Improve Your Algebra Skills with Practice Problems

Division by rational (decimal) number meaning
When I say, that I exchanged 42 CZK into 1,5 euro. Why do I get the rate for one euro by dividing?
1) How do you explain this division in words. Like when you say when doing integer division, that you divide 10 cookies to 2 people, so one get 5. That type of style.
2) Does division always gives back an average rate per 1 unit of somethings?
3) the same for mean average, I bought 100 units of something for 2500 unit of any currency. 2500/100 = 25 per one (25/1).
How do you explain this in words too, please. Something like you want to distribute 2500 into 100 equal parts and the answer is the units per one?
Thanks, All I care is the problems being said in words, meaning of division rational number by rational number, in this case in its decimal form

Let $P$ be a polynomial of degree $n$ , $n\ge <!--\; \ge \; -->2$, $P(X)=\underset{i=1}{\overset{n}{\prod <!--\; \prod \; -->}}{(X-<!--\; -\; -->z_i)}^{m_i}$
with zeros $z_1,z_2,\dots <!--\; \dots \; -->,z_n$
Decompose the rational fraction ${\displaystyle \frac{P^{\prime }}{P}}$
My thoughts:
$P(X)=\underset{i=1}{\overset{n}{\prod <!--\; \prod \; -->}}{(X-<!--\; -\; -->z_i)}^{m_i}$
$P^{\prime }(X)=\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{m}_i{(X-<!--\; -\; -->z_i)}^{m_i-<!--\; -\; -->1}\underset{?}{\overset{?}{\prod <!--\; \prod \; -->}}{(X-<!--\; -\; -->z_i)}^{m_{?}}$
I don't know with what i should fill the ?
Then:
$\begin{array}{rl}{\displaystyle \frac{P^{\prime }}{P}}& ={\displaystyle \frac{\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{m}_i{(X-<!--\; -\; -->z_i)}^{m_i-<!--\; -\; -->1}\underset{?}{\overset{?}{\prod <!--\; \prod \; -->}}{(X-<!--\; -\; -->z_i)}^{m_i}}{\underset{i=1}{\overset{n}{\prod <!--\; \prod \; -->}}{(X-<!--\; -\; -->z_i)}^{m_i}}}\\ & =?\end{array}$
I'm stuck here

Can a greatest common factor be 1?

May I know how to exclude tied rank, and interpret the sum of the Poisson random variables?
for the first part of the question, my interpretation is that:
$\begin{array}{|cc|}\hline Value& Rank\\ 1& 1\\ 2& 2\\ 2& 2& \text{(Tie rank)}\\ 2& 2& \text{(Tie rank)}\\ \hline\end{array}$
So, there are two tied ranks, so I should delete them from the data set?
For the second part of the question, I want to ask how do I interpret the fact that the sum of independent Poisson random variables has a Poisson distribution. my interpretation is that : If $X\sim <!--\; \sim \; -->P_o({\mathrm{\lambda <!--\; \lambda \; -->}}_1),andY\sim <!--\; \sim \; -->P_o({\mathrm{\lambda <!--\; \lambda \; -->}}_2)$. We can conclude that $(X+Y)\sim <!--\; \sim \; -->P_0({\mathrm{\lambda <!--\; \lambda \; -->}}_1+{\mathrm{\lambda <!--\; \lambda \; -->}}_2)$.
Thank you so much for your reply, guys. And sorry for the fact that I don't know how to insert columns in the question, so I can only attach picture for illustration.

Assume $f_1,\dots <!--\; \dots \; -->,f_N:\mathbb{R}\to <!--\; \to \; -->\mathbb{R}$. If $\int <!--\; \int \; -->|f_n|\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}<\mathrm{\infty <!--\; \infty \; -->}$ for all $n=1,\dots <!--\; \dots \; -->,N$, then we always have
$\underset{n=1}{\overset{N}{\sum <!--\; \sum \; -->}}\int <!--\; \int \; -->f_n\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}=\int <!--\; \int \; -->\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}{f}_n\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}.$
Now we assume $f_1,\dots <!--\; \dots \; -->,f_N:\mathbb{R}\to <!--\; \to \; -->[0,\mathrm{\infty <!--\; \infty \; -->}]$ with $\int <!--\; \int \; -->f_1\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}=\mathrm{\infty <!--\; \infty \; -->}$. I would like to ask if
$\underset{n=1}{\overset{N}{\sum <!--\; \sum \; -->}}\int <!--\; \int \; -->f_n\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}=\int <!--\; \int \; -->\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}{f}_n\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}$
still holds. Thank you so much for your explanation!

