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

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5
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4answers
7k views

What's the algebraic property where you can flip the fractions in an equation?

Earlier in algebra, we spent over 20 minutes trying to figure out $$ \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e} \,\,\,\, \text{solve for }R_2 $$ when the teacher said "What you start out with is ...
2
votes
2answers
78 views

For how many integers $a$ is $\frac{2^{10} \cdot 3 ^8 \cdot 5^6}{a^4}$ an integer?

In Mathleague $11316$ Target #$4$, the question is: For how many integers $a$ is $$\frac{2^{10} \cdot 3 ^8 \cdot 5^6}{a^4}$$ an integer?
1
vote
2answers
150 views

Finding modular of a fraction

Im really into cryptography and to find the private key of a message I need to use modular arithmetic. I understand how modular arithmetic using a clock with whole numbers. But I get really stuck when ...
1
vote
1answer
104 views

GRE - Percentage Question

A full glass of juice is a mixture of 20% grape juice and 80% apple juice. The contents of the glass are poured into a pitcher that is 200 percent larger than the glass. The remainder of the ...
0
votes
2answers
240 views

Algebraic Manipulation question - trying to get alternate form

I'm currently working on algebraic manipulation, changing algebraic fractions into a chosen alternate form but I've hit a brick wall. I'm trying to get: $$\frac{2(3^x - 2^x)}{3^{x+1} - 2^{x+1}}$$ ...
0
votes
3answers
56 views

Proof of rational division…

I know that this how it's done but i used to know a simple proof which i have forgotten...so please if you know let me know.. $$\frac{\frac{a}{b}}{\frac{c}{d}} =\frac ab \frac dc.$$ When you ...
1
vote
3answers
68 views

Rationalizing a denominator.

The question instructs to rationalize the denominator in the following fraction: My solution is as follows: The book's solution is which is exactly the numerator in my solution. Can someone ...
0
votes
1answer
85 views

Algebraic Fractions, no I'm not kidding..

I know most of the people on this forum/site are very advanced, and I'm just sitting here wondering how in the world you do this equation, or "simplify" it. I know how to do equations like these when ...
1
vote
3answers
35 views

Equation with fractions

If $P=\frac{h}{1-h}$ then $h$ is equal to? Answer is: $\frac{P}{1+P}$ I understand that $\frac{P}{1+P}$ is the right answer for when I replace $\frac{P}{1+P}$ for h the answer solves the equation, but ...
-3
votes
2answers
34 views

Fractions and invitations

Diane sent out 25 invitations to her birthday party. If 1/6 of the invitations were for her family, about how many invitations were for her family? Please show your work and explain.
1
vote
2answers
60 views

Simple algebra question - separating fractions

How does the following come about? I'm completely lost. Can anyone help me fill in the steps in between? $$ \frac{s+2}{s(s+1)} = \frac{2}{s} - \frac{1}{s+1} $$ I figured that $$ \frac{s+2}{s(s+1)} ...
1
vote
2answers
57 views

How to derive duration of unemployment?

The average monthly flow out of unemployment pool of $7.0$ million people each month is $3.1$ million. Put another way, the proportion of unemployed leaving unemployment equals $\frac{3.1}{7.0}$ or ...
0
votes
0answers
51 views

Transformation of fractions

I have a problem with a certain transformation of a fraction. This is part of a reudctio ad absurdum to show that there are infinit prim numbers. $\mathrm{P} = \prod_{i=1}^{n} p_i$ as the amount of ...
3
votes
1answer
73 views

Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition: ...
1
vote
3answers
56 views

Reducing a fraction, divisibility and indeterminate symbol

Quick question about validity, just to make sure. When I have a fraction in a form: $$\frac{3a + 3b}{a+b}$$ and I extract the common factor 3 out to get: $$\frac{3(a+b)}{a+b} \;=\; 3\frac{a+b}{a+b}$$ ...
1
vote
2answers
94 views

Calculating percentage to compensate for percent discount.

