Answered

Marco prepares 48 ounces of vegetables to use at his restaurant. He uses $3 x$ to represent ounces of vegetables used in each side dish, and $\frac{1}{3}(6 x-2)$ to represent the ounces of vegetables used in each daily special when there are $x$ customers. Marco writes the expression $48-3 x-\frac{1}{3}(6 x-2)$ to predict the number of ounces of vegetables left after $x$ customers order a meal. Which simplified expression is equivalent to the expression Marco wrote?


A$48 \frac{2}{3}-6 x$
B$48 \frac{2}{3}-5 x$
C$46-6 x$
D$46-5 x$

Verified Answer :

  • Start with the expression: \(48 - 3x - \frac{1}{3}(6x - 2)\).
  • Distribute the \(\frac{1}{3}\) term to get: \(48 - 3x - (2x - \frac{2}{3})\).
  • Distribute the negative sign to remove the parentheses: \(48 - 3x - 2x + \frac{2}{3}\).
  • Combine like terms to arrive at the final simplified expression: \(48\frac{2}{3} - 5x\).
  • Final Answer: \(\boxed{\text{B}}\)

Explanation

  1. Analysis of the Expression

The problem asks for the simplification of an algebraic expression that represents the amount of vegetables remaining in a restaurant after serving some customers. The total amount of vegetables prepared is \(48\) ounces. The expression to simplify is:
\[48 - 3x - \frac{1}{3}(6x - 2)\]
where \(x\) represents a variable associated with the number of customers. We need to perform the algebraic operations following the order of operations, which involves distributing the constant \(\frac{1}{3}\) across the parentheses and then grouping like terms.

  1. Distributing the Constant Term

First, we focus on the part of the expression containing the parentheses. We distribute the \(\frac{1}{3}\) across the term \((6x - 2)\):
\[\frac{1}{3}(6x - 2) = \frac{1}{3} \cdot 6x - \frac{1}{3} \cdot 2 = 2x - \frac{2}{3}\]
Substituting this result back into the original expression gives:
\[48 - 3x - \left( 2x - \frac{2}{3} \right)\]

  1. Combining Like Terms and Final Simplification

Now, we distribute the negative sign across the terms inside the parentheses. This changes the sign of each term inside:
\[48 - 3x - 2x + \frac{2}{3}\]
To simplify further, we group the variable terms involving \(x\) together and the constant terms together:
\[48 + \frac{2}{3} + (-3x - 2x)\]
Combining these, we get:
\[48\frac{2}{3} - 5x\]
Thus, the simplified expression is \(48\frac{2}{3} - 5x\). Comparing this to the given options, we identify it as choice B.

Examples

Algebraic simplification is essential in resource management and business planning. For instance, a small business owner must track inventory usage, such as ingredients in a kitchen, to manage costs and prevent waste. By modeling consumption with algebraic expressions like the one Marco used, the owner can easily predict remaining stock levels throughout the day and optimize their purchasing orders, ensuring they always have enough supplies for their customers without overbuying.