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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.

0
votes
1answer
519 views

$\;32\displaystyle\left(\sum_{c}\frac{1}{7+(x-3)^2}\right)\leq \sum_{c} \frac{x^2+yz}{y+z}+6$

Let $x, y, z > 0$ Prove that $\;32\displaystyle\left(\displaystyle\sum_{c}\frac{1}{7+(x-3)^2}\right)\leq \displaystyle\sum_{c} \frac{x^2+yz}{y+z}+6$ My work, $\displaystyle\sum_{c} \frac{x^2+...
1
vote
3answers
52 views

expressing $\frac{a_0 + a_1 + \cdots}{b_0 + b_1 +\cdots}$ in terms of $\frac{a_0}{b_0}, \frac{a_1}{b_1},\ldots$

Is it possible to rewrite fractions of the form $\frac{a_0 + a_1 + \dots + a_N}{b_0 + b_1 + \cdots + b_N}$ in terms of $\frac{a_0}{b_0}, \frac{a_1}{b_1},\ldots, \frac{a_N}{b_N}$? under which ...
-3
votes
6answers
43 views

Does there lie a logic/reason behind precedence rule of mathematical operators? [closed]

To add two fractions (let's say, 4/7 and 2/3), instead of using LCM, why I can't simply add numerators first (4+2), and divide it by sum of denominators (7+3) ? Well, I know the division has more ...
5
votes
0answers
189 views

Multi-dimensional Integration

Let $C=\sum_{s=1}^tx_sx_s'+aD^{-1}$, which is symmetric positive definite, vector $x,w\in\mathbb{R}^n$ and a scalar $y\in\mathbb{R}$. The integral I am trying to solve is as follows: $$\sqrt{\frac{a\...
16
votes
2answers
1k views

How many slices of pizza do we have?

At the beginning, we had a whole piece of pizza and two people. So we cut the pizza into two slices of the same size. But at the same time, there is another guy joined us. We had to re-cut the pizza ...
1
vote
3answers
38 views

One-step equations with negatives (Fractions)

I am solving the one-step equation: $$-\frac{3}{4}\alpha = \frac{5}{4}$$ In the answer sheet it says to multiply both sides by the "reciprocal" $-\dfrac{4}{3}$. In one step equations you are supposed ...
4
votes
5answers
110 views

Solve $8\sin x=\frac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}$

Solve $$8\sin x=\dfrac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}$$ My approach is as follow $8 \sin x-\frac{1}{\sin x}=\frac{\sqrt{3}}{\cos x}$ On squaring we get $64 \sin^2 x+\frac{1}{\sin^2 x}-16=\...
0
votes
2answers
58 views

Find $\mathcal{L}^{-1} \frac{9}{(s+3)^3} $

Find $$\mathcal{L}^{-1}\left[ \frac{9}{(s+3)^3}\right].$$ How do I go about with the fraction inside? There is no fixed formula for this expression. I did a partial fraction of the repeated linear ...
1
vote
1answer
45 views

How many reduced fractions a/b such that ab=20! and 0 < a/b < 1?

(a, b have to be integers) Assuming a,b both positive, we get $ab<b^{2}$ Therefore, $b> \sqrt{20!}$ Similarly, $a < \sqrt{20!}$ I am stuck after this. Help?
-1
votes
4answers
68 views

If $x:y=2:1$ and $y:z=2:1$, then $x$,$y$,$z$ are continued proportional? and/or $z:x=1:4$? and/or $y^2+z x=4yz$?

I will prove that I attempted this math by listing the things that I know. $y=2z$ $x=2z^2$ $x=yz$ Also, did you notice something? If $x=8, y=4,z=2$ then everything works out. I figured this while ...
0
votes
0answers
19 views

Approximation for fractional function for given boundary conditions

I try to find an approximation for the following expression $$\tan\phi(x)=\frac{\mathrm{si}(x)\sin(kx)-\mathrm{si}(x-\tau)\sin(k(x-\tau))}{\mathrm{si}(x)\cos(kx)-\mathrm{si}(x-\tau)\cos(k(x-\tau))} \...
3
votes
0answers
43 views

