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
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1answer
66 views

General solution for $x$ of $C = 100/(1+aX) + 100/(1+bX)+ \cdots + 100/(1+zX)$

Please can someone help me find a general solution for $X$: $$ C = \frac{100}{(1+aX)} + \frac{100}{(1+bX)}+ \cdots + \frac{100}{(1+zX)} $$ UPDATE Its not ideal but if we make $C = 350$ would this ...
0
votes
2answers
32 views

variables to the power of a fraction

I have this question for advanced math, I can't seem to get my head around. $$\frac{x^{5/2}}{(x^{1/3})^4}$$
5
votes
1answer
137 views

Will 0.99999999 eventually become equal to 1?

I am currently learning about fractions, and there is something that I am finding it hard to make sense of. When a fraction it added to the right of the decimal point, the number becomes slightly ...
0
votes
2answers
20 views

what is the default order and direction of operation?

I have a division like this 16/8/4/2 what is the default way to do calculations when the bracket is not specified . Method 1 : Is it correct to go from right to left like [16/ (8 / { 4 / ...
4
votes
2answers
103 views

How do I simplify this expression about factorization?

I am trying to simplify this $$\frac{9x^2 - x^4} {x^2 - 6x +9}$$ The solution is $$\frac{-x^2(x +3)}{x-3} = \frac{-x^3 - 3x^2}{x-3} $$ I have done $$\frac{x^2(9-x^2)}{(x-3)(x-3)} = ...
1
vote
3answers
75 views

How do I solve this, first I have to factor $ 2x\over x-1$ + $ 3x +1\over x-1$ - $ 1 + 9x + 2x^2\over x^2-1$?

I am doing calculus exercises but I'm in trouble with this $$\frac{ 2x}{x-1} + \frac{3x +1}{ x-1} - \frac{1 + 9x + 2x^2}{x^2-1}$$ the solution is The only advance that I have done is factor $ ...
2
votes
3answers
61 views

Why does $\sqrt{x} / y =\sqrt{x/y/y}$?

Sorry for the awkward title, hard to to sum a mathematical problem with words alone. Having said that, I recently learned that the root of any value, $x$, and then that over value $y$, is identical ...
7
votes
3answers
130 views

Why does partial fraction decomposition always work?

Say you have a function $p(x)/q(x)$ for some polynomials $p(x)$ and $q(x)$ and $p$ has a lower degree than $q$. Say $q$ has degree three and $p$ has degree two. If you partially decompose it, you'll ...
1
vote
0answers
28 views

exponential integration with fractional powers

I am trying to solve the following integral $$\int_{-\infty}^a \frac{\beta_1 \beta_2}{y^2(c-y)^2} e^{-\beta_1/(c-y)} e^{-\beta_2/y} \, dy$$ where $a<0$, $c>0$, $\beta_1>0$, $\beta_2>0$ I ...
1
vote
3answers
110 views

What is $\frac{9}{3} - \frac{1}{2}$?

I need to compute $\frac{9}{3} - \frac{1}{2}$. I got an answer of $\frac{8}{6}$ but that is incorrect. $\frac{5}{2}$ is the correct answer. How is this possible?
0
votes
0answers
45 views

How to find maximum and minimum of $\frac{x+y+z}{ax+by+cz}$ where $0\leq x \leq y \leq z \leq 1$ for given positive real numbers $a,b,c$

How do I find the maximum and minimum of $$\frac{x+y+z}{ax+by+cz}$$ where $0\leq x \leq y \leq z\leq 1$ for given positive real numbers $a,b,c$? I guess those are one of $\frac{3}{a+b+c}$ or ...
-4
votes
2answers
42 views

Find out students [closed]

As a part of learning journey program 73 pupils traveled on two buses, one air conditioned and one non air conditioned to the Singapore Discovery center. Three fifth of the pupils on the air ...
82
votes
12answers
10k views

Can you be 1/12th Cherokee?

