To determine which situation represents a proportional relationship, we need to identify whether the relationship between the two quantities is directly proportional. In a proportional relationship, the ratio between two quantities remains constant.
Let's analyze each option:
A. The cost of [tex]\( m \)[/tex] months of a gym membership, where there is a \[tex]$250 startup fee and a \$[/tex]25 per month fee.
The total cost [tex]\( C \)[/tex] of the gym membership for [tex]\( m \)[/tex] months is given by:
[tex]\[ C = 250 + 25m \][/tex]
Here, the cost includes a fixed startup fee of \[tex]$250 plus \$[/tex]25 per month. Because of the fixed startup fee, the relationship is not proportional. The total cost does not simply vary directly with the number of months because of the initial fee.
B. The cost of [tex]\( n \)[/tex] t-shirts, where each t-shirt costs \[tex]$19.99 plus an additional \$[/tex]1.75 each for tax.
The total cost [tex]\( C \)[/tex] for [tex]\( n \)[/tex] t-shirts is given by:
[tex]\[ C = n(19.99 + 1.75) = n \times 21.74 \][/tex]
In this case, the total cost is directly proportional to the number of t-shirts purchased. There are no additional fixed fees or varying costs. The cost per t-shirt (including tax) remains constant at \[tex]$21.74. Thus, this represents a proportional relationship.
C. The cost of riding \( r \) rides at the fair, where the admission to the fair is \$[/tex]12.75 and each ride costs \[tex]$1.50.
The total cost \( C \) for \( r \) rides is given by:
\[ C = 12.75 + 1.50r \]
Here, there is a fixed admission fee of \$[/tex]12.75 plus an additional \[tex]$1.50 per ride. Due to the fixed admission fee, the total cost is not directly proportional to the number of rides.
D. The cost of purchasing \( b \) books at a bookstore, where each paperback book is \$[/tex]12 and each hardbound book is \$20.
The total cost [tex]\( C \)[/tex] depends on the type and number of books purchased. Let's break it down:
- If you only purchase paperback books, the cost is:
[tex]\[ C = 12b \][/tex]
- If you only purchase hardbound books, the cost is:
[tex]\[ C = 20b \][/tex]
However, without knowing the exact number of paperback and hardbound books separately, we cannot establish a clear proportional relationship for general [tex]\( b \)[/tex] books that include different types. The costs are not uniformly constant.
Conclusion:
Option B represents a proportional relationship, as the total cost per t-shirt including tax remains constant regardless of the number of t-shirts purchased.
