#### Answer

The number we need to find is 18.

#### Work Step by Step

Let X represent the number we need to find.
Two-thirds of the number means $\frac{2}{3}$X.
2 less than two-thirds of the number means $\frac{2}{3}$X - 2.
One-half of the number means $\frac{1}{2}$X
1 more than one-half of the number means 1 + $\frac{1}{2}$X.
Since both of these expressions are equal, we solve the following equation:
$\frac{2}{3}$X - 2 = 1 + $\frac{1}{2}$X
$\frac{2}{3}$X - $\frac{1}{2}$X = 3
$\frac{2}{3}$X $\times$ $\frac{2}{2}$ - $\frac{1}{2}$X $\times$ $\frac{3}{3}$= 3
$\frac{4X}{6}$ - $\frac{3X}{6}$ = 3
$\frac{X}{6}$ = 3
Multiply both sides by 6:
X = 18