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

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-2
votes
3answers
84 views

Find the value of $\frac{ x_1^2}{1-x_1}+\frac{x_2^2}{1-x_2}+…+\frac{x_n^2}{1-x_n}.$ [on hold]

Suppose, $x_1+x_2+...+x_n=1\space (x_i\in \mathbb{R}, x_i ≠1)$ and $\frac{ x_1}{1-x_1}+\frac{x_2}{1-x_2}+...+\frac{x_n}{1-x_n}=1.$$\\$ Find the value of $\frac{ ...
0
votes
3answers
21 views

Comparison of two values

I have to figure out the relation between the quantity $(0.9/1.1)^2 +(1.1/0.9)^2$ and 2. How can i do this without explicitly calculating the first value, by using some laws of exponents?
1
vote
3answers
29 views

Rationalization of a fraction

Can someone please explain how i can rationalize the fraction $\frac{2 +\sqrt{3}}{(2-\sqrt{3})^3}$ so that i obtain an answer that has an exponent of 6. I basically need to compare this value with ...
6
votes
4answers
530 views

How can I simplify this complex number to get a real number?

$$\large \frac {e^{i \frac{\pi a}{2}}[1-e^{i\pi a}]} {[1-e^{i2\pi a}]}$$ I am trying to arrive at $$\frac {1}{2\cos\left(\frac{\pi a}{2}\right)}$$ I've tried dividing top and bottom by one of the ...
1
vote
3answers
112 views

Solving a rational equation with multiple and nested fractions

This is the equation to solve: $\dfrac{\dfrac{x+\dfrac{1}{2}} {\dfrac{1}{2}+\dfrac{x}{3}}}{\dfrac{1}{4}+\dfrac{x}{5}}=3$ What I did: $x+\dfrac{1}{2}=\dfrac{2x+1}{2}$ ...
2
votes
4answers
82 views

Is $(x^2 + 1) / (x^2-5x+6)$ divisible?

I'm learning single variable calculus right now and at current about integration with partial fraction. I'm stuck in a problem from few hours given in my book. The question is to integrate $$\frac{x^2 ...
-4
votes
0answers
25 views

If 25=m^w, then 125m= [closed]

also these: If 20% of 45% of a positive number is equal to x percent of 0.3 percent of the same number what is the value of x? Sarah washes her car interior ever 14 days and vacuums the ...
-2
votes
0answers
38 views

Getting fractions [closed]

Just wondering if there is a better algo/formula to get fractions (I don't know the correct Math term): Given: x and y Get this list: ...
0
votes
1answer
19 views

Removing a fraction in the denominator

What steps do I need to take to simplify $\frac{1}{\frac{1}{\sqrt{2}}}$ to $\frac{\sqrt{2}}{1}$? Can you please explain the steps i need to take in this problem and explain generally how to remove ...
0
votes
4answers
40 views

I can't understand the algebraic simplification below

I can't understand the last step of simplification in the following algebraic expression. I am aware that $\sqrt{3}=3^{\frac{1}{2}}$. Can't see how they get the $6$ in the denominator, and how ...
4
votes
1answer
36 views

Question on recurring decimal digits

In my discrete maths class, I have come across an interesting phenomenon for which I can't find an explanation! If we divide $1$ by $13$ we obtain $0.07692307\ldots$ If we divide $3$ by $13$ we ...
-1
votes
1answer
19 views

Fractions and decimals

Ten companies sponsored a tournament and decided to give M rupees collectively.Two companies dropped and remaing agreed on paying their share equally.What was the increase in share of each company? ...
0
votes
4answers
41 views

How to rearrange: $16 = \frac{1}{n} 25 + \frac{n−1}{n} 218.75$

Can anyone here help me out to rearrange the following formula and solve for $n$? $$16 = \frac{1}{n} 25 + \frac{n−1}{n} 218.75$$
0
votes
3answers
76 views

What is the purpose of mixed numbers outside of common usage? [closed]

I am wondering if there is indeed any real usage of mixed numbers such as $3 \frac{1}{2}$ meaning $3+\frac{1}{2}$ in standard Mathematics. Personally I dislike the use of such conventions in schools ...
2
votes
1answer
24 views

What would 480:15 be in simplest form? [closed]

So I am doing fraction division and I have gotten stuck. 18/5 divided by 3/25 so 18/5 mutiplyed by the reciprocal of 3/25 which is 25/3. which gets me 480/15. What is 480/15 in simplest form? ...
1
vote
2answers
73 views

Compare $A=\frac{1.0\,000\,004}{(1.0\,000\,006)^2}$ and $ B=\frac{(0.9\,999\,995)^2}{0.9\,999\,998}$

My work: $1.0\,000\,004 = 1+\frac{4}{10^7}=1+\frac{1}{125\cdot 10^6}$ $ (1.0\,000\,006)^2=(1+\frac{6}{10^7})^2=(1+\frac{3}{5\cdot 10^6})^2$ $ ...
-7
votes
5answers
172 views

Difference between ${2\over 9}$ and ${22\over 99}$? [closed]

