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.

224 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
1
vote
0answers
32 views

Economics Math: Optimal allocation of free time from Core Econ website. MRT = MRS

CONTEXT: I have the following equation for MRT=MRS (marginal rate of transformation = marginal rate of substitution): $\frac{αy}{24-t} = \frac{ay}{bt}$ I am asked to "multiply through" by $\frac{t}{y}...
1
vote
0answers
44 views

A quick way for decomposing fractions

The complete method for decomposing fractions is obvious . For example $y = \frac{2x+1}{(x-1)(x+3)} = \frac{a}{x-1} + \frac{b}{x+3} = \frac{a(x+3) + b(x-1)}{(x-1)(x+3)} \Rightarrow $$ \left\{ \begin{...
1
vote
2answers
63 views

What's the value of $x$ if $\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}$ and…

Given $x, y, z\in \mathbb{R}$, such that $\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}$ and $ \frac{1}{y}+\frac{1}{z+x}=\frac{1}{3}$ and $\frac{1}{z}+\frac{1}{x+y}=\frac{1}{4}$. Find the value of $x$.
1
vote
1answer
226 views

Anomalous Cancellation definition

Quoting mathworld: Anomalous cancellation is a "canceling" of digits of $a$ and $b$ in the numerator and denominator of a fraction $a/b$ which results in a fraction equal to the original. Note ...
1
vote
0answers
43 views

Name of maths technique?

I'm currently looking at runs of identical outputs, and looking at the fractions I get back. When I was running through it, I had a feeling I'd come across something similar before, and it would be ...
1
vote
0answers
24 views

the same convergents among 3 irrotations

Let a, b, c be irrational numbers with a < b < c. If a and c have identical convergents $\frac{p_0}{q_0}$, $\frac{p_1}{q_1}$, . . . , up to $\frac{p_n}{q_n}$, how to prove that b also has these ...
1
vote
2answers
71 views

Two Positive Real Numbers $m$ and $m$ such that $m<n$ and $\sqrt{m}<\sqrt{n}$

This is a homework problem I came across for my discrete math class and I can't manage to figure out the answer. I know it is asking for an example where $m$ and $n$ are positive real numbers, $m$ is ...
1
vote
1answer
30 views

Cancelling (prime) roots of unity

I have to show that the fraction $$\frac{u^{pq}-1}{(u^p-1)(u^q-1)}$$ is of the from $$\frac{f_{p,q}(u)}{u-1}.$$ I'm searching for a short proof. If we factorize, the p'th and q'th roots of unity ...
1
vote
0answers
81 views

What type of relationship intercede between fractions and division?

Here's what I thought: Let's start with a fraction, it is just a number which represent a value by telling us how many pieces of a certain size we have. Now, what about another way to represent this ...
1
vote
1answer
28 views

How do I describe this relation?

I have an equation and I would like to express in words the relationship between a and b, where $$a = \frac {1}{x - b}$$ in a similar (physics-y) vein to "a Is directly/inversely proportional ... to ...
1
vote
0answers
44 views

How does one prove the following inequality is true?

I can prove that if $\frac{a_{i}}{b_{i}} \geq \frac{a_{j}}{b_{j}} $, it follows that $\frac{a_{i}}{b_{i}} \geq \frac{a_{i} + a_{j}}{b_{i} + b_{j}} $. However, I'd like to generalise this result by ...
1
vote
0answers
68 views

Simplified Fraction.

My son has been set homework with half-completed fractions along a line of $30$. He has to complete the missing half. For example, there is an indicator at the $10$th line and the fraction is half ...
1
vote
0answers
51 views

What is the reduced ratio for $\frac{\left\{\begin{array}{c}n\\k\end{array}\right\}}{\binom n k}$?

Looking at the table of $\frac{\left\{\begin{array}{c}k+m\\k\end{array}\right\}}{\binom{k+m}k}$, $1\leqslant k,m\leqslant10$, where $\left\{\begin{array}{c}k+m\\k\end{array}\right\}$ is the Stirling ...
1
vote
1answer
43 views

Make $t$ the Subject

Could someone explain how to get to an answer for this GCSE question, please? Thank you. "Make $t$ the subject of the formula:" $$k = \frac{2(t+3)}{t-3}$$
1
vote
2answers
134 views

Surds question grade 9

I am a student and I need help answering this question. Simplify: $\frac{6}{\sqrt{28}}$-$\frac{9}{\sqrt{63}}$ What I did: $\frac{6}{\sqrt{28}}$- $\frac{9}{\sqrt{63}}$ =$\frac{6}{\sqrt{7×4}}$ - $...
1
vote
0answers
33 views

Inequality involving irreducible fractions and constants

Let $\frac m n$ be an irreducible fraction such that $1 \gt \sqrt 2 + \sqrt 3 - \frac m n \gt 0$. Show that there is a constant $c \gt 0 \space$ such that $$\sqrt 2 + \sqrt 3 - \frac m n \gt \frac {1} ...
1
vote
0answers
77 views

Given a fraction a, what's the terminology for a fraction b, which when added to a equals 1?

