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

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-3
votes
0answers
24 views

Basic maths questions involving ratio and fraction. [on hold]

why the ratio a:b is written as a/b when a/b means taking 'a' equal parts from an object which is divided into 'b' equal parts and a:b means the object is actually divided into 'a+b' parts. why a ...
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 ...
3
votes
2answers
38 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 ...
0
votes
2answers
66 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.
1
vote
1answer
28 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 ...
-5
votes
0answers
34 views

Algebra 2 help!… [closed]

If a certain number is added to the numerator and denominator of 17/20, the result is 8/9. Find the number. A. 7 B. 9 C. 14 D. -7 E. -8
2
votes
1answer
33 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
25 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
37 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 ...
0
votes
1answer
121 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 ...
-1
votes
2answers
66 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
31 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. $${1000\over \pi\sqrt[3]{500\over \pi}^2}=2\sqrt[3]{500\over \pi}$$ ...
1
vote
1answer
544 views

Compute operations with fractions using calculator

I have a CASIO fx-350MS, and I need to make fraction computations, like $\frac{7}{2} \cdot \frac{4}{5}$ for an exam where I have to compute a lot matrix multiplications (programmable calculators ...
2
votes
5answers
84 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
81 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
474 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
30 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
34 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 ...
0
votes
1answer
40 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 ...
1
vote
1answer
57 views

Integral exponential and fraction of powers

I am trying to solve the following integral $$ \int_0^y \frac{x^{m-1}}{(1+x)^{m+k}}\, \exp\left(-\frac{m}{\gamma} x \right) \,dx. $$ I tried to look into different books such as Gradshteyn and ...
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
31 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
34 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 ...
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 $h$.
-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: $$ \frac{4mlt(c^2 - c \sqrt{c^2-v^2})}{\sqrt{1-\frac{v^2}{c^2}}} $$ ...
7
votes
2answers
109 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 ...
7
votes
1answer
99 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 ...
4
votes
2answers
528 views

Length of period of decimal expansion of a fraction

Each rational number (fraction) can be written as decimal periodic number. Does exists a method or hint that show how long will be the period of arbitrary fraction. For example $1/3=0.3333...=0.(3)$ ...
241
votes
13answers
29k views

Find five positive integers whose reciprocals sum to $1$

Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was ...
1
vote
2answers
165 views

Understanding the concepts of division and fractions

$\require{cancel}$ I'm having some issues regarding division so I will start by asking how this concept was developed throughout the ages: What was the first civilization to introduce the idea of ...
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
11k 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 ...
1
vote
3answers
69 views

How do I simplify $\frac{\sqrt{21}-5}{2} + \frac{2}{\sqrt{21} - 5}$?

How do I simplify the following equation? $$\frac{\sqrt{21}-5}{2} + \frac{2}{\sqrt{21} - 5}$$ I have no idea where to start. If I multiply either fraction by its denominator I will still end ...
2
votes
1answer
43 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: ...
39
votes
0answers
2k views

If the decimal expansion of $a/b$ contains “$7143$” then $b>1250$

I recently stumbled upon this really interesting problem: If we have a fraction $\frac{a}{b}$ where $a,b \in \mathbb{N}$ and we know that the decimal fraction of $\frac{a}{b}$ has the numerical ...
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 ...
2
votes
3answers
103 views

Get rid of the square roots of the denominator: $\dfrac{1}{\sqrt{7}-2\sqrt{5}+\sqrt{3}}$

How to get rid of the square roots of the denominator: $\dfrac{1}{\sqrt{7}-2\sqrt{5}+\sqrt{3}}$? I squared the whole denominator, but that didn't help. Also I searched for a propriety or ...
2
votes
1answer
43 views

When we can change the sign of denominator

Suppose $z=\frac{-x_1}{x_2-x_3}$, find $-z$. Which one is correct $$-z=\frac{x_1}{x_2-x_3}\ \ \ \text{or}\ \ \ -z=\frac{x_1}{-x_2+x_3}$$
0
votes
1answer
97 views

Adding a natural number to a normalized fraction

I am currently writing yet another rational number class where the fraction should always be normalized. When adding a natural number to a normalized fraction, it possible to get a non-normalized ...
14
votes
2answers
495 views

proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$ [duplicate]

i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows: $$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
3
votes
2answers
36 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 ...
2
votes
2answers
41 views

decimal to fractions

When being asked how to solve the Arithmetic Means of 8, 7, 7, 5, 3, 2, and 2, I understand that adding these numbers then dividing by 7 (the amount of numbers) gives me the decimal 4.85714... But ...
2
votes
2answers
79 views

Here are two fractions, $\frac{2}{3}$,$\frac{7}{8}$, which of these fractions are closer to $\frac{3}{4}$?

I've been throwing this question around my family. No one has a clue, therefore can someone help? I'm pretty sure this will be easy to do
0
votes
2answers
68 views

Fractions of an amount [closed]

I need help with the following problem: Catalin works in an office. One week he divides his time between these tasks: $\frac{1}{4}$ of his time in meetings $\frac{5}{8}$ of his time writing ...
2
votes
1answer
75 views

Are fractions with zero divisors in the denominator never well defined?

Are fractions with zero divisors in the denominator never well defined? I know that for a fraction in modular arithmetic to be well defined, the denominator must not be a zero divisor, e.g: $$ x ...
0
votes
4answers
95 views

Dividing fractions in real life scenario / application

First of all sorry if this question sounds too stupid or offends anyone. One apple divide by two you get half an apple. $\large{\frac{1}{2} = 0.5}$ I couldn't get my head around with dividing ...