Missing something very basic here and cannot pin point it. We need to charge a client \$100 for a product. Let's say our payment processor charges us 10% on every transaction. We make this ...
2
votes
8answers
442 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
0
votes
2answers
61 views

Is the product of two numbers both less than one less than one

I'm bad at mathematics, and I wanted to know something. Say there are two numbers $a$ and $b$ where $a, b \in \Bbb R$ $-1 < a < 1$ and $-1 < b < 1$ Is it necessary that $a \times b < ...
1
vote
1answer
116 views

Simplifying square root with fraction

I'm not sure about this equality $$4(-3+\sqrt {15})/4)^2 = (9-6 \sqrt{15} +15)/4$$ Hope some one can enlighten me. I will be facing more of such fractions, please guide me on how to solve/simplify ...
1
vote
0answers
26 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 ...
0
votes
1answer
32 views

If $\sum_{k=1}^{p-1}\frac{1}{k}=\frac ba$ where $\frac ba$ is an irreducible fraction, then $b$ can be divided by $p^2$?

Question : Is the following true for any prime number $p\ge 5$ ? If $\sum_{k=1}^{p-1}\frac{1}{k}=\frac ba$ where $\frac ba$ is an irreducible fraction, then $b$ can be divided by $p^2$. ...
5
votes
1answer
147 views

$\frac{\prod_{i=1}^n (1+x_i)-1}{\prod_{i=1}^n (1+x_i/\delta)-1} \stackrel{\text{?}}{\le} \frac{(1+x_n)^n-1}{(1+x_n/\delta)^n-1} $ .

Let $x_1 \le x_2 \le \cdots \le x_n$. Let $\delta>1$ be some positive real numbers. I assume that $0\le x_i <1$, for $i=1,\ldots,n$ and $x_n >0$. Does the following expression hold? $$ ...
0
votes
1answer
109 views

Fractional-Recursive Sequence

Here from the fraction set we have a really hard question to be answered...suppose that a sequence is defined as a(n) = a(n-1) - 1/a(n-1), where a(0) is given. ...you already know what I'm asking you ...
0
votes
1answer
43 views

How to work out these fractions?

$\dfrac{-x^4 + 4x^2 + 6}{x}$ $\dfrac{7x^8 - 5x^5 + 9x^3 + x^2}{x}$ I have no idea how to do this. I was first thinking of doing $-x$ or collecting up the $x's$ but I'm not sure as I haven't dealt ...
0
votes
1answer
57 views

Calculating a Fraction's Reciprocal

Is there any way or equation that allows me to calculate the reciprocal of any fraction? I mean if i have 5/6 and i need it's reciprocal by using a formula or an equation to calculate it. Is there or ...
0
votes
1answer
57 views

Embarrassingly-basic fraction question

I'm trying to do a calculation at work to figure out what the average # of pages a visitor is viewing. I am given: 47,000 visits 12% of visits do not bounce (that is, 12% navigate the site at least ...
1
vote
4answers
114 views

How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$?

How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$? What else? Well. Well can you use long division?
23
votes
4answers
722 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
0
votes
2answers
27 views

Total Time Expression Question

How do I do this question? Thanks!
0
votes
1answer
161 views

What are the big ideas needed to develop conceptual understanding of fractions?

In order to be able to perform arithmetic on fractions, students need to understand what fractions are and how they operate. Just teaching rules (e.g. "to add fractions you must have common ...
2
votes
1answer
53 views

Dividing rational expressions

How come I am able to divide the following: $$\frac{2}{2} = 1$$ Yet I am not allowed to divide the $2x$'s in the following: $$x^2 + \frac{2x}{2x} = x^2 + 1$$ Why can't I divide the $2x$ in ...
2
votes
1answer
38 views

special addition on fractions.

I would like to define a function $\oplus$ on $\mathbb Q$ ($\oplus:\mathbb Q^2\mapsto \mathbb Q$), such that (for all $a,b,c$): (commutativity) $a\oplus b=b\oplus a$ (associativity) $(a\oplus ...
0
votes
2answers
41 views

Percentage of Amounts

I'm studying and I'm not that sure how to answer this question. Is $97.1%$ $=$ $650,000,000$? I was going to find $2.9%$ of $650,000,000$ however this would be wrong as I would finding our the ...
1
vote
2answers
23k views

How do you simplify radical expressions with fractions in them [closed]

I need to know for algebra 2 homework ex. Radical 5/32
0
votes
3answers
95 views

Grade calculation

My teacher said that $\frac{1}{3}$rd of the final grade will be based on Exam 1, $\frac{1}{12}$th each based on Exam 2-5, and $\frac{1}{3}$rd again based on Exam 6, how will my teacher calculate my ...
2
votes
2answers
42 views

How to identify whether a fractional part of a number contains more that 2 digits.