Dirac delta from poles of a function

Suppose we are given the simple expression $$ F(k) = \frac{1}{E^2-E(k)^2} $$ which has a pole when $E^2 = E(k)^2$ and where $E, E(k)$ are real numbers. When working with this expression (e.g. inside ...
1
vote
4answers
98 views

Sum of n terms of this series

$\frac{1}{1.3} + \frac{2}{1.3.5} +\frac {3}{1.3.5.7} + \frac{4}{1.3.5.7.9}........ n $ Terms. I Know the answer to this problem but I couldn't find any proper way to actually solve this question. ...
3
votes
2answers
98 views

Evaluating $\lim_{x\to\infty}\frac{\int_0^{2x}\sqrt{1+t^2}dt}{x^2}$

How do I evaluate the following limit? $$\lim_{x\to\infty}\frac{\int_0^{2x}\sqrt{1+t^2}dt}{x^2}$$ What I've noticed so far is that due to the limit going to infinity, the integral is indefinite. ...
0
votes
0answers
31 views

An approach for this question accompanied by a solution

This question is very elementary when compared to the level of problems asked on this site. However, I am preparing for the Joint Entrance Examination in India and I needed some help in finding an ...
0
votes
1answer
48 views

Nested fraction with factorial

In a statistics book I'm studying, I'm given the following: $\frac{\binom{8}{3}}{\binom{10}{3}}=\frac{\frac{8!}{3!5!}}{\frac{10!}{3!7!}}=\frac{8\times7\times6}{10\times9\times8}=\frac{42}{90}=0.47$ ...
1
vote
1answer
21 views

Extending a ring hom from $R \to L$ to $K \to L$, where $K$ is fraction field of domain $R$.

Will this work as a proof? Let $R$ be a domain and $L$ a field. Let $f : R \to L$ be a ring hom. Let $K$ be the field of fractions of $R$. Then to extend $f$ to $K$ means there is a map $f^* : K ...
0
votes
1answer
99 views

Reciprocal of $\frac {x}{y} - 1$

What is the reciprocal of $\frac {x}{y} - 1?$ Isn’t it $\frac {y}{(x - y)}?$ But I don’t see this answer in the multiple choice. $\frac {x}{y} - 1$ = $\frac {x - y}{y}$ and reciprocal is $\frac {y}{...
0
votes
6answers
138 views

Prove that inequality $\frac{2ab}{a+b}+\sqrt{\frac{a^2+b^2}{2}}\ge \sqrt{ab}+\frac{a+b}{2}$

Let $a;b\ge 0$. Prove that inequality $$\frac{2ab}{a+b}+\sqrt{\frac{a^2+b^2}{2}}\ge \sqrt{ab}+\frac{a+b}{2}$$ My try: $LHS-RHS=\frac{2ab}{a+b}-\frac{a+b}{2}+\sqrt{\frac{a^2+b^2}{2}}-\sqrt{ab}\ge 0$ ...
0
votes
1answer
52 views

Prove that $\frac{a}{2a+\beta b}+\frac{b}{\alpha b+\beta a}\ge \frac{2}{\alpha +\beta }$

Let $a;b;\alpha;\beta>0$ and $\beta>\alpha $. Prove that $$\frac{a}{2a+\beta b}+\frac{b}{\alpha b+\beta a}\ge \frac{2}{\alpha +\beta }$$ $$LHS-RHS=\frac{a^2\alpha \beta+a^2\beta^2-4a^2\beta+ab\...
0
votes
1answer
44 views

Find the maximum constant such that the inequality

Let $a;b>0$. Find the maximum constant such that the inequality $$\frac{1}{a^2+b^2}+\frac{1}{a^2}+\frac{1}{b^2}\ge \frac{8+2k}{\left(a+b\right)^2}$$ Let $a=1$ then we have: $-\frac{k-1}{2a^2}\ge 0\...
14
votes
6answers
194 views

Why is it that $\frac ab\times\frac1c=\frac a{bc}$?

I know it may sound stupid but I'm just genuinely wondering about it.... $$\frac ab\times\frac1c=\frac a{bc}$$ where $b,c\ne0$. How can we multiply numerators by numerators and denominators by ...
1
vote
3answers
46 views

$n < m$ implies $\frac{n-j}{m-j} \leq \frac{n}{m}$?

