Let the unknown number be denoted by $x$.

According to the problem, one sixth of the number is 3 less than one half of the number. We can express this as:

$\frac{1}{6}x = \frac{1}{2}x - 3$

Solving the equation:

1. To eliminate the fractions, multiply both sides of the equation by 6:

$x = 3x - 18$

2. Move all terms involving $x$ to one side:

$x - 3x = -18$

3. Simplify the equation:

$-2x = -18$

4. Divide by -2:

$x = 9$

Thus, the number is $\boxed{9}$.

Would you like a detailed explanation or have any other questions?

Relative questions:

1. What is the next step if the equation had different constants?
2. How would the problem change if the fractions involved were different?
3. Can you solve a similar problem with different coefficients?
4. What happens when both sides of the equation are equal without subtraction?
5. Can you set up an equation for other types of proportional relationships?

Tip: Multiplying both sides of an equation by the least common denominator (LCD) helps clear out fractions quickly.