Question 48·Hard·Linear Equations in One Variable
Mika joined a basketball league that charges a one-time registration fee of $25 plus the same fee for each game played. This season, she played 8 games and also incurred a $12 late payment penalty. Her total payment for the season was $109.
Which equation must be true for , the fee in dollars for each game?
For word problems that ask you to write an equation, first identify the variable and what it represents, then turn each phrase into an algebra term: fixed fees become constants, and repeated charges like "p dollars for each of 8 games" become a product such as . Decide whether each amount increases or decreases the total (add or subtract accordingly), then combine all the parts and set them equal to the given total. Finally, scan the answer choices for the one whose terms and signs exactly match your expression.
Hints
List the different charges
Break the story into separate amounts of money: one-time charge, per-game charges, and any extra fees. How many distinct money amounts are there, and what are they?
Turn words into algebra pieces
How can you write "p dollars for each of 8 games" using multiplication with ? Think: number of games times dollars per game.
Decide on plus or minus
Does a late payment penalty make Mika pay more money or less money? Should that be added to or subtracted from the total when you write the equation?
Connect to the total
Once you have expressions for the registration fee, the total game fees, and the penalty, add them together and set that sum equal to the total amount she paid, . Which option shows that correctly?
Desmos Guide
Model the total cost as an expression
In Desmos, define a function for Mika’s total cost in terms of by adding: the fixed registration fee (25), plus the number of games (8) times , plus the penalty (12). Type this as a single expression like T(p) = followed by those three parts added together.
Match your expression to a choice
Look at the structure of the expression you just typed in Desmos: a fixed number, plus a multiple of for the 8 games, plus another fixed number for the penalty. Compare that expression term-by-term with each answer choice and select the choice that has the same three parts in the same way.
Step-by-step Explanation
Identify all the money amounts in the story
From the problem:
- One-time registration fee:
- Fee for each game: dollars per game
- Number of games:
- Late payment penalty:
- Total payment for the season:
We want one equation that connects all these pieces using the variable .
Write an expression for each part of the total cost
Turn each part of the description into algebra:
- Registration fee is just .
- Playing games at dollars for each game is (multiply number of games by the fee per game).
- The late payment penalty adds another dollars, so that contributes to the total.
These three parts together make up the total cost.
Combine the parts and set them equal to the total
Add all the cost parts together and set their sum equal to the total payment of :
- Registration:
- Games:
- Penalty:
So the equation that must be true is: