Skip to main content

Questions tagged [fractions]

Questions on fractions, i.e. expressions (not values) of the form $\frac ab$, including arithmetic with fractions. Not to be confused with the tag (rational-numbers): fractions denote rational numbers, but the same rational number may be written in different ways as a fraction.

Filter by
Sorted by
Tagged with
0 votes
0 answers
25 views

Prove that $x^xy^y+x^yy^x \le 1$ if $x+y=1$ and $x,y$ are positive [duplicate]

Given $x+y=1$, $x>0$, $y>0$, prove that $$x^xy^y+x^yy^x \le 1.$$ I first tried using lagrangian, $\mathcal{L}=f(x,y)-\lambda(g(x,y)-c)$ where $f(x,y)=x^xy^y+x^yy^x $ ,$g(x,y)=x+y$ and $c=1$, ...
Xiaobao's user avatar
  • 21
4 votes
4 answers
140 views

Expressing $\frac{2}{n}$ as the sum of two unit fractions

Consider fractions such as $\frac{2}{5}$ and $\frac{2}{7}$ expressed as the sum of two unit fractions. Respectively, they can be expressed as $\frac{1}{3}+\frac{1}{15}$ and $\frac{1}{4}+\frac{1}{28}$. ...
James Chadwick's user avatar
2 votes
2 answers
219 views

How can i do the following partial decomposition?

I need to prove that: $$ \frac{1}{(x-a)(x-b)} = \frac{1}{(b-a)(x-b)}- \frac{1}{(b-a)(x-a)}, $$ and I must note that I need to go from the left expression to the right (because of the exercise). So, I ...
Miguel Simões's user avatar
0 votes
0 answers
18 views

Is the following mixed fraction equation correct?

I am wondering if the plus sign is understood in mixed fractions. So it is not written.
Dave's user avatar
  • 21
0 votes
0 answers
63 views

Weird Property of Irrational Numbers and their "Conjugate"

Let us have a irrational number $a$. Let $a'$ be $\frac{1}{a'} + \frac{1}{a} = 1$. If we let $A$ and $B$ be sets such that $A =$ {$\lfloor a \rfloor, \lfloor 2a \rfloor, \lfloor 3a \rfloor, \lfloor 4a ...
bob john's user avatar
3 votes
7 answers
164 views

Compare $3^4 \times 6^5 \times 7^8 \bigcirc 4^3 \times 5^6 \times 8^7$

Compare $$3^4 \times 6^5 \times 7^8 \bigcirc 4^3 \times 5^6 \times 8^7$$ Options: $\text{(A)} > \space\space\space\space\space \text{(B)} <\space\space\space\space\space \text{(C)} =$ Notes: $1....
Hussain-Alqatari's user avatar
1 vote
0 answers
35 views

Method to difference between two continuous fractions and keep continued fraction form

I warmly welcome an approach on differencing between two continuous fractions Without applying the appropriate algebra to reduce the continuous fraction. I cannot find a way without completing the ...
Nishi's user avatar
  • 31
1 vote
0 answers
53 views

Finding the number of fractions with unique values where the range for the denominator and numerator is in $\{1,2,\dots, n\}$

Imagine that you have a fraction, where both the numerator and denominator take values in the set $\{1,2,\dots, n\}$. Let us assume that the fraction is smaller then and or eqal to 1. The question is ...
jurcistan's user avatar
1 vote
1 answer
54 views

The Equivalence Transformation (??) for Generalized Continued Fractions

The equivalence transformation says that any sequence of non-zero complex numbers satisfy the general continued fraction in the following manner. Here is the link: https://en.wikipedia.org/wiki/...
Lucien Jaccon's user avatar
1 vote
1 answer
78 views

Continued Fraction Representation of sin(x)

