Select the correct answer.
The science club is comparing options for purchasing new T-shirts. One company charges a $100 base fee plus an additional $3.50 per shirt. A second company charges a $120 base fee but only an additional $2.50 per shirt. The given expression represents the difference of the two companies' average cost per shirt.
[tex]$\frac{3.5 x+100}{x}-\frac{2.5 x+120}{x}$[/tex]
Which expression is equivalent to the expression representing this situation?
A. [tex]$\frac{x-20}{x}$[/tex]
B. [tex]$x-120$[/tex]
C. [tex]$\frac{x+220}{z}$[/tex]
D. [tex]$6 x+220$[/tex]
Answer (1)
The problem requires simplifying the given expression. The steps are:
Combine the two fractions into a single fraction: x 3.5 x + 100 − x 2.5 x + 120 = x ( 3.5 x + 100 ) − ( 2.5 x + 120 ) .
Simplify the numerator by combining like terms: x 3.5 x + 100 − 2.5 x − 120 = x ( 3.5 x − 2.5 x ) + ( 100 − 120 ) = x x − 20 .
The simplified expression is: x x − 20 .
Explanation
Understanding the Problem We are given the expression x 3.5 x + 100 − x 2.5 x + 120 which represents the difference of the two companies' average cost per shirt. We want to find an equivalent expression.
Combining Fractions Since the two fractions have the same denominator, we can combine them into a single fraction: x 3.5 x + 100 − x 2.5 x + 120 = x ( 3.5 x + 100 ) − ( 2.5 x + 120 )
Simplifying the Numerator Now, we simplify the numerator by combining like terms: x 3.5 x + 100 − 2.5 x − 120 = x ( 3.5 x − 2.5 x ) + ( 100 − 120 ) = x x − 20
Final Answer Therefore, the equivalent expression is x x − 20 .
Examples
Understanding average costs is crucial in many real-world scenarios. For example, imagine you're running a small business and need to decide between two suppliers for your materials. One supplier might have a lower per-unit cost but a higher initial setup fee, while the other has a higher per-unit cost but a lower setup fee. By analyzing the average cost per unit at different production volumes, you can determine which supplier offers the best deal for your specific needs. This type of analysis helps optimize your spending and maximize profits.
