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

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-1
votes
1answer
16 views

Simplify the numerical expression [on hold]

9 1/3-12 1/2+(-4 1/6)-(-1 1/6) Simplify the numerical expression
35
votes
1answer
727 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
4
votes
1answer
67 views

Prove the equality $\frac{x+y+z}{a+b+c}=\frac{ax+by+cz}{a^2+b^2+c^2}$

If $\displaystyle\frac{x+y}{3a-b}=\frac{y+z}{3b-c}=\frac{z+x}{3c-a}$ then prove that $\displaystyle\frac{x+y+z}{a+b+c}=\frac{ax+by+cz}{a^2+b^2+c^2}$ I tried to prove this in many ways. First, I tried ...
0
votes
2answers
24 views

multiply fraction with what number to get a whole number?

I'm solving some programming puzzle and it has come down to this: I've a fraction, say 12/13, and I need to multiply it with a smallest possible natural number (say x) to get a whole number. How do I ...
1
vote
2answers
28 views

How do I convert a fraction in base 10 to a quad fraction (base 4)?

I am totally confused when it comes to converting fractions or floating point numbers to a different base. I have no problem converting whole numbers to any base but when it comes to fractions or ...
0
votes
0answers
38 views

Order of Operations, performing operations, fractions [on hold]

I don't understand my homework. How do I perform a fraction operation?
1
vote
1answer
27 views

Fractions and stretch your thinking. [on hold]

Carol's book shelf has 4 shelves with 6 books on each shelf. Her brother Robert has 3 shelves and 7 books on each. They want to combine their books. They put 9 books on a shelf; how many shelves will ...
-3
votes
1answer
28 views

Rationalising Top Heavy Surds [on hold]

What is $$\dfrac{12 - 5\sqrt{3}}{\sqrt{3}}$$ expressed in the form $$a + b\sqrt{3}$$ where $a$ and $b$ are integers? Give the correct answer and your method.
6
votes
3answers
181 views

Why do these fractions give $99…9$?

Today, as usual, we were doing all those boring numerical computations in our calculators. It all started when my professor replaced $0.2$ with $1/5$. I got into calculating the unit fractions one by ...
0
votes
1answer
18 views

Solving $\frac12 (3y+2)-\frac58=\frac34y$ for $y$ using LCD method

I am solving $$\frac12 (3y+2)-\frac58=\frac34y$$ for $y$ using LCD method. Can't figure out what I did wrong! The answer in the back of the book is $-1/2$. PS: In the first line that is a $1/2$ in ...
17
votes
1answer
513 views

Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$

Simplify $$\frac{\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}{\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}$$ I don't have any good idea. I need your help.
0
votes
4answers
66 views

Is $\frac{4x + 2}{12 x ^2}$ simplifiable?

I'd like to know what methods can I apply to simplify the fraction $\frac{4x + 2}{12 x ^2}$ Is it valid to divide above and below by 2? (I didn't know it but Geogebra's Simplify aparantly does ...
0
votes
1answer
31 views

How can I find $x$, $y$ values for $\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i$

$$ \frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i $$ I believe the format I need in order to solve this problem should be such that the real parts and imaginary parts are separated, ...
3
votes
2answers
728 views

How do I get the integer part of a number by using basic arithmetic?

While it is trivial to simply remove the fractional part of an irrational or rational number, and in programming I could just use the floor() or ...
0
votes
3answers
64 views

Divison of Fractions

Intuitively answer of $(1/1)/(1/(5^{-2}))=25$ But assuming this mathematical logic of evaluating $(a/b) /(c/d) = (a*d) / (b*c)$ equation evaluates to $1/25$. Is there any specified rule to put ...
0
votes
0answers
38 views

Do parts of a fraction always have to be equal? [closed]

Do parts of a fraction always have to be equal? At first I think "of course!" -- but can different things about a fraction be equal? For example, when talking about a "half" of a piece of paper, ...
2
votes
3answers
32 views

Simplifying nested/complex fractions with variables

I have the equation $$x = \frac{y+y}{\frac{y}{70} + \frac{y}{90}} $$ and I need to solve for x. My calculator has already shown me that it's not necessary to know y to solve this equation, but I ...
0
votes
2answers
27 views

