$\frac{5}{8} x+\frac{1}{2}\left(\frac{1}{4} x+10\right)$
To simplify the expression \(\frac{5}{8} x + \frac{1}{2} \left(\frac{1}{4} x + 10\right)\), we first analyze the structure. The expression consists of a term \(\frac{5}{8} x\) added to a product involving a scalar multiplication of a binomial. Our goal is to expand the parentheses and collect like terms to arrive at the simplest form.
The first step is to apply the distributive property to the term \(\frac{1}{2} \left(\frac{1}{4} x + 10\right)\). By multiplying \(\frac{1}{2}\) by each term inside the parentheses, we get:
\[\frac{1}{2} \times \left(\frac{1}{4} x\right) + \frac{1}{2} \times 10 = \frac{1}{8} x + 5\]
Now, substitute this back into the original expression:
\[\frac{5}{8} x + \frac{1}{8} x + 5\]
Now, we combine the terms involving \(x\). Adding the coefficients \(\frac{5}{8}\) and \(\frac{1}{8}\) gives:
\[\frac{5}{8} x + \frac{1}{8} x = \left(\frac{5+1}{8}\right) x = \frac{6}{8} x\]
Simplifying the fraction \(\frac{6}{8}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2, we obtain \(\frac{3}{4}\). Thus, the combined term is \(\frac{3}{4} x\). Adding the constant 5, the fully simplified expression is \(\frac{3}{4} x + 5\).
Simplifying algebraic expressions is a foundational skill used in fields like physics and engineering to reduce complex formulas into manageable ones. For example, if you are calculating the total cost of a project involving a base fee and a variable cost per unit, such as \(\text{Cost} = \frac{5}{8}x + \frac{1}{2}(\frac{1}{4}x + 10)\), simplifying it to \(\frac{3}{4}x + 5\) makes it much easier to estimate the cost for any number of units \(x\). This helps in quick decision-making and budgeting.