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.

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problem canceling in multiplication of fractions

now that i have a grasp on formatting mathematical fractions and whatnot i present you with my latest confusion $$3\frac15 \times 1\frac23 \times 2\frac 34$$ converted into improper fractions $$\frac{...
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0 answers
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Confusion multiplying fractions: $1\frac35 \times 2\frac13 \times 3\frac 37$ [closed]

I'm studying basic engineering mathematics. I'm having trouble when multiplying fractions and canceling by division. Here's what's in the text book Problem: $1\frac35 \times 2\frac13 \times 3\frac 37$....
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-1 votes
0 answers
40 views

Find general formula for $x$ in terms of $k$. [closed]

Find general formula for $x$ in terms of $k$. $k=0$ $$x=1$$ $k=1$ $$x=\frac{1+4e^3}{1+e^3}$$ $k=2$ $$x=\frac{1+4e^3+7e^6}{1+e^3+e^6}$$ $k=3$ $$x=\frac{1+4e^3+7e^6+10e^9}{1+e^3+e^6+e^9}$$ $k=4$ $$x=\...
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0 votes
0 answers
24 views

What would be a good metric that puts heavier weight on a longer time period?

I'm currently trying to think of a metric to measure a performance of a product that gives higher points to longer contracts. For example, let's say that one contract A is a year long contract and the ...
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1 vote
0 answers
52 views

Can ratios of unsigned Stirling numbers of the first kind be simplified?

Motivation I am looking into the series $$\begin{align} \sum_{n=2}^{\infty} (\zeta(n)^{2}-1) &= 1+ \sum_{m=2}^{\infty} \frac{H_{-\frac{1}{m}}}{m} \\ &= 1+ \sum_{k=1}^{\infty} \bigg{(} \frac{1}{...
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  • 5,135
-1 votes
1 answer
33 views

Moving denominator $x$ to numerator for algebra fraction [closed]

For the question, $x=\frac{a}{{3b}}$, can the denominator of "$b$" be brought up to the numerator such that it becomes $x=\frac{a/b}{3}$ ? If not, what are the laws that I have to apply? The ...
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  • 3
0 votes
5 answers
100 views

Why does $\frac{3}{4} \times 1\frac{1}{3}$ equal $1$?

Can someone please explain how this works and how to apply it to other fractions to get 100%? I thought it might have something to do with the Dividend in the first Part (In this Example the 3 in $\...
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0 votes
0 answers
34 views

Why can we treat the numerator as a constant in partial fractions?

I'm learning the method of partial fraction decomposition as a 'useful dodge' (Silvanus Thompson, Calculus Made Easy) for calculus problems, but I'm not quite following the reasoning. According to ...
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  • 31
0 votes
1 answer
41 views

Dividing by square root of zero equals infinity?

So, my calculator app produced a result that doesn't seem correct to me. According to my calculator, $\frac{1}{\sqrt{0}}=\infty$. By my understanding, $\sqrt{0}=0$ (since $0^2=0$). So, shouldn't $\...
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0 votes
1 answer
49 views

How do you consistently reduce fractions?

in math class we are given fractions like 632/6241 and are expected to know what they reduce to in under 10 seconds. Currently, I can only guess by trial and error until I find that the common factor ...
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10 votes
2 answers
136 views

If $\sum_n \frac{1}{a_n} = 2$ where $a_n$ are positive integers, is there a subset such that $\sum_{n\in S} \frac{1}{a_n} = 1$? [duplicate]

As the title says: I'm wondering, out of curiosity, whether any (weak) Egyptian fraction decomposition of 2 always splits into two Egyptian fraction decompositions of 1. By a "weak" ...
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1 vote
0 answers
48 views

How to know without factoring whether a division will result in recurring decimals [duplicate]

Assumption: When dividing an integer n by another integer m, and representing the result as a decimal in base ...
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1 vote
3 answers
86 views

Show $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{1999}-\frac{1}{2000} =\frac{1}{1001}+\frac{1}{1002}+\ldots+\frac{1}{1999}+\frac{1}{2000}$

I am trying to show that $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{1999}-\frac{1}{2000} =\frac{1}{1001}+\frac{1}{1002}+\ldots+\frac{1}{1999}+\frac{1}{2000}. $$ It seems there is some ...
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2 votes
1 answer
80 views

The integer next above $(\sqrt3+1)^{2m}$ contains $2^x$ as a factor. Find $x$.