Simplifying Fractions with Radicals

How would I simplify a fraction that has a radical in it? For example: $$\frac{\sqrt{2a^7b^2}}{{\sqrt{32b^3}}}$$
3
votes
1answer
92 views

How can I do this? $\int\frac{dx}{x^4+1}$ [duplicate]

I tried to integrate this: $\displaystyle\int \dfrac{dx}{x^4+1}$ I tried to do it with the partial fractions method (after factoring the denominator), but the process is really large, and I got a lot ...
4
votes
4answers
499 views

What is the non-trivial, general solution of these equal ratios? [closed]

Provide non-trivial solution of the following: $$\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}$$ $a=?, b=?, c=?$ The solution should be general.
1
vote
4answers
162 views

Can a fraction be simplified like this?

Ridiculously embarrassing question, but can $\frac{x^2-x}{x^2-25}$ be simplified to simply $\frac{1-x}{1-25}$? Full thought process here is that this is essentially $\frac{x*x-x}{x*x-25}$ so the $x$s ...
25
votes
14answers
5k views

Logic behind dividing negative numbers

I've learnt in school that a positive number, when divided by a negative number, and vice-versa, will give us a negative number as a result.On the other hand, a negative number divided by a negative ...
0
votes
0answers
30 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
0
votes
4answers
84 views

breaking up fractions

I have these two fractions ${11 \over 31 }+{-11 \over 61}$ Adding them gives $330 \over 1891$ But how do I go back to the two fractions, once I've added them? I can get the denominators just by ...
0
votes
1answer
29 views

Spending fraction of salary

I have this question and kind of confused... Mary spent $1/4 $ of her salary in for rent and $1/4$ more than rent for car payment. Which of the following could be the fraction of her savings if ...
3
votes
3answers
86 views

How to combine ratios? If $a:b$ is $2:5$, and $c:d$ is $5:2$, and $d:b$ is $3:2$, what is the ratio $a:c$?

How would I go about solving this math problem? if the ratio of $a:b$ is $2:5$ the ratio of $c:d$ is $5:2$ and the ratio of $d:b$ is $3:2$, what is the ratio of $a:c$? I got $a/c = 2/5$ but that is ...
4
votes
3answers
70 views

Ratios as Fractions

I’m having trouble understanding how fractions relate to ratios. A ratio like 3:5 isn’t directly related to the fraction 3/5, is it? I see how that ratio could be expressed in terms of the two ...
2
votes
4answers
50 views

Simplify $\frac{x}{c} - \frac{x}{c-d}$

There's a long time that I don't solve questions like this one. I'm having problems to simplify this one: $$\frac{x}{c} - \frac{x}{c-d}$$
2
votes
1answer
73 views

Is there a mathematical concept of fractions using transfinite numbers as numerators and denominators?

http://de.wikipedia.org/wiki/Cantors_erstes_Diagonalargument (German) http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument (English) While looking at Cantors method of proof, which he used to ...
0
votes
1answer
41 views

how to tell a fraction in denominator or numerator should be substituted with its integer equivalent

Suppose we have equations as follows (A, C and B are all integers and $\gcd$=greatest common divisor). $$R_1 = \frac{A\times C}{B} \hspace{2cm} R_2 = ...
3
votes
3answers
121 views

What is $\lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$?

These are two little questions that came to mind while I was looking at this problem. What is $\displaystyle \lim_{n \to \infty} \sum_{x=0}^{n-1} \frac{n-x}{n+x}$? I am fairly certain that the ...
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 ...
8
votes
1answer
384 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
3
votes
4answers
99 views

Why does $\dfrac{8}{\frac{8\sqrt{145}}{145}} = \sqrt{145}$?