I was watching an old Daily Show clip and someone self-identified as "one twelfth Cherokee". It sounded peculiar, as people usually state they're "1/16th", or generally $1/2^n, n \in \mathbb{N}$. ...
-4
votes
1answer
48 views

How much laundry is done? [closed]

On Saturday morning, Bambi washed 3/5 of 50 clothes in the laundry bag. In the afternoon, she washed 4/5 of the clothes. How many clothes did Bambi washed?
1
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2answers
240 views

Determine if $\frac{k-1}{k}+\frac{1}{k(k+1)}=\frac{k}{k+1}$ holds

How to prove if the following equality holds? $$\frac{k-1}{k}+\frac{1}{k(k+1)}=\frac{k}{k+1}$$ Maybe finding a common denominator would work, but I have no idea how to do it in this example. I see ...
1
vote
1answer
20 views

Function that maps a rational number to its numerator and denominator

Question: Is there a simple way to represent a function $f:\mathbb Q\to \mathbb Z^2$ that maps a rational number in lowest terms $r=\frac ab$ to the ordered pair of its numerator and denominator ...
1
vote
3answers
90 views

Can $1\over 1$, $1\over 2$, $1\over 3$, $1\over 4$, etc. be calculated by the added fractions below?

About $1\over 1$, $1\over 2$, $1\over 3$, and $1\over 4$, can $1\over 4$ also be written as $1\over 5^1$+$1\over 5^2$+$1\over 5^3$+$1\over 5^4$+...=$1\over 5$+$1\over 25$+$1\over 125$+$1\over ...
0
votes
4answers
131 views

What does Pi equal to [duplicate]

What is the approximation of pi in a fraction form. I am very curious to know what it is. I have been seeing pi almost everywhere.
0
votes
1answer
27 views

Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$

Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$ So need to show $\frac c {1+c} \le \frac {a+b+2ab} {1+a+b+ab}$ We have $\frac c {1+c} \le \frac {a+b} ...
3
votes
2answers
233 views

How do I prove that any unit fraction can be represented as the sum of two other distinct unit fractions?

A number of the form $\frac{1}{n}$, where $n$ is an integer greater than $1$, is called a unit fraction. Noting that $\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$ and $\frac{1}{3} = \frac{1}{4} + ...
0
votes
1answer
35 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 Prudnikov ...
0
votes
1answer
29 views

power of a fraction

I am having trouble understanding where the numbers are coming from in this question. John and Melissa wonder about the potential increase in the value of their house. Assuming a 6% appreciation per ...
4
votes
5answers
99 views

$-\frac76=-\frac {4n}{3}$ struggling on solving this equation

I was wondering how to do this equation step by step. I forgot how to but it would be awesome if someone could help me out $$-\frac76=-\frac {4n}{3}$$
0
votes
2answers
26 views

Solve using Proportions/Multiplication?

How do I solve this using proportions and multiplication? How much is $\dfrac 1{200}$ of $50$ percent? I know that the answer is $0.25$, however, how would I solve that using proportions and ...
6
votes
2answers
59 views

Finding the derivative $f(x)=\sqrt{x^2 -9}$,

I need to find the slope at a=5, using the definition for the function $f(x)=\sqrt{x^2 -9}$, $$f'(x) = \lim_{\Delta x \to 0} {f(x+\Delta x)\over \Delta x}$$ The answer book says the slope is ...
1
vote
2answers
57 views

Finding derivative $f(x)={2\over x^3}$

I have to find the derivative and the slope at $a=6$ The function is $f(x)={2\over x^3}$ I have to find the answer using the formula, $$f'(x)= \lim_{\Delta x \to 0} {f(x+ \Delta x) - f(x) \over ...
1
vote
2answers
27 views

Fractions with Hours and Days

So the question is: The number of hours left in a day on Mars was $\frac{1}{4}$ on the number of hours that had already passed. How many hours were left in the day? Day on Mars: $40$ hours. I did ...
0
votes
3answers
164 views

Explanation of series for $\sin(x)$ and $\cos(x)$.

Can anyone explain me what is this equation telling us? I need to implement it in my computer program. I do not need a proof of these, but an explanation of notation used here. $$ \sin x = ...
1
vote
2answers
63 views

Is a prime to the power of a fraction always irrational?

Let $p$ be a prime number and let $x$ be a faction, i.e. $x \in \mathbb{Q} - \mathbb{N}$. It seems to be the case that $p^x$ is always irrational. How do I prove this?
4
votes
4answers
507 views

Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?

Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication? I had also got a clue: it's related to e. Please help! ...
1
vote
1answer
84 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 ...
3
votes
7answers
177 views

Why Is $y^{-1}$ = $\frac{1}{y^1}$?

Basically, I'm asking 'Is there any place where I can access a compendium of formal mathematical proofs'? I need to know what processes mathematicians went through to declare $(-1)(-1)=1$ and so on. I ...
0
votes
1answer
25 views

How to calculate a whole amount with fractions?

A contractor first completes $7/16$ of a building. Then he completes $1/4$ of it. And finally completes $2/5$th of the remainder of the building. If there is $36$ days left to finish the construction ...
0
votes
1answer
38 views

How to calculate a distance with different fraction ratios?