To provide context, the continued fraction in the form $\frac{a_0}{1-\frac{a_1}{1+a_1-\frac{a_2}{1+a_2-...}}}$ evaluated to the $n$th denominator equals $\sum_{k=0}^{n}\prod_{j=0}^{k}a_j$. If one ...
Lucien Jaccon's user avatar
-1 votes
1 answer
66 views

What is the formula for converting an improper fraction to a mixed number [closed]

There are methods for converting improper fractions to mixed numbers, but I am interested on finding a formula to which I can input the numerator and denominator of an improper fraction and get an ...
Aceffad's user avatar
  • 39
1 vote
1 answer
89 views

Reduce two variables to a ratio [closed]

I was playing around with these two functions : $$ f_1(x)=\frac{a}{ax+b} \\f_2(x)=\frac{\sqrt{a}\sqrt{a+b}}{ax+b} $$ and I realized that I can rewrite $f_1(x)$ as $$ f_1(x)=\frac{c}{cx+1} \quad \text{...
Xavier's user avatar
  • 13
0 votes
1 answer
69 views

Can every positive real number $n$ be expressed as a sum of infinite non-equivalent and unlike fractions?

One time I was pondering about expressing irrational numbers as infinite series of fractions. I came to the conclusion that, If we allow an infinite number of fractions, we can express every positive ...
user avatar
1 vote
2 answers
62 views

Write an equation for the line with inclination 30 degrees, 𝑦-intercept 2 [closed]

I've decided to re-learn my college years. I have text with answers. The problem: Write an equation for the line with inclination $30$ degrees, $y$-intercept $2$. Finding the answer with a table of ...
Kinesava's user avatar
7 votes
2 answers
279 views

Find the smallest real number $M$ for which $\{a\} + \{b\} + \{c\} ≤ M$

This is a problem from an Ukrainian math competition which was held on 28 January 2024. Find the smallest real number $M$ for which $$\{a\} + \{b\} + \{c\} ≤ M$$ for any positive real numbers $a, b, c$...
sleexed's user avatar
  • 181
0 votes
1 answer
57 views

Simplify $\frac{2e^nm^2 + 2e^nm - e^n -6m^2 - 6m +3}{e^nm^2 + e^nm}$ into a sum of two fractions

For this problem, I must simplify the fraction so I end up with a sum of two different fractions: $$1 - \frac{3}{e^n}$$ and $$2 - \frac{1}{m(m+1)}$$ and ultimately end up in a sort of sum between them....
integralna_žitarica's user avatar
0 votes
1 answer
77 views

How can I calculate a Blue-Red Hackenbush position value using the Simplest Number Tree?

In this document (pages 5-7), the Simplest Number Tree is used to explain how to assign a value to any arbitrary Blue-Red Hackenbush position. I'm having trouble following its approach. I think I ...
user10478's user avatar
  • 1,862
0 votes
2 answers
67 views

Is my probability correct

Suppose you draw $1$ athlete at random from a group of $100$ athletes such that: $\{$$30$ swim$\}$, $\{$$44$ run$\}$, $\{$$9$ swim and run$\}$,$\{$$5$ swim, bike and run$\}$, $\{$$11$ swim and bike$\}$...
gen ragland's user avatar
2 votes
2 answers
99 views

Determine the minimum value of $S=p+q$

question For a fixed natural number $ n \geq 1$ we consider all rational numbers of the form $\frac{p}{q}$, with $p,q \in N*$ and $\frac{n - 1}{n } < \frac{p}{q }< \frac{n}{ n+1}$ Determine the ...
IONELA BUCIU's user avatar
  • 1,389
0 votes
0 answers
49 views

Is it possible for two simplified non integer fractions to have an integer sum and an integer product?

