If $\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15\;$ then which of the following equals $a×b×c$?

Question

The problem is:

If $\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15,\;$ then which of the following equals $a×b×c$ ?

A) $\frac {1}{375}$

B) $\frac{1}{125}$

C) $\frac{27}{125}$

D) $\frac{3}{125}$

E) $\frac{27}{5}$

I think the question is wrong.

My counterexample:

Let $x=m,\; y=m-3,\; z=m.$

Then $a=b=c=\frac{3^{m-1}}{5}.$

So, $a×b×c=\frac{3^{3m-3}}{125},\; m\in\mathbb{R}.$

Am I right?

Answer 1

Yes some information is missing, indeed we have that

$$\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15$$

then

$$abc=\frac{3^{(x+y+z)}}{125}$$

then we need a condition for $t=x+y+z\in \mathbb R$, since $3^{t}$ can assume any value $\in(0,\infty)$.