$\lfloor (25x-2)/4 \rfloor =(13x+4)/3$

Question

This problem is designated for those who have the basic knowledge of floor brackets

I know the answer will be 6. But tell me how.

Answer 1

$$\lfloor (25-2)/4 \rfloor =\lfloor (23)/4 \rfloor=\lfloor 5.75 \rfloor=5=(13x+4)/3$$

$$\Rightarrow (13x+4)/3=5$$ $$\Rightarrow (13x+4)=15$$

$$\Rightarrow (13x)=11$$ $$\Rightarrow x=\frac{11}{13}$$

It's not $6$!

Answer 2

It is clear that $\dfrac{13x + 4}{3}$ must be an integer so $x$ must have the form of $2 + 3k$.

Substitute this into the equation and now we must solve:

$$\left\lfloor\frac{48+75k}{4}\right\rfloor = \left\lfloor 12 + \frac{75k}{4}\right\rfloor = 12 + \left\lfloor \frac{75k}{4}\right\rfloor = 13k + 10$$

$$ \left\lfloor \frac{75k}{4}\right\rfloor = 13k - 2$$

Find an approximate solution by solving $\frac{75k}{4} = 13k - 2$ and reaching $k = -\frac{8}{23}$.

We will now look for solutions on intervals around this point.

$$ \left.\left\lfloor \frac{75k}{4}\right\rfloor\right|_{k = -8/23} = -7$$

Let $13k -2 = -8,$ $k = -\dfrac{6}{13}.$

Let $13k -2 = -7,$ $k = -\dfrac{5}{13}.$

Let $13k -2 = -6,$ $k = -\dfrac{4}{13}.$

Let $13k -2 = -5,$ $k = -\dfrac{3}{13}.$

From these possible values of $k$ we find that $x = \dfrac{8}{13}, \dfrac{11}{13}, \dfrac{14}{13}, \dfrac{17}{13}$.

Plugging in to the original equation to verify we find two extraneous solutions and that $\boxed{x = \dfrac{14}{13}, \dfrac{17}{13}}$.

This method will not work well in all cases. Because $\frac{75}{4} = 18.75 \not\approx 13$ I assumed that there are a maximum of four solutions. In a case where we must solve $ \lfloor \frac{75k}{4}\rfloor = 19k - 2$ there are five solutions. In $ \lfloor \frac{75k}{4}\rfloor = 18.75k - 2$ there will be infinitely many.

Answer 3

$\left[{p}\right]=\left[\frac{\mathrm{25}{x}−\mathrm{2}}{\mathrm{4}}\right]=\frac{\mathrm{13}{x}+\mathrm{4}}{\mathrm{3}}={n}\in{Z} \\ $ ${x}=\frac{\mathrm{3}{n}−\mathrm{4}}{\mathrm{13}}\Rightarrow\frac{\mathrm{25}{x}−\mathrm{2}}{\mathrm{4}}=\frac{\mathrm{75}{n}−\mathrm{74}}{\mathrm{52}} \\ $ $\left\{{p}\right\}=\frac{\mathrm{75}{n}−\mathrm{74}}{\mathrm{52}}−{n}=\frac{\mathrm{23}{n}−\mathrm{74}}{\mathrm{52}} \\ $ ${n}\in\left\{\mathrm{4},\mathrm{5}\right\}\Rightarrow{x}\in\left\{\frac{\mathrm{8}}{\mathrm{13}},\frac{\mathrm{11}}{\mathrm{13}}\right\} \\ $

Yes, it's not 6