Questions on fractions, numbers of the form $p/q$ where $p$ and $q$ are integers, and $q$ is not zero.

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2
votes
2answers
92 views

Simplifying fractions

How to transform $$\left({1+\sqrt{1-4ab}\over2a}\right)^n-\left({1-\sqrt{1-4ab}\over2a}\right)^n\over \left({1+\sqrt{1-4ab}\over2a}\right)^m-\left({1-\sqrt{1-4ab}\over2a}\right)^m$$ into ...
8
votes
1answer
653 views

Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$

Let $x\in\mathbb{Q}$ with $x>0$. Prove that we can find $n\in\mathbb{N}^*$ and distinct $a_1,...,a_n \in \mathbb{N}^*$ such that $$x=\sum_{k=1}^n{\frac{1}{a_k}}$$
1
vote
6answers
454 views

Proof of dividing fractional expressions

For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot ...
1
vote
3answers
2k views

Distributive Property on Fractions: Swapping Denominators

I'm learning Algebra and am curious about some methodological fundamentals here. One, in particular is why the following equation: 6(2x + 1 / 3) = 6(x + 4 / 2) ...
3
votes
3answers
3k views

How to convert a fraction into ternary?

In a ternary system how is $\dfrac{1}{2}=0.\bar1$ ,$\dfrac{1}{3}=0.1=0.0\bar2$??, etc. In general, how does one write the ternary expression for a given fraction?
0
votes
1answer
246 views

Simplify a equation containing factorial, summation, and fraction

I really need some help on simplifying this math equation. Please help me reduce it as simple as possible! Thanks in advance! $$\large P_0=\frac{1}{\left[\sum_{i=0}^{M-1} ...
1
vote
2answers
85 views

Why is nothing being done to right side numerator?

From what I understand is that you have to multiply both sides by $2$, so that on the left side 2 cancel out and you are left with 3(t-7) but why does right side turn into 2t-12? So now you have: ...
0
votes
0answers
164 views

Partial fraction expansion when the degree of the numerator is unknown

Hope it's not too stupid: is there any general approach to partial fraction expansion when the degrees of polynomials in the numerator are unknown?
0
votes
0answers
109 views

Fractions for use of determining ratios

When you have two processes and you want to compare the results of them, is it: (original process)/(new process) or (new process)/(old process)? There isn't a particular context for this question, ...
10
votes
2answers
519 views

Is there any toy for learning algebraic manipulation of fractions?

Is there any toy for learning algebraic manipulation of fractions? If you don't know of any, how would you design one? What I'm imagining is something similar to a Rubik's cube whose manipulation ...
1
vote
2answers
319 views

Simplifying this algebraic fraction

I don't understand how to simplify this fraction: $$\frac{12ab-12b}{8ac-24c}$$ This is my idea to what should be done, but I think it is totally wrong: $$\frac{12ab-12b}{8ac-24c}=\frac{12\cdot a\cdot ...
1
vote
1answer
214 views

How would I solve this fraction division problem?

5/8 ÷2 1/2 I understand how to divide it by flipping the numbers (reciprocal) and what not. I just don't know what to do with the 2 next to the division sign. Help?
3
votes
1answer
179 views

Counting fractions with $n$ digits in the numerator and denominator

Playing around with fractions, I eventually had to consider the following question: Is there a formula for counting how many proper fractions in lowest terms with $n$ base-$b$ digits in both the ...
2
votes
2answers
167 views

Reducing fractions?

I want to reduce the two following fractions: $$ \frac{2x + 2y}{x + y} $$ $$ \frac{3ab^2}{12ab} $$ I fully understand the concept of reduce fractions of this type: $$ \frac{15}{20} $$ but i ...
0
votes
2answers
107 views

Solve $\frac{x-a}{b}-\frac{x+c}{d}=0$ for x

I need to solve $$\frac{x-a}{b}-\frac{x+c}{d}=0$$ for x. The answer is: $$x=\frac{ad+bc}{d-b}$$ But i can't figure out how to get there, I think i have to start by making the fractions have the ...
1
vote
5answers
649 views

How to shorten this fraction?

How to shorten this fraction? $R_1+R_2$ divided by $\frac1{R_1} + \frac1{R_2}$ The answer is $R_1R_2$. I just don't know how to get there.
1
vote
2answers
87 views

Question about a Question: Simplifying Fractions

In a question I asked several weeks ago an interim step reached was a.): $$\frac{1}{(x-6)!6!}=\frac{1}{(x-4)!4!}$$ hence b.): $$ \frac{(x-4)!}{(x-6)!}=\frac{6!}{4!}$$ I'm not following how we got ...
2
votes
2answers
304 views

Approximation of irrationals by fractions

If $\alpha$ is an irrational, and I'm trying to judge the suitability of of a rational $p/q$ as its approximation by the error $\Delta = |\alpha - p/q|$. For a given denominator $q$, I am finding a ...
22
votes
3answers
5k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
0
votes
1answer
110 views

Calculate percentage between two numbers

I have two numbers which summed up are considered to be $100\%$. The first number $n_1$ should decide the percentage amount. For e.g., $$n_1 = 5$$ $$n_2 = 15$$ so the percentage in this case ...
-1
votes
1answer
1k views

Can weighted average be used to calculate percentage increase? [duplicate]

Possible Duplicate: Is this a weighted average/percentage problem? Let's say a Marketing company has a total turnover of 10000 \$ There are 3 salesmen A,B,C with the following turnovers: ...
0
votes
5answers
405 views

Is this a weighted average/percentage problem?