The fractions ${2\over 9}$ and ${22\over 99}$ both have the same decimal value $0.22222\ldots$ But obviously they are not equal. What causes this situation? And also, what is the correct rational ...
0
votes
2answers
20 views

Simple algebraic manipulation with 2 equations

My first equations is this: $ d_2 = d - 30.$ My second equations is this: ${1\over d_2 }= {1\over12} - {1\over1.066(d-30)}$ I am trying to solve for $d_2$ in the second equation and then set the ...
0
votes
2answers
67 views

Help with Calculate integral

Find $\int^a_0 \dfrac{3x^2-ax}{(x-2a)(x^2+a^2)} dx$ I tried using partial fractions and the substitution $u=a-x$ but I haven't made any real progress. Please help.
3
votes
2answers
44 views

How to conceptualize “dividing out” a number (e.g. in permutations, Bayes' Theorem)?

I'm trying to achieve a better conception of what it means to "divide out" a variable/number, because I'm currently have a lot of trouble justifying to myself why it actually works the way it does in ...
1
vote
1answer
31 views

Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home? Data provided is the amount of books each ...
2
votes
1answer
36 views

Weird square root disappearing and flipping fraction upside down?

So here I was, making 2 math problems, I was able to solve them, but 2 operations seem a bit intractable to me. Maybe you can help me understand why this is true: The first problem: $$x = \frac{1}{5} ...
1
vote
2answers
27 views

Simplification of rational expressions

I have the following expression: $${2\over x-2} + {2 \over{x^2} -5x +6}$$ So I can simplify this as: $${2 \over x -2} + {2 \over (x -3) (x-2)}$$ I make the common denominator to be ${(x-3)(x-2)}$ ...
0
votes
1answer
20 views

Trending proof for fairly simple fraction

(Please humour the physicist!) Why does $\left(\frac{1-a}{1-a^b}\right) \to \frac{1}{b}$ as $a \to 1$? This came from a calculation involving flow measurement of gases, and although I can see and ...
1
vote
2answers
39 views

Exponential function negative: $\left(\frac{81}{4}\right)^{1/4}\left(\frac{1}4\right)^{-3/4}$

This is another example. $\left(\dfrac{81}{4}\right)^{1/4}\left(\dfrac{1}4\right)^{-3/4}$ Multiply on both sides equals $\dfrac{81^{1/4}}{4^{1/4}}\cdot \dfrac{1^{-3/4}}{4^{-3/4}}$ This should be ...
-1
votes
2answers
79 views

How to show $\frac12\cdot\frac34\cdot\frac56\cdots\frac{99}{100}<\frac{1}{12}$? [closed]

How can I show that $$\frac12\cdot\frac34\cdot\frac56\cdots\frac{99}{100}<\frac{1}{12}?$$
1
vote
4answers
33 views

Simplifying a fraction with a cubed root in the denominator

I have an equation the following equation in my textbook, but I don't understand how it's legal for it to be simplified this way. $${1000\over \pi\sqrt[3]{500\over \pi}^2}=2\sqrt[3]{500\over \pi}$$ ...
0
votes
1answer
123 views

Four mathematical notations for fraction $1/999$ and how to show/present they are equal

I need some help for notation. I need to present fractions in four different format and I'd like to get it right. I just take $1/999$ for example, but of course it could be any fraction with positive ...
2
votes
5answers
87 views

The expression $(1+q)(1+q^2)(1+q^4)(1+q^8)(1+q^{16})(1+q^{32})(1+q^{64})$ where $q\ne 1$, equals

The expression $(1+q)(1+q^2)(1+q^4)(1+q^8)(1+q^{16})(1+q^{32})(1+q^{64})$ where $q\ne 1$, equals (A) $\frac{1-q^{128}}{1-q}$ (B) $\frac{1-q^{64}}{1-q}$ (C) $\frac{1-q^{2^{1+2+\dots +6}}}{1-q}$ ...
2
votes
5answers
83 views

If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ then..

If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ then (A) $a=c$ (B) either $a=c$ or $a+b+c+d=0$ (C) $a+b+c+d=0$ (D) $a=c$ and $b=d$ I solved $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ and got $a(a+b+d)=c(c+b+d)$ and ...
9
votes
1answer
477 views

Primes and certain unit fractions [closed]

Are there primes $p,q$ and a natural number $a$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{a}$?
0
votes
1answer
31 views

Can $\dfrac{b_0}{a_0} + \dfrac{b_1}{a_1} + \dfrac{b_2}{a_2} + \dfrac{b_3}{a_3} + … + \dfrac{b_n}{a_n}$ be represented as …

Is this correct? (Last step $\rightarrow$ After taking L.C.M.) $\large \dfrac{b_0}{a_0} + \dfrac{b_1}{a_1} + \dfrac{b_2}{a_2} + \dfrac{b_3}{a_3} + ... + \dfrac{b_n}{a_n} = \sum\limits_{k=0}^{n} ...
6
votes
1answer
62 views

Certain Fraction between Fractions

Is there always a fraction $\frac{r}{s}$ with $\frac{p}{q}<\frac{r}{s}<\frac{p+1}{q}$ and $s<q$ for $0<p<q-1\in\mathbb{Z}$ and $r,s\in\mathbb{Z}$?
0
votes
1answer
41 views

Convergent conjecture: What is the proof?

