Which of the following could be the possible value of $x$ for which each of the fractions is in its simplest form, where $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to $x$ ?
$\dfrac{\lfloor{x\rfloor}+7}{10},\ \dfrac{\lfloor{x\rfloor}+18}{11},\ \dfrac{\lfloor{x\rfloor}+31}{12},\ \dfrac{\lfloor{x\rfloor}+46}{13},\cdots\cdots\cdots\cdots,\ \dfrac{\lfloor{x\rfloor}+1489}{39}$
and
$\dfrac{\lfloor{x\rfloor}+1567}{40}$
$a.)\ 95.71\quad \quad \quad \quad \quad b.)\ 93.71 \\ \color{green}{c.)\ 94.71}\quad \quad \quad \quad \quad d.)\ 92.71 \\$
I simplified this into
$\dfrac{\lfloor{x\rfloor}+7}{10},\ 1+\dfrac{\lfloor{x\rfloor}+7}{11},\ 2+\dfrac{\lfloor{x\rfloor}+7}{12},\ 3+\dfrac{\lfloor{x\rfloor}+7}{13},\cdots\cdots\cdots\cdots,\ 38+\dfrac{\lfloor{x\rfloor}+7}{39}$ and
$39+\dfrac{\lfloor{x\rfloor}+7}{40}$
I am also struggling to interpret the question.
This question if from chapter quadratic equations though it doesn't look like.
I look for a short and simple way.
I have studied maths up to $12$th grade.
Interpreting the question:
The greatest integer function and the non-integral answers are a red herring, meant to distract you. Everything in the question except $x$ is an integer, so the fractional part of $x$ is immediately thrown away and not used. Your simplification is also important, since the integral parts of the expression are irrelevant as to whether the fraction is in lowest terms. Therefore, the question could be more simply phrased:
Which of these are all in simplest terms: $$\frac{x+7}{10},\ \frac{x+7}{11},\ \frac{x+7}{12},\ \ldots,\ \frac{x+7}{40}$$
with the answer choices (a) $95$; (b) $93$; (c) $94$; (d) $92$
A fraction is not in lowest terms if the greatest common divisor for the numerator and denominator is not one: i.e. if there is any common divisor greater than one.
Solving the problem:
For each possible value of $x$, let's look at the factors of $x+7$.
(a) $95+7=102=2\cdot 3\cdot 17$. Many denominators in the list of $10$ through $40$ have a factor of $2$ or $3$, and $17$ is itself in the list. So this is not the answer.
(b) $93+7=100=2\cdot 2\cdot 5\cdot 5$. Many denominators in the list of $10$ through $40$ have a factor of $2$ or $5$. So this is not the answer.
(c) $94+7=101$ which is prime. None of those denominators can have any factor in common, so this is the answer.
(d) $92+7=99=9\cdot 11$. Many denominators in the list of $10$ through $40$ have a factor of $9$ or $11$, and $11$ is itself in the list. So this is not the answer.
Therefore the only answer is (c), $94.71$ which becomes $94$.
Quadratic equations:
Why is this in a chapter about quadratic equations? The fractions given are all in the form
$$\frac{\lfloor x\rfloor + (n+4)^2-18}{n+9}$$
for $n=1,2,\ldots, 31$. You can see the quadratic expression in $n$ in the numerator.