Let's say a Marketing company has a total turnover of 10000 \$ There are 3 salesmen A,B,C with the following turnovers A = 2000 $ B = 3000 $ C = 5000 $ Now, ...
3
votes
3answers
924 views

How to add compound fractions?

How to add two compound fractions with fractions in numerator like this one: $$\frac{\ \frac{1}{x}\ }{2} + \frac{\ \frac{2}{3x}\ }{x}$$ or fractions with fractions in denominator like this one: ...
32
votes
7answers
1k views

Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$ This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
4
votes
3answers
335 views

Fractions with radicals in the denominator

I'm working my way through the videos on the Khan Academy, and have a hit a road block. I can't understand why the following is true: $$\frac{6}{\quad\frac{6\sqrt{85}}{85}\quad} = \sqrt{85}$$
2
votes
2answers
3k views

Exponents in the denominator?

I'm having trouble understanding exponents in the denominator. For example: I have the expression: $\displaystyle 1 - \frac{1}{3^n} + \frac{2}{3^{n+1}}$. I know that this simplifies to $\displaystyle ...
3
votes
7answers
2k views

Effect of adding a constant to both Numerator and Denominator

I was reading a text book and came across the following: If a ratio $a/b$ is given such that $a \gt b$, and given $x$ is a positive integer, then $$\frac{a+x}{b+x} ...
2
votes
1answer
473 views

Interesting problem on “neighbor fractions”

This is from I. M. Gelfand's Algebra book. Fractions $\displaystyle\frac{a}{b}$ and $\displaystyle\frac{c}{d}$ are called neighbor fractions if their difference $\displaystyle\frac{ad - bc}{bd}$ ...
1
vote
2answers
98 views

Are these fractions all equal?

Are the following expressions all equal to one another? (2ab+c)/y (2ab)/y + c/y (2a)/y * (2b/y) + c/y ...
3
votes
1answer
164 views

How to add coins fast as fractions

When you have a problem that requires you to add coins together. Is it better to use fractions? For example, you have 14 one dollar bills, 24 quarters, 12 dimes, 78 nickles, and 20 pennies. So I ...
5
votes
4answers
251 views

Writing a percent as a decimal and a fraction

I am having a problem understanding some manipulations with recurring decimals. The exercise is Write each of the following as a decimal and a fraction: (iii) $66\frac{2}{3}$% (iv) ...
2
votes
1answer
333 views

Trivial: Rationalize fraction with a third-degree root

This is a pretty trivial question. How do you rationalize a function with a denominator that contains a third degree root? Edit: My expression is $\displaystyle{\frac{1}{\sqrt[3]{2}-1}}$.
3
votes
1answer
135 views

Is this a correct proof of a fact about rational numbers?

Prove the following statement: if $\frac{p}{q} (p \in \mathbb{N}, q \in \mathbb{N})$ is a rational number that corresponds to an infinite periodic decimal fraction $\alpha$, then the rational ...
17
votes
7answers
975 views

Simple proof that $8\left(\frac{9}{10}\right)^8 > 1$

This question is motivated by a step in the proof given here. $\begin{align*} 8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\ &=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\ &\geq ...
3
votes
3answers
276 views

How can $\frac{1}{a/x-b/x}$ be equal to $\frac{1}{a-b}$?

In an exercise asking to mark true or false, it shows: $$\frac{1}{a/x-b/x}=\frac{1}{a-b}$$ It really look like false to me. But the answer is true! How can it be?
2
votes
2answers
306 views

How does this denominator cancel out to create the next step?

I was following an example in my text, and there was one step I got stuck on. Given $\frac{2x(\Delta x) + (\Delta x)^{2}}{\Delta x}$, how does the denominator cancel out to produce $2c+\Delta x$? My ...
2
votes
1answer
334 views

Upper bound/exact length of decimal expansion of simple fraction

E.g. 1/8=0.125 has three decimals when written out in base 10, but what is a good example of a simple fraction where the decimal sequence terminates but is very large? Is there some sort of rule ...
7
votes
4answers
1k views

How to simplify $\frac{\sqrt{4+h}-2}{h}$

The following expression: $$\frac{\sqrt{4+h}-2}{h}$$ should be simplified to: $$\frac{1}{\sqrt{4+h}+2}$$ (even if I don't agree that this second is more simple than the first). The problem is ...
5
votes
7answers
1k views

Best way to exactly solve a linear system (300x300) with integer coefficients

I want to solve system of linear equations of size 300x300 (300 variables & 300 equations) exactly (not with floating point, aka dgesl, but with fractions in ...
6
votes
1answer
3k views

Ratios make me feel like an idiot - help me mix up some Coca-Cola

I may be over-complicating things, but something doesn't seem right (and I swear this isn't homework, I'm friggin 30 years old :P). I want to see what it will cost me to make a TWO liter of Coca-Cola ...
4
votes
3answers
517 views

Computing a rational number between two others, minimizing numerator and denominator

Given two positive rational number $\frac{a_1}{b_1}$ and $\frac{a_3}{b_3}$ (written in lowest terms) such that $$\frac{a_1}{b_1} < \frac{a_3}{b_3},$$ I want to find a rational number ...
6
votes
2answers
184 views

How to solve this rational equation for y

Greetings! On a test recently I ended up having to solve this for y: $$ x = \frac{2y}{y + 1} $$ But I kept getting stuck in cirlces... $$ \begin{aligned} x(y + 1 ) = 2y \\ xy + x = 2y \\ ...
19
votes
2answers
1k views

Why is the decimal representation of $\frac17$ “cyclical”?

$\frac17 = 0.(142857)$... with the digits in the parentheses repeating. I understand that the reason it's a repeating fraction is because $7$ and $10$ are coprime. But this...cyclical nature is ...