X, Y and Z

Algebra Level 4

$\begin{aligned} x+\left[ y \right] +\left\{ z \right\} &=& 4.3\\ y+\left[ z \right] +\left\{ x \right\} &=& 5.5\\ z+\left[ x \right] +\left\{ y \right\} &=& 6.4\\ \end{aligned}$

Given that $x,y,z$ satisfy the equations above.

What is the value of ${ \left( \frac { 20xy }{ z } \right) }^{ 2 }$ ?

Details and Assumptions

$\left [ A \right ]$ denote the greatest integer function of $A$

$\{ A \}$ deonte the fractional part of $A$

The answer is 729.

1 solution

Chew-Seong Cheong
Jan 13, 2015

It is given that:

$\begin{cases} x + \lfloor y \rfloor + \{z\} = 4.3 &...(1) \\ y + \lfloor z \rfloor + \{x\} = 5.5 &...(2) \\ z + \lfloor x \rfloor + \{y\} = 6.4 &...(3) \end{cases}$

$Eq.1 + Eq.2 + Eq.3 \Rightarrow 2(x+y+z) = 16.2 \quad \Rightarrow x+y+z = 8.1 \quad...(4)$

$\begin{cases} Eq.4-Eq.1 \Rightarrow \{y\} + \lfloor z \rfloor = 3.8 & \Rightarrow \lfloor z \rfloor = 3, \quad \{y\} = 0.8 \\ Eq.4-Eq.2 \Rightarrow \{z\} + \lfloor x \rfloor = 2.6 & \Rightarrow \lfloor x \rfloor = 2, \quad \{z\} = 0.6 \\ Eq.4-Eq.3 \Rightarrow \{x\} + \lfloor y \rfloor = 1.7 & \Rightarrow \lfloor y \rfloor = 1, \quad \{x\} = 0.7 \end{cases}$

$\Rightarrow \begin{cases} x = 2.7 \\ y =1.8 \\ z = 3.6 \end{cases} \quad \Rightarrow \left( \dfrac {20xy}{z} \right)^2 = 27^2 = \boxed{729}$

I too used the same method sir :)

A Former Brilliant Member - 6 years, 5 months ago

Same method!!

Harsh Khatri - 6 years, 5 months ago

Note, to get { } to display in Latex, you need to type it in as { }, otherwise it will simply be interpreted as you're trying to define a term.

I've edited your solution. Can you confirm that I've got all of the instances?

Calvin Lin Staff - 6 years, 4 months ago

It appears okay to me. I thought I keyed in correctly earlier. As shown here $\{ \}$ .

Chew-Seong Cheong - 6 years, 4 months ago

X, Y and Z

The answer is 729.

1 solution

0 pending reports