Solve for $p$.
$\frac{1}{2} p-\frac{1}{4} p=4$
We are given the linear equation \(\frac{1}{2}p - \frac{1}{4}p = 4\). Our goal is to isolate the variable \(p\) and determine its value. The equation involves two linear terms in \(p\) that can be combined by finding a common denominator for the coefficients.
To combine the terms on the left-hand side, we note that the fractions \(\frac{1}{2}\) and \(\frac{1}{4}\) have a common denominator of 4. Since \(\frac{1}{2} = \frac{2}{4}\), we can rewrite the equation as follows:
\[\frac{2}{4}p - \frac{1}{4}p = 4\]
Subtracting the fractions, we get:
\[\left(\frac{2-1}{4}\right)p = 4\]
\[\frac{1}{4}p = 4\]
Now that the equation is simplified to \(\frac{1}{4}p = 4\), we can solve for \(p\) by multiplying both sides of the equation by 4:
\[4 \times \left(\frac{1}{4}p\right) = 4 \times 4\]
\[p = 16\]
Thus, the solution to the equation is \(p = 16\).
Solving linear equations like this is essential in real-world budgeting and resource allocation. For example, if you are splitting a total budget of \$4,000 between two different departments where one department receives exactly half of the total budget and another receives a quarter, the remaining amount is what you might be trying to solve for, or if you know the remaining amount, you can determine the total budget. This algebraic skill helps in managing finances efficiently and accurately.