In "Introduction to Algorithms" by CLRS, section 11.4 states: $n < m$ implies that $\frac{n - j}{m - j} \leq \frac{n}{m}$ for all $j$ such that $0 \leq j \lt m$ This section of the text assumes ...
2
votes
5answers
136 views

More intuitive solution to simplifying complex fraction?

My problem is this: $$\frac{3 - \frac{1}{x}}{\frac{1}{3x} - 1}$$ This simplifies to $-3$. So to solve this you must get everything with a denominator of $3x$ for each term in the complex fraction. Is ...
1
vote
1answer
22 views

calculating percentages from fraction confusion - business math problem

I have a bit of confusion from a word problem proposed in my business math course. The problem is as follows: "A restaurant manager wants to track attendance in his restaurant. He notes that at 7PM ...
1
vote
2answers
35 views

Value of a fraction independent of x

For $a=4$ it is known that the value of the fraction $\frac{(a+2)x + a^2 - 1}{ax-2a + 18}$ is independent of $x$. The other value of $a$ for which this is the case, belong to the interval _______. My ...
5
votes
2answers
98 views

Powers of Irreducible Polynomials in Partial Fractions

Fractions are normally decomposed by finding the numerator that goes with each individual term of the denominator of the original fraction. However, when a denominator term is repeated, the solution ...
2
votes
1answer
57 views

Find $\frac{1}{x-y+z}$ from given fraction equations.

$$\begin{align} \frac{3}{x}\,-\,\frac{4}{y}\,+\,\frac{2}{z}\quad&=\quad3\\ \frac{2}{x}\,-\,\frac{8}{y}\,-\,\frac{1}{z}\quad&=\,-\,8\\ \frac{4}{x}\,-\,\frac{6}{y}\,-\,\frac{3}{z}\quad&=\...
0
votes
1answer
117 views

Asymptotic of gamma function

I came across a quetion: Let $h$ go to zero. What is the asymptotic of $\Gamma(x+o_{p}(h))$ where $x\in(0,2)$? The difficulty is the limitation of x goes to zero. Can I obtain $$\Gamma(x+o_{p}(h))\...
1
vote
5answers
131 views

Show $\frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)}$.

If $a,b,c,d > 0$ and distinct then show that $$ \frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)} $$ I tried using HM < AM inequality but am ...
-1
votes
3answers
225 views

How to calculate a repeating decimal for any fraction?

I have been struggling for a while to try to code a program to convert any fraction 1/n to a repeating decimal. So far, my program works only for numbers that end in 1, 3, 7, or 9 (n cannot divide 2 ...
0
votes
1answer
49 views

If two fractions add up to 1, are their denominators the same? [duplicate]

If two fraction add up to 1, what is the relation between their denominators? Are their denominators equal? If $\frac{a}{b}+\frac{c}{d}=1$, then, is what is the relation between $b$ and $d$? I was ...
0
votes
0answers
23 views

Confused with a Problem Involving sums and products of numerators/denominators

Here is the problem: if the product of the numerator and denominator of a proper fraction in simplest form is 24, what is the maximum possible sum of its numerator and denominator? When we try to ...
1
vote
1answer
44 views

How to separate denominator?

I am sorry, it's probably very basic question, but I have great hole in head and can't figure it out. I need to separate denominator, like that: $ \frac { 1 } { n _ { 0 } + n _ { 1 } + n _ { 2 } + \...
1
vote
2answers
47 views

Easy expression, I’m stuck here

This is an expression obtained for resolving a circuit. I tried for 1 hour now and I’m close but I’m doing something wrong. I’m trying to find detA. (details under the photo) I’m new to this so be ...
0
votes
0answers
24 views

Creating an equation from a group of numbers and the answer

sigh... I've been trying for three days to figure out this one last question. Any help will be much appreciated, thank you. The objective is to create an equation/formula with the given information ...
1
vote
1answer
43 views

Want to Confirm Answer for the Sum of Fractions with Triangular Number Sequence as the Denominators

I am trying to compute the sum of these fractions: $$\frac{3}{1}+\frac{3}{1+2}+\frac{3}{1+2+3} + \dots + \frac{3}{1+2+3+\dots+100}.$$ I believe the denominators are a triangular number sequence, ...
3
votes
5answers
108 views

