Verified Answer :

  • Distribute the fraction \(\frac{1}{4}\) to both terms inside the parentheses.
  • Multiply \(\frac{1}{4} \times 32x\) to get \(8x\).
  • Multiply \(\frac{1}{4} \times 48\) to get \(12\).
  • Sum the terms to obtain the final result: \(\boxed{8x + 12}\).

Explanation

  1. Problem Analysis and Method Selection

The expression to be simplified is \(\frac{1}{4}(32x + 48)\). The problem asks us to distribute the factor \(\frac{1}{4}\) across the terms inside the parentheses to reach a simpler linear form. We apply the distributive property, which states that \(a(b + c) = ab + ac\). Here, \(a = \frac{1}{4}\), \(b = 32x\), and \(c = 48\).

  1. Simplification Process

To simplify the expression, we multiply \(\frac{1}{4}\) by each term inside the parentheses:
\[\frac{1}{4}(32x + 48) = \left(\frac{1}{4} \times 32x\right) + \left(\frac{1}{4} \times 48\right).\]
First, we multiply \(\frac{1}{4}\) by \(32x\):
\[\frac{1}{4} \times 32x = 8x.\]
Next, we multiply \(\frac{1}{4}\) by \(48\):
\[\frac{1}{4} \times 48 = 12.\]
Combining these two results, we get the simplified expression:
\[8x + 12.\]

  1. Final Conclusion

The expression \(\frac{1}{4}(32x + 48)\) simplifies to \(8x + 12\). Thus, the final simplified form is \(\boxed{8x + 12}\).

Examples

Simplifying algebraic expressions is a fundamental skill in mathematics, used daily in many fields. For example, if you are calculating the cost of a purchase where you buy 32 items for a price \(x\) and an additional flat shipping fee of \$48, but then you get a 75% discount (meaning you pay 25% or 1/4 of the original cost), you can use the expression \(\frac{1}{4}(32x + 48)\) to calculate your final cost, which simplifies to \(8x + 12\). This makes it much easier to quickly determine the total price for any base item price \(x\).

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