt{x}/y$?

The solution to the quadratics is given by $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this ...
3
votes
3answers
7k views

Detecting that a fraction is a repeating decimal

Given any fraction where both the numerator (N) and denominator (D) are both positive and are both whole numbers. Without manually dividing N by D, is it possible to pre-determine if the resulting ...
1
vote
3answers
44 views

A fraction problem [closed]

$$a = x + \frac1x \\b =y + \frac1y \\ c = xy +\frac1{xy} $$ Express $c$ in terms of $a$ and $b$
2
votes
2answers
36 views

Property of fractions

Given two fractions $\frac{h}{k}$ and $\frac{h^{'}}{k^{'}}$ both in reduced form. I am unable to find a case when $\frac{h+h^{'}}{k+k^{'}}$ does not lie in the interval $\big[ \frac{h}{k},\frac{h^{'}}{...
0
votes
1answer
26 views

simple integration artimethic error

I am trying to integrate a polynomial but I couldn't get the correct answer somehow. I feel like I'm making a mistake when evaluating the integral. $$\pi\int_{-1}^1{1-2x^2+x^4}dx=[{x-{2x^3\over3}+{x^...
2
votes
3answers
77 views

Can fractions actually be converted to decimals?

I was working on a spreadsheet in Excel (I'm a plebe, I know), and I came across a fraction that actually equated to 33.3% of a total number. While looking at it, and looking at the number that went ...
0
votes
1answer
26 views

Working out cost based on time spent - simple math [closed]

I did a task, and my hourly rate is $£25$ , I spent a total of $36$ minutes on it, how can I work out the total amount of time spent on the task? My attempt: I can do this for simple sums such as ...
0
votes
1answer
44 views

Help with some algebra

Can anybody explain to me how they went from this $$y − y_1 = \frac{y_2 − y_1}{x_2 − x_1} (x − x_1)$$ to this. $$(y_1 − y_2 )·x − (x_1 − x_2)·y − (y_1 − y_2 )·x_1 + (x_1 − x_2)·y_1 = 0$$ Its ...
3
votes
2answers
104 views

Evaluate $1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{…}}}}$ when you see $15$ fraction lines

Evaluate $1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}}$ when you see $15$ fraction lines. I have solved this problem but using a quater calculating I come from down to up 15 times and ...
-1
votes
2answers
59 views

Find the integer closest to $ a - b$ [closed]

Let $$a = \frac{1^{2}}{1} + \frac{2^{2}}{3} + \frac{3^{2}}{5} + \ldots + \frac{1001^{2}}{2001}.$$ Let $$b = \frac{1^{2}}{3} + \frac{2^{2}}{5} + \frac{3^{2}}{7} + \ldots + \frac{1001^{2}}{2003}.$$ ...
0
votes
1answer
38 views

How to solve: $\ \lim_{n \to +\infty} \frac{n^n + \frac {1}{n}}{(n + \frac {1}{n})^n} \ t^n $

How can I solve: $$\ \lim_{n \to +\infty} \frac{n^n + \frac {1}{n}}{(n + \frac {1}{n})^n} \ t^n $$ tis a whole number. Thank you very much! Please tell me your ...
4
votes
5answers
463 views

What is the fastest method to find which of $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ and $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $ is bigger manually?

What is the fastest method to find which number is bigger manually? $\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $
0
votes
1answer
38 views

Pre-Algebra Fractional Exponent Question

Why does $t^{\frac{3}{2}} \cdot t^{\frac{1}{2}} = t^2$? What I tried to do was multiply the exponents together $\frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$ so my final answer was $t^{\frac{3}{4}}$ ...
0
votes
1answer
56 views

Proof of $\frac{q_n}{q_{n-1}} = [a_n,a_{n-1},a_{n-2},…,a_2,a_1]$?

Proof of continued fractions axiom. Let $c=[a_0,a_1,a_2,\dots,a_n,\dots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \ddots}}$ be a continued fraction which could be finite or infinite. By $\frac{p_n}{...
-1
votes
1answer
35 views

when the sum of some fractions equal to 1. [duplicate]

$r$ is a number such that $r=p^a$. If the sum of some fractions equal to $1$ and one of the denominators is divisible by $r$ then there is another denominators that is exactly divisible $r$. It seems ...
2
votes
1answer
34 views

when the sum of some fractions be $1$

prove if we want that the sum of some fractions be $1$ and the denominators of one of them is $d$ then another denominators should divisible by $d$ or $d$ should be divisible to another denominators. ...
0
votes
3answers
92 views

Very Basic Math question?

How can I prove that $$\frac{r}{(1-x)^2} + \frac{rx}{x(1-x)} = \frac{r}{x(1-x)^2}$$ I have tried to prove that , but I could not , can someone help me please ? Thanks
0
votes
2answers
35 views

Please help me understand this simple fraction

I got started learning about fractions a few days ago, The tutorial I'm using for this, is limited to fractions like this Now as I'm trying to find further exercices, I keep stumbling upon ...
8
votes
2answers
106 views

Does $\frac{x-2}{3x-6}$ really equal $\frac{1}{3}$?

In my maths lesson today we were simplifying fractions by factorising. One question was something like this: $\frac{x-2}{3x-6}$, which I simplified as $\frac{x-2}{3x-6}=\frac{x-2}{3(x-2)}=\frac{1}{3}$....
3
votes
3answers
65 views

multiplication and addition fractions

Try to visualize process of multiplication fraction addition is obvious, need to split each part to the same size - "reduce to a common denominator" for example $$\frac23 +\frac24 = \frac{8}{12}...
1
vote
3answers
47 views

fraction division understanding [duplicate]

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
1answer
96 views

For which $a,b\in \mathbb{N},$ is $\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is a rational number.

I found the following problem on a Olympiad question paper: For which $a,b\in \mathbb{N},$ is $$\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$$ a rational number. I am unable to solve it. Any help ...
2
votes
2answers
90 views

How to remove parentheses from $x/(y-z)$

To remove parentheses from $x(y-z)$ I reword it to $xy-xz$. How do I remove parenthesis from $x/(y-z)$?
1
vote
1answer
65 views

Estimation of fraction of integrals

(edited for more clarity) For a given function $f$, which is continuous, and $a < b$ real numbers, I need to make an estimation of the type $ \Bigg| \frac{\int_a^b f(t) (-t)dt}{\int_a^b f(t)dt} \...