EX. I want to accept numbers which have maximum of 2 digits after decimal points. i, e, 10.23 should be allowed and 10.233 should not be allowed. What mathematical operations can be done to ...
0
votes
3answers
98 views

How are fractions involving three numbers simplified.

This is something thats been bugging me since my high school. Is A/B/C = A/BC or AC/B
5
votes
2answers
1k views

Is 9/1 an improper fraction?

My son took a test in school. The teacher told them that they did not need to simplify improper fractions in their answers. On one question, for example, the answer of 28/3 was marked as correct. ...
0
votes
1answer
65 views

Unknown terms of the proportion

please help me solving this problem. The question is, find the unknown terms of the proportion $$\frac 23 = \frac x{12} = \frac y{15}.$$
6
votes
1answer
109 views

$\left\lfloor\left(\sum_{k=n}^{\infty}\frac 1{k^3}\right)^{-1}\right\rfloor=2n(n-1)$ is true for any $n\in\mathbb N$?

Question : Is the following true for any $n\in\mathbb N$? $$\left\lfloor\left(\sum_{k=n}^{\infty}\frac 1{k^3}\right)^{-1}\right\rfloor=2n(n-1).$$ Note that $\lfloor x\rfloor$ is the largest integer ...
0
votes
2answers
84 views

Simplifying Multiple Summations for worst case analysis

I'm figuring out a worst case analysis on a function. After converting it to a set of summations, and changing the sigma notations into summation formuale I ended up with: ...
4
votes
1answer
136 views

Correctness of proof of division in ceiling function

My professor asked me to present a proof to my fellow students tomorrow that $$\left\lceil\frac{n}{2^k}\right\rceil = \left\lceil\frac{\left\lceil\frac{n}{2^{k-1}}\right\rceil}{2}\right\rceil$$ I ...
2
votes
1answer
150 views

$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$ is true for any $n,N\in\mathbb N$?

Is the following true for any $n,N\in\mathbb N$? $$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$$ Motivation : I've known the $N=3$ case. ...
0
votes
3answers
306 views

simplify fraction with radical in numerator, first steps?

$(\sqrt x−2)/(x−4)$ Looking to simplify a fraction with radical in the numerator.
3
votes
1answer
148 views

About two 'negative' continued fractions whose sum equals $1$

Letting $a_1,a_2,\cdots,a_r$ be integers which are larger than or equal to $2$, let us define $$[a_1,a_2,\cdots,a_r]=\frac{1}{a_1-\frac{1}{a_2-\frac{1}{\ddots-\frac{1}{a_r}}}}$$ (Note that the ...
1
vote
1answer
112 views

Any 'odd unit fraction' whose denominator is not $1$ can be represented as the sum of three different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is 1 'odd unit fraction'. Then, here is my question. Question : Is the following ...
0
votes
3answers
72 views

Fractions and roots

I have this problem: $$\frac{\sqrt{18}+\sqrt{98}+\sqrt{50}+4}{2\sqrt{2}}$$ I'm able to get to this part by myself: $$\frac{15\sqrt{2}+2}{2\sqrt{2}}$$ But that's when I get stuck. The book says ...
12
votes
3answers
224 views

Any 'odd fraction' can be represented as the finite sum of different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is $1$ 'odd unit fraction'. Then, here is my question. Question : Is the ...
0
votes
4answers
43 views

Reducing a fraction with exponents

How do I reduce $$ \frac{(k+1)^2}{k^2} $$ to simplest terms? My algebra is really rusty... Thanks!
2
votes
1answer
122 views

About the set of all the solutions $\mathbf x=(x_1,x_2,\cdots,x_m)$ to $\sum_{j=1}^m\frac{1}{x_j}=\frac1n$

Let $m,n$ be natural numbers, and let $S_{m,n}$ be the set of all the natural number solutions $\mathbf x=(x_1,x_2,\cdots,x_m)$ to the following equation : $$\sum_{j=1}^m\frac{1}{x_j}=\frac1n.$$ ...