Finding a number with certain properties

Question

I am having a hard time finding this number.

Here is my work:

$RLPNM$

$3(RLPNM)=RLPNM-1$

However, how am I supposed to solve this equation of $5$ unknowns? Are there any different approaches to this problem?

Answer 1

Your expansion of the number into powers of ten is unnecessary.

Let's call this number $N$.

If we add a $1$ before it, we are effectively adding $10^5$ to $N$.

If we add a $1$ after it, we are effectively multiplying $N$ by $10$ then adding $1$

Put this all together and we have:

$$3\cdot \left ( 10^5 + X \right ) = \left (10\cdot X\right ) +1$$

Rearranging yields:

$$10^5 -1 = 7 X$$

Thus, it is clear that:

$$ X = \frac{10^5-1}{7}=42857$$