The science club is comparing options for purchasing new T-shirts. One company charges a [tex]$100[/tex] base fee plus an additional [tex]$3.50[/tex] per shirt. A second company charges a [tex]$120[/tex] base fee but only an additional [tex]$2.50[/tex] per shirt. The given expression represents the difference of the two companies' average cost per shirt.
[tex]\frac{3.5 x+100}{2}-\frac{2.5 x+120}{2}[/tex]
Which expression is equivalent to the expression representing this situation?
A. [tex]$x-120$[/tex]
B. [tex]$6 x+220$[/tex]
C. [tex]$\frac{z+220}{z}$[/tex]
D. [tex]$\frac{z-20}{z}$[/tex]
Answer (1)
Combine the fractions: 2 3.5 x + 100 − 2 2.5 x + 120 = 2 ( 3.5 x + 100 ) − ( 2.5 x + 120 ) .
Simplify the numerator: 2 3.5 x + 100 − 2.5 x − 120 .
Combine like terms: 2 ( 3.5 x − 2.5 x ) + ( 100 − 120 ) = 2 x − 20 .
The equivalent expression is: 2 x − 20 .
Explanation
Problem Analysis Let's analyze the problem. We are given an expression that represents the difference in average cost per shirt between two companies. Our goal is to simplify this expression and find an equivalent one among the given options. The given expression is:
Combining Fractions The given expression is: 2 3.5 x + 100 − 2 2.5 x + 120 Since both fractions have the same denominator, we can combine them:
Simplifying the Numerator Combining the fractions, we get: 2 ( 3.5 x + 100 ) − ( 2.5 x + 120 ) Now, we simplify the numerator by distributing the negative sign:
Combining Like Terms Simplifying the numerator: 2 3.5 x + 100 − 2.5 x − 120 Next, we combine like terms in the numerator:
Comparing with Options Combining like terms: 2 ( 3.5 x − 2.5 x ) + ( 100 − 120 ) 2 x − 20 Now, we compare this simplified expression with the given options.
Finding the Equivalent Expression The simplified expression is 2 x − 20 . Let's look at the options:
A. x − 120 B. 6 x + 220 C. x x + 220 D. 2 x − 20
Option D, 2 x − 20 , matches our simplified expression.
Final Answer Therefore, the equivalent expression is 2 x − 20 .
Examples
Understanding how to simplify algebraic expressions like this is useful in many real-world scenarios. For example, when comparing costs from different vendors, you might have a base fee and a per-item charge. Simplifying the expression that represents the difference in cost helps you quickly determine which vendor offers the better deal based on the number of items you need. This is applicable in business, personal finance, and even in scientific research when comparing costs of different experimental setups.