As the title says... Given a fraction x, what's the terminology for a fraction y, which when added to x equals 1? I'm assuming there must be a name for this. Something similar 'reciprocal' in ...
1
vote
0answers
341 views

Splitting a fraction

Question: How do you split$$\dfrac {1}{\left\{1+\dfrac {x^2}{a^2}\right\}\left\{1+\dfrac {x^2}{(a+1)^2}\right\}\left\{1+\dfrac {x^2}{(a+1)^2}\right\}\cdots\left\{1+\dfrac {x^2}{(a+n-1)^2}\right\}}\tag{...
1
vote
1answer
506 views

Prime numbers which 1/p has decimal period of p-1

How to find prime numbers where $\frac{1}{p}$ has decimal period of $p-1$? Is there any formula? please help. Example: $\frac{1}{7}=0.142857142857...$ with decimal period of $7-1=6$ also $17$ ...
1
vote
1answer
23 views

Equation transposition

I am struggling to see how an equation has been transposed. It is probably pretty easy, so apologies in advance. If possible; please show the transposition stages: $V_{out} = V1 . \frac{R2}{R1 + R2} ...
1
vote
2answers
180 views

Factorial simplification with fractions

I am attempting to simplify the expression $$\frac{\left(\frac{x+y}{2}\right)!\left(\frac{x-y}{2}\right)!}{y!}$$ I'm familiar with expanding expressions like $$y! = (y)(y-1)(y-2)\ldots$$ but I ...
1
vote
0answers
295 views

What is the difference between a fraction, ratio, quotient, rational number?

They are mixed everywhere and I can't find a single article that treats them right and with the required rigor they should get. Can someone explain them?
1
vote
1answer
17 views

Finding x with other variables involved

How do you find $x$ when there are also other variables like $P, A, t$ $$P=\frac{A}{1+xt}$$ Here's what I did, I'm not sure if I'm correct LCD: 1 + xt $$1+xt(P=\frac{A}{1+xt})$$ $$(1+xt)P =1+xt(\...
1
vote
0answers
55 views

Is this fraction a natural number only in case ${n_1}^2+{n_2}^2+{n_3}^2={m_1}^2+{m_2}^2+{m_3}^2$?

Suppose that $m_1,m_2,m_3,n_1,n_2,n_3 \in \mathbb N$ and $m_1<m_2<m_3$ and $n_1<n_2<n_3$. If $\dfrac {{n_1}^2+{n_2}^2+{n_3}^2-3}{{m_1}^2+{m_2}^2+{m_3}^2-3}$ is a natural number, ...
1
vote
0answers
140 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 ...
1
vote
5answers
184 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-...
1
vote
0answers
77 views

unique ringhomomorphism from the Field of fractions to another field

$R$ is a ring, $L$ a field and $K$ the fraction field constructed from $R$. For any injective ring homomorphism $f=R \rightarrow L$, there is a unique ring homomorphism $\tilde{f}:K \rightarrow L$ ...
1
vote
1answer
88 views

Partial Fraction Decomposition - Multiple Answers-Question

Now I do understand how partial fraction decomposition works and why you can do it, but there is one case that I don´t understand. And that is, the following: $$\frac{A_{1}}{(x-x_{1})} + \frac{A_{2}}{(...
1
vote
1answer
55 views

Auto loan calculator website widget

I was going to try my luck on StackOverflow, but I have a feeling my issue here is on order of operations. I'm using the loan calculation found here to build a loan calculator for a clients website. ...
1
vote
0answers
49 views

How much information is missing?

If we know the value of $\frac{(a-b)}{(c-d)}$, can we calculate the value of $\frac{(a-d)}{(c-b)}$ That is : Let $\frac{(a-b)}{(c-d)}=k$ , can we calculate $\frac{(a-d)}{(c-b)}$ in terms of $k$ And if ...
1
vote
1answer
67 views

When graphing both X, and Y are fractions

In my instructions, I am told to place the point on the coordinate system. My X, and Y value are $$(\frac 52, \frac 72)$$ at this point I am a little lost. Would I flip it, and multiply it like so? ...
1
vote
0answers
532 views

quotient of two differentiable functions is differentiable

I have two functions $k(t)$ and $l(t)$ in a certain closed interval $[a,b]$ both functions are continuous and differentiable in the interval. In addition we have: Both functions are increasing with ...
1
vote
0answers
143 views

three fractions between π and 22/7

three fractions between π and 22/7 π=355/113 =3.14159 22/7=3.1428 using a/b 355/113<22/7 then a+b/c+d 1) 355+22/113+7 =377/120~ 3.14166 2) 377+22/12+7 =399/127~3.14173 3) 399+22/127+7 =421/134~3....
1
vote
0answers
648 views

How many elements are in the following set?