Is it possible for two simplified non integer fractions to have an integer sum and an integer product? Let’s say the fractions are $\frac{a}{b}$ and $\frac{c}{d}$. For them to multiply to an integer, $...
MathingPenguin's user avatar
0 votes
0 answers
46 views

GCD and LCM of n Fractions – Proof of the Formulas [duplicate]

I need help proving the following two formulas: $$\gcd\left(\frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n}\right) = \frac{\gcd(a_1, a_2, \dots, a_n)}{\operatorname{lcm}(b_1, b_2, \dots, b_n)}...
Bolzano's user avatar
  • 81
0 votes
1 answer
69 views

How do you convert fractions from one number base system to another? Like for example 31/44 base 5 to a base 10 number... [closed]

The original question was to convert 0.31(repeating) base 5, to a fraction in base 10 in its lowest form, however, after solving about half of it, you would get this fraction and you must try to ...
Balla_2007's user avatar
1 vote
1 answer
99 views

Solution verification - show that $a=b=c$ knowing an equality.

Question Let the positive real numbers $a, b, c, x$ be such that the numbers $ax + b , b*x + c$ and $cx + a$ are directly proportional to the numbers $c, a$ and$ b$. Show that $a = b = c$. My idea So ...
IONELA BUCIU's user avatar
  • 1,389
1 vote
0 answers
58 views

$ \iint_D \frac{x^3}{x^2 + y^2} dA; D := {(x − 1)^2 + y^2 ≤ 1} ∩ {y ≥ 0}$

I encountered the following question online: Evaluate $$ \iint_D \frac{x^3}{x^2 + y^2} dA$$ where D is the region $$D := {(x − 1)^2 + y^2 ≤ 1} ∩ {y ≥ 0}$$ Could you begin by stating your approach ...
Nick's user avatar
  • 6,804
12 votes
3 answers
1k views

Is it ok to cancel out fractions while integrating?

Is it ok to cancel out fractions while integrating? If yes, how? For example: $$\int \frac{x^3+1}{x+1}dx = \int \frac{(x+1)(x^2-x+1)}{x+1}dx = \int (x^2-x+1)dx = \frac{x^3}3-\frac{x^2}2+x + C$$ Wouldn'...
hyun's user avatar
  • 165
1 vote
1 answer
56 views

Combining Fractions with Delta Function

The Fourier Transform of $u(t)$ is $\frac{1}{jω} + \piδ(ω)$. However I'm troubled with the following fact: $\frac{1}{jω} + πδ(ω) = \frac{1+jωπδ(ω)}{jω} = \frac{1+j0πδ(ω)}{jω} = \frac{1}{jω}$ So is the ...
Nick Tsiodras's user avatar
1 vote
2 answers
92 views

Is there another way of solving $\frac{\frac{\sqrt{a}}{\sqrt{2}}}{\frac{a}{2}}$. Without the method of multiplying by the inverse?

So, with this: $$\frac{\frac{\sqrt{a}}{\sqrt{2}}}{\frac{a}{2}}$$ I could just do $${\frac{\sqrt{a}}{\sqrt{2}}}\times{\frac{2}{a}}$$ But I was wondering if there was a way of simplifying the $\sqrt{a}$ ...
Manuel's user avatar
  • 55
0 votes
0 answers
23 views

Monotonicity of quotient of power series

How can it be proved that $$ f(x) = \frac{\sum_{n=0}^{\infty} \frac{n}{2 x} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}}{\sum_{n=0}^{\infty} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}} $$ is monotonic? The plot of $f(x)$ ...
Guillermo's user avatar
-1 votes
1 answer
34 views

Finding the value of a fraction according to a given information

$a,b$ and $c$ are real numbers such that: $abc=ab+ac+bc$. Find the value of $\frac{a+b}{b}+\frac{a+c}{c}+\frac{b+c}{a}$. I've tried simplifying the expression and eventually I got: $\frac{a}{b}+\frac{...
Adrien's user avatar
  • 29
1 vote
1 answer
65 views

How to logically represent a smaller number being divided by a large number on a number line?

