Floors, Fractions & Numbers

Algebra Level 5

$\large{\begin{cases} 3x+\{ y\} +\lfloor z \rfloor=23.3 \\ \lfloor x \rfloor +2y+\{ z\} =9.5 \\ 2\{ x\} +3\lfloor y \rfloor +z=14.1 \end{cases}}$

Let $x,y$ and $z$ be real numbers , whose decimal part is formed by only one digit, satisfying the system of equations above. Find $x+y+z$ .

Notations :

$\lfloor \cdot \rfloor$ denotes the floor function .
$\lceil \cdot \rceil$ denotes the ceiling function .
$\{ \cdot \}$ denotes the fractional part function .

The answer is 14.8.

1 solution

Matteo Monzali
May 31, 2016

Analysing the possible values of the fractional parts, we obtain two choices:

$\\1)\ x=\lfloor x \rfloor + 0.4; y=\lfloor y \rfloor + 0.1; z=\lfloor z \rfloor + 0.3\\ 2)\ x=\lfloor x \rfloor + 0.9; y=\lfloor y \rfloor + 0.6; z=\lfloor z \rfloor + 0.3$

Trying to solve the system with the two different triplets, we find that only the first is possible, so we solve it and find the values $\\x=5.4;\ y=2.1;\ z=7.3\\ x+y+z=14.8$

Another possible solution is $x = 5.025$ , $y = 2.225$ , and $z = 8.05$ . Then $x + y + z = 15.3$ .

Jon Haussmann - 5 years ago

That is true.

Billy Sugiarto - 5 years ago

Right, now I correct thanks

Matteo Monzali - 5 years ago

You cannot conclude the fractional part just by "looking" at it.

$k = [k] +$ { $k$ } $<=>$ { $k$ } $= k -[k] \forall k \in R$ . Here, you only consider the case where { $k$ } $= m.10^{-1}$ for a positive integer $m$ , which isn't the definition of { $k$ }.

Billy Sugiarto - 5 years ago

Yes thanks I've corrected

Matteo Monzali - 5 years ago

Floors, Fractions & Numbers

The answer is 14.8.

1 solution

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