The set is $$\{ x \in Q:x^2 =64/25 \} $$ I thought the answer was $\{ \frac{8}{5}, -\frac{8}{5} \}$ but I am told there are in fact 4 distinct elements: $$\{ \frac{8}{5}, \frac{8}{-5}, \frac{-8}{5}, ...
1
vote
1answer
258 views

How to calculate the Integer portion of a fraction using only +, -, $\div$ and *?

I made something in excel that calculates the days left until a given date, and from that how many weeks were left. I had it so that 9 days displayed as 1.2 using this formula: $$\frac{\left(\frac{A}{...
1
vote
0answers
23 views

Extracting a function of a variable from an expression

I have this expression: $\frac{d+2wd}{2w+3wd-3d-w^2-1}$ Is there anyway I can write it just as a function of f(d)? [To me this looks like it is already a function of d, but I want to confirm if ...
1
vote
0answers
263 views

exponential integration with fractional powers

I am trying to solve the following integral $$\int_{-\infty}^a \frac{\beta_1 \beta_2}{y^2(c-y)^2} e^{-\beta_1/(c-y)} e^{-\beta_2/y} \, dy$$ where $a<0$, $c>0$, $\beta_1>0$, $\beta_2>0$ I ...
1
vote
0answers
66 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 $ \sum_{i=...
1
vote
0answers
44 views

Rational exponentiation?

Consider the following operation: $\left(\frac{a}{b}\right)^\frac{n}{m}$ where $a, n\in\mathbb{Z}$ and $b, m\in\mathbb{N^*}$. My question is: when the result is a rational number, how (formula or ...
1
vote
0answers
92 views

Does there exist an operation which partitions any fraction into the sum of the minimum number of unit fractions?

Motivation : I've been interested in finding an operation which partitions a fraction into unit fractions. The following is one of the operations which I've found. Let's start a rational number $q_0$ ...
1
vote
0answers
95 views

Turn a number $x$ into a fraction with a denominator with no more than $k$ digits

Is there a function for turning any number $x$ into a fraction with a denominator that has a maximum of $k$ digits? (I'm sure there is, since Excel has one built in, I just can't figure out what it ...
1
vote
1answer
313 views

If I add a constant $c$ to each fraction's numerator and denominator in a sequence of fractions, how is the sequence affected?

Given a sorted ascending sequence of fractions, if I add a constant $c$ to each fraction's numerator and denominator, how is the sequence affected? For example, if I have a sequence in ascending ...
1
vote
0answers
2k views

Contribution (weighted average) of change in rate over time

I'm trying to determine the weighted average impact of one customer's change in rate on the total change in effective rate. Let's say I have two customers and two time periods: ...
1
vote
2answers
48 views

Stuck on candy bowl fraction

I am really stuck on this problem because I'm not even sure where to start. Larissa has a bowl of candies. On the first day, she eats 1/2 of the candies plus one more. On the second day, she eats 1/3 ...
1
vote
1answer
95 views

Simplify into partial fractions $\frac{x^8}{x(x^2+1)^2}$

Simplify into partial fractions $$ \frac{x^8}{x(x^2+1)^2}$$ I tried many times but after a point a $"x"$ of denominator vanishes. Can't all fraction be shown in its partial fraction? If not ...
0
votes
0answers
23 views

Approximation of binomial

I have the following quantity: $$\frac{gp+1-(gp+1)^{\frac{x}{x+1}}}{gp}$$ I have the value of $g$ and $x$ but I do not have the value of $p$. Another issue is that $gp$ is greater than $1$. Any ...
0
votes
0answers
38 views

Expression evaluation/Book

I have a problem with deriving the second displayed formula in the snippet (which comes from the book Game Theory by Tirole and Fudenberg): in the square brackets should be $$\Bigg [\frac{3(1-\delta)+...
0
votes
0answers
13 views

Equation to find when two cycles of different periods synchronize naturally

I am trying to find an equation that will predict exactly when two cycles of different periods will re-synchronize naturally. Here is a pic of some cycles with different periods: https://i.stack....
0
votes
0answers
28 views

Integral of a squared sum of sin + exponential

What is the best way to solve this integral? $$\int_{0}^{\pi/2}\bigg(\frac{\sin^{2}\left(x\right)}{\sin^{2}\left(x\right)+c_{1}}e^{m\frac{\sin^{2}\left(x\right)}{\sin^{2}\left(x\right)+c_{1}}} + \...
0
votes
0answers
14 views

How to rewrite the following equation as x in terms of y?

How to convert the following function y of x $y=f(x)=\frac{x}{x+a}+\frac{x}{x+b}$ into a function x of y $x=g(y)=?$ in its simplest form. This is part of an integration problem so having no ...