Recently I asked this question on this platform: When we divide a number by another number ($x \div y$), we can interpret it in two ways: $x$ is divided in equal groups, where each group consists of $...
Steve's user avatar
  • 85
0 votes
1 answer
100 views

Why can't $\frac{1}{\not0}\cdot \frac{\not0}{1}=1$?

I would assume that this question should have the obvious answer, that since $\frac{1}{0}$ is undefined, this expression is undefined also. However, given that we can solve arithmetical problems with ...
Alexander Pasha's user avatar
2 votes
1 answer
97 views

Manipulating Integrals involving Inverse exponential

I was studying integration when I came across mannipulating natural logarithms. It is known that: $$ \int \frac{1}{x}dx = \ln{|x|} +c \forall x>0 \wedge \int \frac{1}{ax+b}dx =\frac{1}{a} {\ln{|ax+...
skyfall's user avatar
  • 181
0 votes
0 answers
31 views

Is this example of scaling fractions correct?

Suppose we have $\frac{5\times2}{3\times5}$. If we consider it as $\frac{5}{5}\times\frac{2}{3}$, then $\frac{5}{5}$ is being scaled down and $\frac{2}{3}$ is neither being scaled down, nor being ...
Steve's user avatar
  • 85
0 votes
2 answers
51 views

How to prove a circle in the complex plane tangent to the real axis at the origin is invariant by the function $\frac{\omega}{1-\omega}$? [closed]

Let $C$ be a circle in the complex plane $\mathbb{C}$ whose centre is $(0,ir)$ (where $r$ is a non-zero positive real number) and is tangent to the real axis at the origin $(0,0)$ (the radius of the ...
Neil hawking's user avatar
  • 2,478
0 votes
1 answer
79 views

Simplifying a complex fraction

Simplify the complex number: $\frac{(\sqrt{3} + 9i)^{181}}{(-12 + 48\sqrt{3}i)^{90}}$ I've tried a few things such as turning the top and bottom of the fraction into their trigonometric form or trying ...
Marin's user avatar
  • 187
2 votes
1 answer
77 views

What is the Difference between these two expressions: $3$ $\frac {1}{7}$ and $3 + \frac{1}{7}$?

What is the difference between $3$ $\frac {1}{7}$ and $3 + \frac{1}{7}$? In the first expression, $3$ seems to be multiplied by $\frac {1}{7}$ using juxtaposition, but while doing the calculation, we ...
Steve's user avatar
  • 85
3 votes
0 answers
40 views

Behaviour and limits of $f(n+1) = \frac{f^5(n)}{2} - f(n-1)$

Let $f(0) = 0,f(1) = \frac{1}{2}$ and $$f(n+1) = \frac{f^5(n)}{2} - f(n-1)$$ where $*^5$ is a power. Then it seems $$ \sup f(n) = \lim \sup f(n) = \frac{1}{2}$$ and $$ \inf f(n) = \lim \inf f(n) = \...
mick's user avatar
  • 16k
1 vote
1 answer
86 views

Fraction decomposition, how to find coefficients?

How to decompose? Can anybody give a hint $\frac{1}{(x^n-1)^2}$. $\frac{1}{(x^n-1)^2}=\sum_{i=1}^{n}\frac{a_{i}}{x-\zeta_{i}}+\sum_{i=1}^{n}\frac{b_{i}}{(x-\zeta_{i})^2}$. $1=\sum_{i=1}^{n}\frac{a_{i}(...
bob's user avatar
  • 39
0 votes
1 answer
95 views

Proving that two distinct rational fractions where all integers are bounded by $N$ are at least $1/N^2$ apart

I recently came across an interesting problem in number theory, namely, Given distinct integers $a, b, c, d < N$ ( with the condition that $a<b$ and $c<d$ ) , where $N$ is also an integer : ...
requiemman's user avatar
1 vote
2 answers
51 views

What if the number ten was a single digit in the base 10 number system? How would that work? [closed]