Lets say that $\def\nn{\mathbb{N}}$$\def\rr{\mathbb{R}}$$K : \nn \to \rr$ and $\displaystyle \sum_{i=1}^\infty \frac{K(i)}{K(i+1)}$ is a convergent sum. My conjecture is that the function $K$ must be ...
0
votes
1answer
38 views

Choosing suits of cards in a row

Three cards from a standard deck are dealt. What is the probability that the first is a heart, the second is a spade, and the third is another heart? I have figured out so far that you can use ...
1
vote
3answers
61 views

Mathematical Induction getting the right side

So I 've been doing Mathematical Inductions but I seem to have a issue in simplify and getting the right side. So I have this on the L.H.S $$\frac{k(k + 1)(2k +1)}{6} + (k + 1)^2 $$ And I'm trying ...
1
vote
2answers
32 views

Expression for binomial coefficient denominator

I'm trying to find an analytical expression for the denominator of $\pmatrix{-1/2\\k}$ in terms of $k$ when the fraction is fully reduced. E.g., the first several such denominators, starting with ...
0
votes
1answer
30 views

How to calculate $n$th term in terms of constants?

The expression is $$\large t_n=\frac{(x\times t_{n-1})^2}{((x-t_{n-1}\times y)^2+4\times x\times t_{n-1})\times t_{n-2}}$$ where $x$ and $y$ are constants. $t_0$ , $t_1$ , $t_2$ , $t_3$ and $t_4$ ...
2
votes
1answer
35 views

Congruence of 2 fractions—how to properly rewrite in terms without modulo?

EDIT: Following Theo's comment, the equivalence holds since one can (must) rewrite $1/a$ as $(1+23k)/a$. Provided that $$\frac{1}{25} \equiv \frac{1}{2}\pmod {23}$$ is true, why can I not rewrite ...
7
votes
1answer
101 views

Distribution of the sum reciprocal of primes $\le 1$

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots \le 1 $$ This is an interesting infinite summation. This is very closely resembling my other problem with has to do with the distribution of ...
7
votes
2answers
115 views

Does the sum of the reciprocals of composites that are $ \le $ 1

The sum itself: $$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$ These are all sums of reciprocals of composites that ...
-1
votes
2answers
69 views

Find all integer numbers $n$ such that $\frac{11n-5}{n+4}$ is a perfect square.

Find all integer numbers $n$, such that, $$\sqrt{\frac{11n-5}{n+4}}\in \mathbb{N}$$ I really tried but I couldn't guys, help please.
0
votes
0answers
43 views

Can This Expression Be Simplified? (Involves Square Roots)

I started with the expression $$ \frac{4mlt(1-\sqrt{1-\frac{v^2}{c^2}})c^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ and have ended up at: $$ \frac{4mlt(c^2 - c \sqrt{c^2-v^2})}{\sqrt{1-\frac{v^2}{c^2}}} $$ ...
0
votes
3answers
77 views

Problem with simplifying $\frac{(3+h)^2-9}{(3+h)-3}$ [closed]

I need help simplifying $$ {(3+h)^2-9\over (3+h)-3}. $$ The answer is $6+h$. I keep getting $h$.
0
votes
2answers
48 views

How can I approximate a decimal with two fractions where denominator is less or equal to $d$

I was looking for a way to approximate a decimal number with a fraction, whose denominator is less or equal to $d$. Basically, having a decimal $X$, I want to find two fractions such that ...
62
votes
14answers
12k views

Express 99 2/3% as a fraction? No calculator

My 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is ...
2
votes
1answer
44 views

How to simplify a diabolical expression involving radicals

A friend and I have been working on this problem for hours - how can the following expression be simplified analytically? It equals $\frac{1}{2},$ and we have tried the following to no avail: ...
2
votes
1answer
45 views

When we can change the sign of denominator

Suppose $z=\frac{-x_1}{x_2-x_3}$, find $-z$. Which one is correct $$-z=\frac{x_1}{x_2-x_3}\ \ \ \text{or}\ \ \ -z=\frac{x_1}{-x_2+x_3}$$
3
votes
2answers
40 views

SAT math problem about solute and solution

There are two solutions $P$ and $Q$. There are $50 g$ of $P$, which has $30\%$ benzene by mass, and $200 g$ of $Q$, which has $70\%$ benzene by mass. If $20 g$ of solution $P$ is added to $20 g$ of ...
4
votes
3answers
150 views

Why does $\frac{49}{64}\cos^2 \theta + \cos^2 \theta$ equal $\frac{113}{64}\cos^2 \theta $?

I have an example: $$ \frac{49}{64}\cos^2 \theta + \cos^2 \theta = 1 $$ Then what happens next: $$ \frac{113}{64}\cos^2 \theta = 1 $$ Where has the other cosine disappeared to? What operation ...