This is the math question for my sixth grader son. The answer is $136$ meters but could not figure it out. Can someone please explain how to solve it. Thank you. Adam first walks $3/8$th of a road. ...
0
votes
1answer
38 views

Maple, simplyfing ODEs questions

I'm a novice using Maple 16. I'm using it mostly to check my DE homework solutions. And it happens a lot that I get stuff like in the picture. I mean (if I'm not missing anything important) that ...
56
votes
14answers
3k views

Why rationalize the denominator?

In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "there ...
1
vote
3answers
73 views

Which one is less than others?

Which one is less than others? $\frac{3}{5} , \frac{2}{3} , \frac{6}{13} , \frac{23}{38}$ Yes the answer is $\frac{6}{13}$ but the real question is this: I've a 12 years old brother and he just ...
0
votes
3answers
52 views

Given that $yz:zx:xy = 1:2:3$ and $\tfrac{x}{yz}: \tfrac{y}{zx} = 1:k$, find $k$

Given that $yz:zx:xy = 1:2:3$ and $\dfrac{x}{yz}: \dfrac{y}{zx} = 1:k.$ Find $k$. I understand that $ k = \frac{y^2}{x^2}, y = 1,$ and $x = 2$. Therefore $k = \frac{1}{4}$. This also brings me ...
13
votes
2answers
171 views

Why do I get $0.098765432098765432…$ when I divide $8$ by $81$?

I got this remarkable thing when I divided $16$ by $162$, or, in a simplified version, $8$ by $81$. It's $0.098765432098765432\cdots$, or more commonly known as $0.\overline{098765432}$, with all the ...
20
votes
4answers
622 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
-1
votes
3answers
59 views

How to show that show that $\frac{v+u}{1+ uv/c^2}=c$ when $u=c$?

I am trying to show that $\dfrac{v+u}{1+\dfrac{uv}{c^2}}=c$ when $u=c$. Context It's needed for a physics proof that I'm working on. This is the formula for relative velocity, $u$ represents the ...
12
votes
4answers
981 views

Is Cantor's diagonal argument dependent on the base used?

Applying Cantor's diagonal argument to irrational numbers represented in binary, one and only one irrational number can be generated that is not on the list. Wikipedia image: But if you change ...
5
votes
6answers
158 views

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$?

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$? I got answer $\frac{1}{\sqrt{2}}$, but the real answer is $\frac{\sqrt{2}}{2}$. Anyway, calculator for both answers return same numbers. ...
0
votes
2answers
81 views

How to simplify this expression with radicals? $3\sqrt2 - \sqrt{32} + \sqrt{\frac{80}{16}}$

I don't understand how I could calculate this: $3\sqrt2 - \sqrt{32} + \sqrt{\dfrac{80}{16}}$ My answer is $-\sqrt2 + \sqrt5$, but the real answer should be $\dfrac{9-4\sqrt2}{4}$.
0
votes
1answer
19 views

How to differentiate for critical points with variable in denominator

sorry for posting a particular problem, this is maybe more of an algebra problem than a calculus problem, but it does involve differentiating so I thought I would state the problem as one. I am ...
14
votes
1answer
226 views

Closed-form of infinite continued fraction involving factorials

Is there a closed form of this: $$ 1!+\dfrac{1}{2!+\dfrac{1}{3!+\dfrac{1}{4!+\ldots}}} $$
1
vote
3answers
125 views

How to solve following limit

I've been struggeling a bit with the following limit: $\lim\limits_{x \to 0} \frac{a- \sqrt{a^2 - x^2}}{x^2}$ The solution is: If a < 0 then -$\infty$ . If a > 0 then $\frac{1}{2a}$ But I don't ...
2
votes
2answers
75 views

Problem with slopes.

I currently have a slope that looks like this: $\frac{-5}{10}$ However, I need to bring it down to it's lowest terms, so I divided the numerator and denominator by -5 and I got: $\frac{1}{-2}$ ...
1
vote
1answer
45 views

Are some infinite fractions in one counting system non-infinite in another?

I'm curious whether some infinite periodic fractions in one counting system (e.g. decimal - 10/3 = 3.33333...) turn out to be non-infinite in another system and vice - versa. Please excuse me if my ...
0
votes
3answers
34 views

Split a whole number into fractions, find which of thoses fractions another belongs in

I want to split a number into $n$ parts and then take another number (which is less than or equal to the first number) and see which of the fractions the other number belongs to. For example, split ...