I've recently been studying about math, and when learning about decimals. I found myself with this question. An explanation I got from Quora was: Arithmetic operations such as addition, subtraction, ...
Stim Roe's user avatar
0 votes
2 answers
45 views

Prove that : A < 1/5 < B for a Product of Fractions Sequence

Consider the numbers A and B such that: $$A = \frac{1}{2}\cdot\frac{3}{4} \cdots \frac{23}{24}$$ $$B = \frac{2}{3}\cdot\frac{4}{5} \cdots \frac{24}{25}$$ Prove that $A < 1/5 < B$. My approach: ...
Adrien's user avatar
  • 29
4 votes
2 answers
98 views

Is converting a mixed-fraction to an improper fraction is just adding the remaining part to the whole?

For example, if you're simplifying $3 \frac{1}{2}$, the conventional way is to do $3 \cdot 2$ and then add the numerator $1$. Isn't this the same as adding $3/1$ and $1/2$?
Curiousnoob72's user avatar
2 votes
0 answers
59 views

Egyptian fraction of a number in the interval (0.5,1)

Assume the real number $a$ such that $0.5 < a < 1$ and $a$ can be expressed as an Egyptian fraction of length $l$, wich means for natural numbers $n_1$ to $n_l$ we have: $$ a = \sum^l_1{\frac{1}{...
Peyman's user avatar
  • 693
1 vote
1 answer
49 views

How to calculate amortized cost of the push operations for a resizing stack

In Algorithms (4th ed.) by Robert Sedgewick and Kevin Wayne at page 199 it's illustrated how the cost of resizing a stack (based on a dynamic array) is amortized among the most recent push operations: ...
giulianopz's user avatar
2 votes
0 answers
81 views

Egyptian fraction of length 3 in the interval (0.5,1)

I've been exploring the properties of Egyptian fractions, which are representations of numbers as sums of distinct unit fractions. Specifically, I'm interested in numbers $a$ such that $ 0.5 < a &...
Peyman's user avatar
  • 693
1 vote
0 answers
36 views

Expressing one as a sum of unique unit fractions whose denominators are increasing?

Given lower bound $N\in\mathbb N$, I am wondering if it is possible to express $1$ as a finite sum of unique unit fractions: $$1=\frac{1}{n_1}+\frac{1}{n_2}+\cdots+\frac{1}{n_k}$$ where $$N\le n_1<...
John Davies's user avatar
1 vote
0 answers
41 views

Should you simplify fractions if it increases the amount of terms?

I was debating with a maths teacher about how to simplify algebraic fractions, and what their simplest form is. From the starting point of $\frac{100x+t(300-x)}{300}=75$, you can easily simplify down ...
Elliott Price's user avatar
1 vote
2 answers
150 views

Power series of $\frac{(1-qx)^k}{(1-qx)^{k+1}-(px)^{k+1}}$

I need to expand the function $$ f(x)=\frac{(1-qx)^k}{(1-qx)^{k+1}-(px)^{k+1}},\quad p>0,\;q>0,\;p+q=1 $$ into a power series $$ \sum_{n=0}^{\infty} r_n x^n. $$ $k$ is an integer $>1$. The ...
mathstudent1's user avatar
0 votes
2 answers
107 views

Could ≡ be used to represent equivalent fractions?

I was thinking about the exact definition of a "triple bar", ≡, which means "identical to". If we are being super formal, wouldn't it be more correct to say that $\frac{1}{2}≡\frac{...
Gordon's user avatar
  • 130
1 vote
0 answers
145 views

Solve for $x$: $\frac{16^x-25^x}{8^x-5^x}$

So I decided to get back to solving equations of the form$$\dfrac{c_1^x-c_2^x}{c_3^x-c_4^x}=c_5$$where $c_1,c_2,c_3,c_4$ and$c_5$ are all some number. After a while, I came up with this:$$\dfrac{16^x-...
CrSb0001's user avatar
  • 2,544

1
2 3 4 5
60