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

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5
votes
1answer
212 views

When is the $lcm$ of a fraction sum the actual denominator.

Consider a sum $$\frac{a}{b}+\frac{c}{d} = \frac{x}{y}$$ where each fraction is reduced. Alternatively using the familiar process of lowest common denominators, we have $$\frac{a}{b}+\frac{c}{d} = ...
2
votes
2answers
154 views

How much can a fraction reduce?

Assume $x/a$ and $y/b$ are positive fractions in it's reduced form. If $x/a+y/b=z/c$, where $z/c$ is also reduced. What can we say about $c$? Does $\frac{ab}{\gcd(a,b)^2}|c$? If it's not true. Is ...
2
votes
1answer
291 views

How to calculate percentage of comment lines in a code?

I have a file within which I have 6 lines of code and 8 lines of comment. What's the formula to calculate how much percent of the whole file comments have?
4
votes
2answers
262 views

Bound on lcm of denominators of rational numbers that sum to 1.

This is related to the question If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators? Suppose $1 = ...
3
votes
1answer
276 views

If a finite set of rational numbers sums to one, does one of the rationals have a denominator equal to the LCM of all the denominators?

I was experimenting with an algorithm for generating random numbers from a discrete distribution and came across an interesting observation. Suppose that you have any finite set of rational numbers ...
2
votes
0answers
530 views

Four candle problem: Using candles as timers

The candles each take one hour to burn completely. Cutting off bits of the candles is forbidden, but the candles are placed on a raft of fork handles so they may be burnt at both ends (e.g. to time ...
3
votes
2answers
316 views

Adding a different constant to numerator and denominator

Suppose that $a$ is less than $b$ , $c$ is less than $d$. What is the relation between $\dfrac{a}{b}$ and $\dfrac{a+c}{b+d}$? Is $\dfrac{a}{b}$ less than, greater than or equal to ...
11
votes
1answer
729 views

Interesting pattern in the decimal expansion of $\frac1{243}$

There appears to be an interesting pattern in the decimal expansion of $\dfrac1{243}$: $$\frac1{243}=0.\overline{004115226337448559670781893}$$ I was wondering if anyone could clarify how this ...
0
votes
0answers
87 views

Point me the primordial and intuitive concepts about this operations on physics

Warning: Layman question. Treat me as a 10 years old child The question was based on this page: ...
3
votes
1answer
131 views

Ways to teach fractions

I'm tutoring elementary-level kids on equivalent fractions and am not doing a very good job of explaining it. I've tried using the example of a pizza or a pie and have shown them how they can come up ...
2
votes
2answers
86 views

How do you convert $(12.0251)_6$ into fractions?

How do you convert $(12.0251)_6$ (in base 6) into fractions? I know how to convert a fraction into base $x$ by constantly multiplying the fraction by $x$ and simplifying, but I'm not sure how to go ...
2
votes
2answers
94 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
677 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
501 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
247 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
165 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
525 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
323 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
215 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
181 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
170 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
672 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
311 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 ...
23
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
417 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
973 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
350 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
4k 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
485 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
100 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
165 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
254 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
362 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
977 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
278 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
307 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
340 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 ...