X, Y and Z are 2-digit numbers. The average of X, Y and Z is 56. X is $\frac{2}{3}$ of Y. Find the smallest possible value of X. Answer: ___

1. Find the total sum of X, Y, and Z.

$\text{Sum of } X, Y, Z = 56 \times 3 = 168$

2. Express X and Y in terms of units.

Since $X = \frac{2}{3}Y$, we can let $X = 2u$ and $Y = 3u$.
$2u + 3u + Z = 168$
$5u + Z = 168$

3. Determine the maximum possible value of Z to minimize X.

To find the smallest possible value of $X$ (and thus the smallest $u$), we must make $Z$ as large as possible.
$Z$ is a 2-digit number, so its maximum theoretical value is $99$.
From $5u = 168 - Z$, $168 - Z$ must be a multiple of $5$.
The largest 2-digit number $Z$ that makes $168 - Z$ a multiple of $5$ is $98$ (since $168 - 98 = 70$).

4. Calculate the smallest possible value of X.

If $Z = 98$, then $5u = 70 \implies u = 14$.
Smallest $X = 2u = 2 \times 14 = 28$.
(Check: $Y = 3 \times 14 = 42$. Both $X$ and $Y$ are 2-digit numbers.)