If $x, y \in \mathbb{N^{*}}$, such that $\frac{x}{y}+\frac{y}{x} \in \mathbb{N}$, show that $x=y$. [duplicate]

If $x, y \in \mathbb{N^{*}}$, such that $\frac{x}{y}+\frac{y}{x} \in \mathbb{N}$, show that $x=y$. My try: If $x$ is a multiple of $y$, such that $x \neq y$, $\frac{x}{y}$ would be greater than $1$, ...
-2
votes
2answers
45 views

A car moves at a constant speed from A place to B. [closed]

A car moves at a constant speed from A place to B. By $8$ o'clock in the morning the car covered $1/6$ part of the planned route, and by $11$ o'clock in the morning of the same day $8/9$ part. What ...
1
vote
4answers
67 views

Arithmetic doubt while studying limits of sequences

While studying limits of sequences, I came across these expressions. $\left|\frac{-5}{n+2}\right|<\delta \iff\frac{5}{n+2}<\delta \iff n+2>\frac{1}{\delta }$ $n\in \mathbb N$ $\delta\in \...
-1
votes
1answer
24 views

Is it correct to sum a daily rate for a period of time?

I have a table of daily events for a number of locations that I am trying to summarize for a week. For a weekly summary, are both of these methods correct, and how would they be interpreted? SUM(...
9
votes
0answers
89 views

Largest Numerator of Sum of Egyptian Fractions

What is the largest possible numerator when put in reduced form over all sums of the form $$\sum_{k=1}^n\frac{c(k)}{k}$$ where $c(k)\in\{-1, 0, 1\}$? An easy bound is to consider what happens when we ...
0
votes
4answers
24 views

simplifying an equation with fractions

I am trying to understand a proposed solution posted here (by user17762) to a problem in Feller's book Introduction to probability and its applications, and there is a step that I do not understand. ...
2
votes
1answer
233 views

Find the value of $\frac{4x}{y+1} + \frac{16y}{z+1} + \frac{64z}{x+1}$

Let $x,y,z$ be positive real numbers such that $x+y+z = 1$ and $xy+yz+zx = \frac{1}{3}$. Find the vlaue of $$\frac{4x}{y+1} + \frac{16y}{z+1} + \frac{64z}{x+1}.$$ So far using the fact that $(x+y+z)^...
0
votes
1answer
47 views

finding $a,b,c,d$ of $\frac{x^2 +ax + b}{x^2 +cx +d}$ from its plot

We Want to find $a,b,c,d$ of $\frac{x^2 +ax + b}{x^2 +cx +d}$ from its plot. If they can be found accurately, found the exact number. If they can be an interval, find that interval. (note that the ...
4
votes
1answer
112 views

Why does the law of distributivity use the distributive property to prove it exists?

I was looking at the Law of Distributivity and disagreed with how it was proven. It proved that $$\frac ab\bigg(\frac cd + \frac ef\bigg) = \frac ab\cdot \frac cd + \frac ab\cdot \frac ef.$$ It ...
0
votes
1answer
41 views

How to compare these two large fractions without any calculation

I want to compare two following fractions $$ a = \frac{1}{411}\cdot\frac{1}{412}\cdot\frac{1}{413}; b = \frac{1}{63990006} $$ I want to compare these without any calculations, but simply manipulating,...
2
votes
4answers
160 views

a polynomial of degree $4$ such that $P(n) = \frac{120}{n}$ for $n=1,2,3,4,5$

Let $P(x)$ be a polynomial of degree $4$ such that $P(n) = \frac{120}{n}$ for $n=1,2,3,4,5.$ Determine the value of $P(6)$. Let $P(x) = ax^4 + bx^3 + cx^2 + dx + e$. For $n=1,2,3,4,5$ I have plugged ...
0
votes
4answers
89 views

Why is a machine producing 7 nails in 9 seconds faster than one producing 11 nails in 14 seconds?

Machine A Produces 11 nails in 14 seconds. Machine B produces 7 nails in 9 seconds. Which is faster? I thought that $\frac{11}{14}$ is larger than $\frac{7}{9}$ and thus A is the answer. ...
0
votes
3answers
26 views

Question on Fraction Cancellation [closed]

Not able to get a proper explanation for this question anywhere, please help!