Trivial Summation Challenge
Write the repeating decimal $\mathrm{1.367367367367...}$ in the form $a/b$ where both $a$ an $b$ are positive whole numbers, but do this using a converging infinite sum.

Prove that $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le <!--\; \le \; -->\frac{3}{2}$
For positive real numbers with $a+b+c=abc$ prove that
$\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le <!--\; \le \; -->\frac{3}{2}$
I made the substitution $a=\tan <!--\; \; -->(\mathrm{\alpha <!--\; \alpha \; -->}),b=\tan <!--\; \; -->(\mathrm{\beta <!--\; \beta \; -->}),c=\tan <!--\; \; -->(\mathrm{\gamma <!--\; \gamma \; -->})$ with the constraint $\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}+\mathrm{\gamma <!--\; \gamma \; -->}=\mathrm{\pi <!--\; \pi \; -->}$
My inequality reduces to proving,
${\cos }^2<!--\; \; -->(\mathrm{\alpha <!--\; \alpha \; -->})+{\cos }^2<!--\; \; -->(\mathrm{\beta <!--\; \beta \; -->})+{\cos }^2<!--\; \; -->(\mathrm{\gamma <!--\; \gamma \; -->})\le <!--\; \le \; -->\frac{3}{2}$
But I am stuck on it. Any help with either inequality would be appreciated.

I was reading article on Bessel's correction:
The article says that sample variance is always less than or equal to population variance when sample variance is calculated using the sample mean. However, if I create a numpy array containing 100,000 random normal data points, calculate the variance, then take 1000 element samples from the random normal data, I find that many of my samples have a higher variance than the 100,000 element population.
import numpy as np
rand_norm = np.random.normal(size=100000)
# save the population variance
pop_var = np.var(rand_norm, ddof=0)
# execute the following 2 lines a few times and and I find a variance of the
# sample that is higher than the variance of rand_normal
samp=np.random.choice(rand_norm, 1000, replace=True)
# calculate the sample variance without correcting the bias (ddof = 0)
# I thought that the variance would always be less than or equal to pop_var.
np.var(samp,ddof=0)
Why am I getting sample variance which is greater than the population variance?

Let $E\subset <!--\; \subset \; -->\mathbb{R}$ be a Borel set and suppose that $(f_k{)}_{k\in <!--\; \in \; -->\mathbb{N}}$ is a sequence of Lebesgue measurable functions defined on $E$. Assume that $\underset{k\to <!--\; \to \; -->+\mathrm{\infty <!--\; \infty \; -->}}{lim}{f}_k(x)=f(x)$ for almost every $x$ and that there exists a Lebesgue integrable function $g$ such that $|f_k(x)|\le <!--\; \le \; -->g(x)$ for almost every $x\in <!--\; \in \; -->E$ and every $k\in <!--\; \in \; -->\mathbb{N}$. Fix $\mathrm{\epsilon <!--\; \epsilon \; -->}>0$ and define $A_k^{\mathrm{\epsilon <!--\; \epsilon \; -->}}=\{x\in <!--\; \in \; -->E:|f_k(x)-<!--\; -\; -->f(x)|\ge <!--\; \ge \; -->\mathrm{\epsilon <!--\; \epsilon \; -->}\}$. Is it true that
$\underset{j\to <!--\; \to \; -->+\mathrm{\infty <!--\; \infty \; -->}}{lim}\left|\underset{k\ge <!--\; \ge \; -->j}{\bigcup <!--\; \bigcup \; -->}{A}_k^{\mathrm{\epsilon <!--\; \epsilon \; -->}}\right|=0?$
($|\cdot <!--\; \cdot \; -->|$ is the Lebesgue measure)
The only thing that I noticed is that $|A_k^{\mathrm{\epsilon <!--\; \epsilon \; -->}}|\underset{k\to <!--\; \to \; -->+\mathrm{\infty <!--\; \infty \; -->}}{\overset{}{\to }}0$, as a consequence of Lebesgue's dominated convergence theorem.
Does anyone know if this is true or not (and why?)?

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

Can someone explain why this happens? (Dividing variables with exponents)
Alright, so let's say I have
$\frac{x^{-<!--\; -\; -->6}}{-<!--\; -\; -->x^{-<!--\; -\; -->4}}$
The answer is ${\displaystyle \frac{1}{x^2}}$, but why isn't it ${\displaystyle \frac{1}{-<!--\; -\; -->x^2}}$??

