Number Representation and Equation Setup

Let the unknown number be represented by the variable $x$. The problem states that the sum of its half ($\frac{x}{2}$), one-third ($\frac{x}{3}$), and one-fifth ($\frac{x}{5}$) exceeds the number itself by $12$. This relationship can be written as the following algebraic equation:

$ \left( \frac{x}{2} + \frac{x}{3} + \frac{x}{5} \right) - x = 12 $

Solving the Sum of Fractions Equation

To find the value of $x$, we first need to combine the fractional terms. The least common multiple (LCM) of the denominators $2$, $3$, and $5$ is $30$. We rewrite each fraction using this common denominator:

• $ \frac{x}{2} = \frac{15x}{30} $
• $ \frac{x}{3} = \frac{10x}{30} $
• $ \frac{x}{5} = \frac{6x}{30} $

Substitute these equivalent fractions back into the equation:

$ \left( \frac{15x}{30} + \frac{10x}{30} + \frac{6x}{30} \right) - x = 12 $

Combine the numerators of the fractions:

$ \frac{31x}{30} - x = 12 $

To subtract $x$, express it with the same denominator ($x = \frac{30x}{30}$):

$ \frac{31x}{30} - \frac{30x}{30} = 12 $

Simplify the expression on the left side:

$ \frac{x}{30} = 12 $

Isolate $x$ by multiplying both sides of the equation by $30$:

$ x = 12 \times 30 $

$ x = 360 $

Final Number Value

The number is $360$.