Riddle! Riddle! #2

Since it is a number less than $1000$ and has a middle digit, it is a three digit number.

Numbers are either in ones, tens, hundreds, thousands, etc.

Since this is a three digit number, it is written in hundreds, tens and ones. For example, $121$ can be written as $100+20+1$ . In other words, $100(1)+10(2)+1$ . $356$ can also be written as $300+50+6$ . In other words, $100(3)+10(5)+6$ .

Let our three digit number be represented by $xyz$ . Meaning it could also be written as $100(x)+10(y)+z$ .

Remember that $x+z=8$ .

When I am written in the reverse order, my new number is 16 less than three times the original number. That is,

$zyx=3 \times (xyz)-16$ . In other words,

$100(z)+10(y)+x=3(100(x)+10(y)+z)-16$

Since the middle digit, $y$ , is zero, we are left with

$100(z)+x=3(100(x)+z)-16$

$100(z)+x=300(x)+3(z)-16$

$97(z)=299(x)-16$

From the equation, $x+z=8$ , we have $x=8-z$

Placing this in the equation, $97(z)=299(x)-16$

We have $97(z)=299(8-z)-16$

$97(z)+299(z)=2392-16$

$396(z)=2376$

$z=6$

Placing this in the equation, $x+z=8$

$x=2$

Therefore, the number is $206$