Why is 7 handled as a negative number in the following equation?
Here is a very simple multi-step equation that bothered me for a specific reason:
20-7x=6x-6
I started by subtracting 6x from both sides and for a second I thought I would find "1x" as a result of subtracting 6x from the left side of the equation (and write it as 20-1x=-6). then I realized that it should have been "-13 x" (20-13x=-6)
Although I could solve it mechanically, I couldn't internalize the logic behind evaluating a subtraction sign as a negative number symbol in algebraic equations. Why should one get the "-" sign before 7x as a negative sign and conduct the subtraction operation accordingly? I would appreciate an answer that covers the logic behind this action.

Fast way to calculate Fraction
I want to solve this problem not by using common denominator. Is there any innovative and fast way ?.
$\frac{1}{6}+\frac{4}{21}+\frac{5}{84}+\frac{7}{228}+\frac{9}{532}=?$

How do you solve and graph ${\displaystyle \frac{a}{9}\ge -15}$ ?

How do you graph ${\displaystyle x\le 4}$ ?

Basic Algebra combining exponent fractions/simplifying (George F Simmons "Precalculus in Nutshell")
From George F. Simmons 'Precalculus' book, Algebra section, 5(d); Combine and Simplify
$\frac{x}{x{y}^2}+\frac{y}{x^{2}y}$
Combine: =
$\frac{x(x^{2}y)+y(x{y}^2)}{(x{y}^2)(x^{2}y)}$
Simplify: =
$\frac{x(x^{2}y)+y(x{y}^2)}{xy(x^2{y}^2)}$
The given answer, which I can't figure out how he arrives at it is:
$\frac{x^2+y^2}{x^2{y}^2}$

For what x is ${\displaystyle \frac{1}{x+3}>\frac{1}{x-2}}$?

Let's say we conduct a random experiment. The possible outcomes may be too complicated to describe, so we instead take some measurement (in terms of real numbers) from it which are of interest to us. Mathematically we have defined a random variable X from the sample space/measurable space $(\mathrm{\Omega <!--\; \Omega \; -->},\mathcal{A})$ to $(\mathbb{R},\mathcal{B})$ where $\mathcal{B}$ is the Borel sigma field. We then choose an appropriate probability for each Borel set, via a distribution function. Then, how does that distribution function determine the probability space $(\mathrm{\Omega <!--\; \Omega \; -->},\mathcal{A},P)$ in theory or proves its existence? What result is implicitly used here?
I understand the probability space on $(\mathbb{R},\mathcal{B})$ is well defined, and that is what practically concerns us, but in theory we also have a probability space $(\mathrm{\Omega <!--\; \Omega \; -->},\mathcal{A},P)$ which pushes forward its measure to the Borel sets. Hence given the distribution on $X$ we are pulling back to $(\mathrm{\Omega <!--\; \Omega \; -->},\mathcal{A})$ via $P(X\in <!--\; \in \; -->B)=P_X(B)$. But what theorem guarantees that such a space exists?
Edit for clarification:For example, let us take weather $\mathrm{\Omega <!--\; \Omega \; -->}$ of a city as the original sample space and the recorded temperature $X$ as the random variable. Suppose the distribution of $X$ is decided upon, based on empirical data. That's fine and now we can answer questions like 'what is $P(a<X<b)$'. We may not care about the probability space on $\mathrm{\Omega <!--\; \Omega \; -->}$ since all the probabilities we wish to know relate to $X$, but how do we know for sure that there exists a probability space on $\mathrm{\Omega <!--\; \Omega \; -->}$ in the first place, and further, how do we know that a probability space on $\mathrm{\Omega <!--\; \Omega \; -->}$ exists which would push its probability to yield the distribution of $X$? Unless we know that there exists such a probability space in theory, we will not be able to talk about expectation of $X$, so the knowledge that such a space exists is crucial.
*Further edit: Note that we may not have an explicit complete mathematical model of the weather ever, but that does not bother us. We do have the ability however, of taking measurements of different types and thereby getting distributions related to temperature, pressure, wind speed, historical records etc. So practically we can answer questions related to these measurements. This is so what happens in practice I think. What I want is some mathematical theorem which assures me that there exists a model of the weather space, even though it is hard/difficult/impossible to find, which pushes its probability on to these distributions.

We know that a $\mathrm{\sigma <!--\; \sigma \; -->}$-algebra is either finite or not countable. Can I find an algebra (field of sets) that is countably infinite? I know that the proof for said theorem uses the fact that a $\mathrm{\sigma <!--\; \sigma \; -->}$-algebra is closed so countable unions, but I cant find an example.

What is x if ${\displaystyle \frac{5}{x+3}\ge \frac{3}{x}?}$