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------------------------------------------------ The cost to rent a rug-cleaning machine from the supermarket includes a fee of [tex]$45[/tex] for the rental. Tristanne spent [tex]$137[/tex] to rent the rug-cleaning machine. Which equation can be used to find the number of days, [tex]d[/tex], Tristanne rented the machine?

A. [tex]23d = 137[/tex]

B. [tex]23d + 45 = 137[/tex]

C. [tex]45d = 137[/tex]

D. [tex]45d + 23 = 137[/tex]

Answer :

Sure! Let's work through the problem step-by-step to find the correct equation that can be used.

### Problem Details:
- Rental Fee: There is a fixed fee of [tex]$45 to rent the machine.
- Total Spent: Tristanne spent a total of $[/tex]137 on the rental.
- Daily Cost: The cost of renting the machine per day is [tex]$23.
- Goal: Find an equation to represent this situation and determine how many days Tristanne rented the machine for.

### Understanding the Scenario:
When Tristanne rents the machine, she has to pay a $[/tex]45 fee upfront. In addition to this, she pays [tex]$23 for each day she rents the machine. Together, these costs total to $[/tex]137.

### Formulating the Equation:

1. Daily Costs Contribution:
- Let's denote the number of days the machine is rented as [tex]\( d \)[/tex].
- The cost just for renting the machine per day is [tex]\( 23d \)[/tex] (since it's [tex]$23 per day).

2. Total Cost Equation:
- Add the daily costs to the initial fee to find the total cost.
- The total cost can be represented by the equation:
\[
23d + 45 = 137
\]
- Here, \( 23d \) represents the daily rental cost over \( d \) days and $[/tex]45 is the initial rental fee.

### Conclusion:

The equation [tex]\( 23d + 45 = 137 \)[/tex] accurately represents the total expenditure for renting the rug-cleaning machine, including both the flat fee and the daily expenses.

This aligns with option 2 from the choices provided, which correctly describes the situation.