Question The integer next above $(\sqrt3+1)^{2m}$ contains $2^x$ as a factor. Find $x$. Attempt 1 $(\sqrt3+1)^{2m}=(4+2\sqrt3)^m=2^m(2+\sqrt3)^m=I+f$, where $I$ is the integral value and $f$ is the ...
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  • 5,394
0 votes
0 answers
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The Simplest Fraction for a Decimal

Is there a way to find the simplest fraction for a decimal? For example, let us say that I am trying to find the simplest fraction that rounds to $93\%$. I could easily say that $\frac{93}{100} = 0.93 ...
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-1 votes
1 answer
31 views

How can I evaluate this infinite series $\sum_{k=0}^\infty \frac23\left(-\frac34\right)^{k+2}$? [closed]

How can I evaluate this infinite series: $$\sum_{k=0}^\infty \frac23\left(-\frac34\right)^{k+2}$$ It was an interesting problem on a math UIL test that I haven't learned about.
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0 votes
1 answer
57 views

$20/71 \cdot 5/4 = (20+5)/71$. How to algebraically represent this oddity?

While doing math the other day, I ran across this strange equation: $$\frac{20}{71} \cdot \frac{5}{4} = \frac{20 + 5}{71}$$ At first I tried to represent it as $$\frac ab \cdot \frac{\frac ad}{d} = \...
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  • 119
1 vote
1 answer
52 views

$\int{\frac{x^3}{(x-1)(x-2)(x-3)}}dx$

Question: $$Find\int{\frac{x^3}{(x-1)(x-2)(x-3)}}dx$$ What I did: resolved into partial fraction; $$A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2) = x^3$$ $$A = \frac{1}{2},\hspace{0.2cm} B = -8\hspace{0.2cm}...
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1 vote
1 answer
31 views

How to integrate function with divisors?

I need to integrate the following function: $$\int (4x^2 +3x+7+\, \frac5{3x} -\frac3{x^2})dx$$ I'm trying to do it term by term by using the exponent rule: $$\int x^ndx = \frac1{n+1}x^{n+1}$$ So I get:...
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0 votes
0 answers
48 views

How to determine if the series $\sum^\infty_{n=1} \frac{1}{3+\cos(2^n)} $ converges or diverges?

$\sum^\infty_{n=1} \frac{1}{3+\cos(2^n)} $ I know that $2\le 3+\cos(2^n)\le4 \Rightarrow \frac{1}{4}\le\frac{1}{3+\cos(2^n)} \le1/2 \Rightarrow \frac{1}{3+\cos(2^n)} \ge \frac{1}{4}$. Which test can ...
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  • 479
1 vote
0 answers
8 views

Relative or Absolute Percent of Changing Total?

I don't have a math background so am lacking in intuition when it comes to problems like this. I have X green apples out of a total of Y apples (red, green, etc). Let's say X=5 and Y=20, so I have 25% ...
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0 votes
0 answers
25 views

How to convert an improper fraction with variables to a mixed number?

I have fractions like these: $$\frac{12000x}{41}\qquad (1)\\\frac{12000+x}{41}\qquad (2)\\$$ What is the process to convert these to a mixed number (example: $5\frac12$)?
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3 votes
1 answer
70 views

Is there a way to find the $n$th term of a Farey sequence?

I was doing some reading on Farey Fractions and was curious if there is a method to find the $n$th term in a particular Farey sequence? I know you could do this with a computer search, but at large ...
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0 votes
1 answer
69 views

Fastest way to simplify large fractions?

Suppose we have a large fraction like this $\frac{2002}{429}$ which we could simplify by e.g. using the euclidean algorithm to find out the GCD. but is there kind of like a "trick" to find ...
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  • 193
0 votes
2 answers
78 views

How does one express $0.0\overline{1410}$ as a fraction?