I can't seem to work out why this is true: $$\frac{8}{\dfrac{8\sqrt{145}}{145}} = \sqrt{145}$$ Could someone break it down for me?
6
votes
3answers
308 views

Recognizing the sequence 1/16, 1/8, 3/16, 1/4, 5/16, …

What is the missing number? $$\frac{1}{16}, \frac{1}{8}, \frac{3}{16}, \frac{1}{4}, \frac{5}{16}, \ \ \ [?]$$ $$A. \frac{5}{4}\quad B. \frac{3}{4}\quad C. \frac{5}{8}\quad D. \frac{3}{8}$$ ...
3
votes
4answers
111 views

Deeply confused about $\sqrt[5]{a^5}=(a^5)^{1/5}$

So is this correct? $\sqrt[5]{a^5} = \left(a^5\right)^{\frac{1}{5}}$ I need proof why $\left(a^5\right)^\frac{1}{5}$ can or cannot just be $a^\frac{5}{5}$ or just $a$? I think of that rule of ...
0
votes
2answers
75 views

How does $\sqrt {\frac{{4 + \sqrt {15} }}{8}} = \frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$

I have the follow answering to a question from my textbook: $\sqrt {\frac{{4 + \sqrt {15} }}{8}}$ However my textbook simplifies it to: $\frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$ I've checked and my ...
0
votes
5answers
47 views

imaginary number evaluation

Question Let $z_1 = 1 + i$, $z_2 = 2 - i$, evaluate $$\left | \frac{z_1}{z_2} \right |$$ I have this question! Its to evaluate the fraction ! what I did is the following ...
0
votes
2answers
59 views

How to transform $2,(9)$ to form $\frac{a}{b}$

How to transform $2,(9)$ to form $\frac{a}{b}$. My attemp: $$x=2,(9)/\cdot 10$$ $$10x=29,(9)$$ $$10x-x=29,(9)-2(9)$$ $$9x=27$$ $$x=3$$ but do not know if I have done the exact. Please help me
2
votes
5answers
39 views

Decompose a fraction in a sum of two

Let's say that I have this fraction: $$ \frac{2x}{x^2+4x+3}$$ I would like to decompose in two fraction: $$ \frac{A}{x+3} + \frac{B}{x+1}$$ Which is the procedure for that? :)
0
votes
3answers
68 views

Dividing line segments with ratios vs. fractions [closed]

I know that $2:3$ is actually $\frac {2}{3}$. So when you split a line segment by a ratio, you would add $2$ and $3$ to get a fraction of $\frac {2}{5}$ that will be used to solve the problem. I ...
6
votes
7answers
723 views

When the numerator of a fraction is increased by 4, the fraction increases by 2/3…

When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$. What is the denominator of the fraction? I tried, Let the numerator of the fraction be $x$ and the denominator ...
2
votes
2answers
47 views

A property of proportions: if $a/b=c/d$, then $(ma+nb)/(pa+qb)$ is equal to $ (mc+nd)/(pc+qd)$

If $\large\frac{a}{b}=\frac{c}{d}$ how we can obtain $\displaystyle{\frac{ma+nb}{pa+qb}=\frac{mc+nd}{pc+qd}}$? I can get $\large\frac{ma}{qb}=\frac{mc}{qd}$ and $\large\frac{nb}{pa}=\frac{nd}{pc}$ , ...
1
vote
3answers
97 views

How does $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ simplify to $1 - \sqrt 2 $?

I've the answer for a question in my textbook to be: $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ which i've then simplifed to: $-\sqrt {3 - 2\sqrt 2 } $ However my textbook states $-\sqrt ...
0
votes
1answer
52 views

Using all types of fractions

Is their a website that teaches you everything you need to know about fractions, just fractions. I ask this because I do calculus...and I suck at fractions. I hate them so much. I have no idea how to ...
17
votes
12answers
3k views

Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
0
votes
0answers
13 views

proving a fraction with 2 parameters to be small

Hi I have a fraction as below $$\frac{1.623x^4+0.434x^4\sum_iy_iz_i^2+(0.014x^2+0.0027)\sum_iy_iz_i^4}{1.645x^2+(0.083-0.329x^2+0.435x^4)\sum_iy_iz_i^2+0.014\sum_iy_iz_i^4}$$ where $x\in[0, 0.5]$, ...
0
votes
0answers
39 views

L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
1
vote
3answers
29 views

Fraction exponents in division

if I have $\frac{a^{6/5}}{b^{1/5}}$, I know you subtract exponents when dividing so $6/5 - 1/5$ is $5/5$, so since that's just one, is this equal to $a/b$?
1
vote
2answers
63 views

Definition of Rational/ Irrational Numbers reguarding denominators

The definition of a Irrational number is "Irrational numbers don't include integers OR fractions. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern." So ...