A gym charges a one-time sign-up fee of $25 and a monthly fee of $30. If Emma paid a total of $205 after months, which equation models this situation?
For linear word problems about fees, first identify the fixed (one-time) fee and the repeating fee (per month, per item, per hour, etc.). The repeating fee should be multiplied by the variable (like for months), and the fixed fee should be a constant term added to that product. Then set this total expression equal to the given total amount in the problem and select the equation that matches this structure, checking whether each term and sign (+ or −) correctly reflects the situation described.
Hints
Separate one-time and monthly fees
Underline or note the words "one-time sign-up fee" and "monthly fee." Which amount is paid just once, and which is paid every month?
Decide which number multiplies
Ask yourself: does the number of months, , affect the sign-up fee or the monthly fee? The number that repeats each month should be multiplied by .
Think about adding vs. subtracting
Since Emma is paying both the sign-up fee and the monthly fees, should those amounts be added together or should one be subtracted from the other?
Connect the expression to the total
Once you have an expression for Emma's total cost in terms of , remember to set it equal to the total amount she paid, $205.
Desmos Guide
Interpret as the x-variable
In Desmos, let be represented by (Desmos uses as the default variable), so think of each equation as involving instead of .
Enter each option as a function of x
Type the left side of each answer choice into its own line in Desmos, solving each for . For example, for an option like , rewrite it as (ignore the for now) to represent the total cost after months.
Use a table to test key values
For each expression you entered (such as or ), tap the table icon to create a table. Check what equals when and when . The correct model will give the one-time fee when and the sum of the sign-up fee plus one month of charges when ; choose the option whose expression matches those values.
Step-by-step Explanation
Identify the two types of charges
The problem describes two kinds of fees:
- A one-time sign-up fee of $25 (this is paid only once, not multiplied by months).
- A monthly fee of $30 (this is paid every month, so it repeats times).
Write an expression for the total cost
For months:
- The total monthly cost is because Emma pays $30 each month for months.
- The one-time sign-up fee is $25.
So, the total amount Emma pays after months is the sum of these: .
Set the total equal to the given amount
The question says Emma paid a total of $205. That means the expression for her total cost must equal $205.
So we write the equation:
This matches choice C, so the correct answer is .