How does one express $0.0\overline{1410}$ as a fraction? I know the answer is $141/9999$ but I am not sure how to derive it using some formula from below. Wikipedia says assume $x=0.a_1a_2...a_n.$ $...
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  • 383
1 vote
2 answers
58 views

How does $\frac{k^2(k+1)^2}{4} + (k+1)^3$ become $\frac{(k+1)^2(k+2)^2}{4}$?

As part of an induction proof, the authors of a beginner combinatorics text reduce/factor a polynomial as follows, but do not show the minutiae of their algebraic steps: $$\frac{k^2(k+1)^2}{4} + (k+1)^...
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0 votes
1 answer
67 views

Can I divide inequalities if all numbers are positive?

Let P be a real number such that $1<P<2$, and $m,n\in\mathbb{N}$ where $m<n$. Let $y\in\mathbb{N}$, and suppose that $y>1+\frac{1}{\sqrt[n-m]{2/P}-1}$. I am trying to show that $\left(\...
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2 votes
1 answer
62 views

How consistent can solving an equation be if one way of solving it might give an undefined answer?

If solving for an equation one way gives me an undefined answer, does that mean solving it in other ways too will give me an undefined answer? If no, then how consistent is math? (I'm not sure if I'm ...
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-1 votes
1 answer
35 views

On inequality of fractions and exponential expressions [closed]

I am having a hard time proving this proposition: Show that for $x_i, n \in \mathbb{N}$, where $x_i\geq 2$ and for any real number $r>1$, if $\prod\limits_{i=1}^{n}\left(1-\frac{1}{x_i}\right) \leq ...
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0 votes
0 answers
33 views

How to write "concave" equals sign in latex

I am reading a mathematical paper and they use this concave looking equals sign: I was unable to find how to write this in LaTeX. I don't even know what this sign is called. They use it to denote ...
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0 votes
1 answer
62 views

Sum of $\frac{25^x}{25^x+5}$

Find $$\sum_{n=1}^{1999}\frac{25^{\frac{n}{2000}}}{25^{\frac{n}{2000}}+5}$$ My work: $$\frac{25^x}{25^x+5}=\frac{5^{2x}}{5^{2x}+5 }$$ I don't see a pattern here, and I don't think the point of this ...
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  • 763
4 votes
4 answers
112 views

Simplification of $\frac{2\sqrt{21}-\sqrt{35}+5\sqrt{15}-16}{\sqrt7+2\sqrt5-\sqrt3}$

Simplify $$\dfrac{2\sqrt{21}-\sqrt{35}+5\sqrt{15}-16}{\sqrt7+2\sqrt5-\sqrt3}$$ Final solution should have rational denominators. Suppose the solution is $X$, I have tried to make up an equation for $...
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  • 763
0 votes
0 answers
13 views

Percentage of goal, target range

first time question-asker here. Not sure if I'm wording the title right but I'll try to explain. This is a business problem. I have a set of goal percentages (50%, 100%, 150%) and a corresponding set ...
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  • 1
0 votes
1 answer
52 views

What does a field of fractions has to do with localization and also integrallity?

We have discovered today the notion of a field of fractions $(R\setminus \{0\})^{-1}R$ where $R$ is a ring. Somehow I we did not discussed much how this is used in localizations or even more with ...
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  • 1,631
3 votes
2 answers
62 views

How to use the $x^n - a^n$ factoring pattern if $n$ is not an integer?

$$x^n - a^n = (x-a)(x^{n-1} + x^{n-2}a \ + \ ... \ + xa^{n-2} + a^{n-1})$$ This works when $n$ is an integer, for example: $x^3 - a^3 = (x - a)(x^2 + xa + a^2)$. This also works when factoring $x^1 - ...
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  • 33
0 votes
1 answer
24 views

Show that the elements of $\mathbb{K}(T)$ that are algebraic over $\mathbb{K}$ are the elements of $\mathbb{K}$.

Let $\mathbb{K}$ be a field. Show that the elements of $\mathbb{K}(T)$ that are algebraic on $\mathbb{K}$ are the elements of $\mathbb{K}$. Let $f = \frac{P[T]}{Q[T]}$ an algebraic element of degree $...
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  • 337
8 votes
2 answers
276 views

Reduce precision of fraction

Say I have a reduced fraction where the numerator and denominator can only be integers: $$ \frac{1071283}{28187739} $$ and I want to reduce it more, accepting the lose of precision. I could just ...
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  • 191
0 votes
1 answer
20 views

What is the shortest and most preferred way of writing the result?

Let's say I have to calculate 6/4 (i know, it's simple). Would the preferred answer be 3/2 or 1.5? Same with square roots: is sqrt(2) a better answer than 2^(1/2)?
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  • 101
0 votes
3 answers
66 views

Calculation on fraction of square roots raised to the power 4

Calculate $$\left(\frac{\sqrt{13}+\sqrt2-3}{\sqrt2}\right)^4+\left(\frac{\sqrt{13}-\sqrt2+3}{\sqrt2}\right)^4$$ The numerator part can be re-written as $$\left(\frac{\sqrt{13}+(\sqrt2-3)}{\sqrt2}\...
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  • 763
0 votes
1 answer
79 views

Irrational Decimals In Reality? How To Imagine?

We know that the use of rational numbers is easy to imagine when we use them in real life. Like dividing a particular area of land by $2.25$ will give us an exact quantity which can actually be used ...
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0 votes
1 answer
35 views

Percentage of contribution using average

I want to calculate how much an individual employee in a Business Unit (BU) of a company contributed to the overall total expense (salary) of that BU. I have the company wide calculated Average Salary ...
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  • 1
0 votes
0 answers
39 views

HCF and LCM of fractions

If a/b and c/d are two fractions in their simplest forms then there HCF is HCF of numerator/LCM of denominators.My question is that is HCF of numerators coprime with LCM of denominators?
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2 votes
1 answer
32 views

Which of the following CANNOT be the possible number of boys and girls in the class?

The ratio boys to girls is 5/4. If a few boys leave the class, the ratio boys to would become the reciprocal of the earlier ratio. Which of the following CANNOT be the possible number of boys and ...
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  • 1,036
0 votes
0 answers
23 views

When are fractions of factorials equal to whole factorials? [duplicate]

10!/7!=6!. 6!/3!=5!. Is there something interesting happening in these examples, or are they just coincidence? Is there a way of finding other examples like these?
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0 votes
1 answer
47 views

Fraction Word Problem: Please help me Understand it.

Multiple Choice: How many otters are there in a group of 27 penguins and otters if there are four fifths as many penguins as otters? My reasoning is $\frac{1}{5} \cdot \frac{27}{1}$ are otters. So ...
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  • 1
0 votes
1 answer
43 views

How to solve $\frac{8x+3}5-\frac{11x-9}6+\frac{4x+3}{15}=\frac{11x+15}{10}$

Here's an equation: $$\frac{8x+3}5-\frac{11x-9}6+\frac{4x+3}{15}=\frac{11x+15}{10}$$ It looks simple, and I know the answer $- 3/4$, but can't arrive at it. Instead, I arrive at $1.75$ by reducing all ...
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  • 9
1 vote
3 answers
50 views

How does $\frac{1}{2jw(1+jw)}$ become $\frac{-j(1-jw)}{2w(1+w^2)}$, where $j^2=-1$?

I am trying not exactly to solve equation, but just change it from what is on right side to what is on left side. But I didn't do any math for years and can't remember what to. $$\frac{1}{2jw(1+jw)}=\...
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  • 31
0 votes
1 answer
33 views

Simplification error on fraction

$$\frac{7 !-6 !+5 ! \cdot 4 !}{6 !}$$ is the fraction I want to evaluate. Dealing with this, I come up with the following wrong expansion. What point do I forget? $$ = \frac{4! \cdot{\color{Green} (}(...
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  • 351
1 vote
0 answers
37 views

Sum of fractions in lowest terms

Let $a, b, c, d$ be natural numbers. Suppose we want to find the sum $\frac{p}{q}$ of the fractions $\frac{a}{b}$ and $\frac{c}{d}$ in lowest terms, that is, $p$ and $q$ are relatively prime. Suppose ...
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0 votes
0 answers
71 views

Simplifying Products of Factorials

I'm not sure if there's any way to do this, but I figure it's worth a shot. Is there any way to simplify the number of products of factorials into a smaller expression? For example, I have the ...
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