#### 3. Many grocery stores have machines that will give you cash for your change but charge a fee.

a. Coinstar machines charge an average fee of 11.9% of your coins’ value. Write an equation that models the total fee you pay (y) based on the value of your coins (x) if you use a Coinstar machine.

b. A new company, ChangeMaker, has a coin machine that charges a flat fee, plus 5% of your coins’ value. Write an equation to model the fee you pay (y) based on the value of your coins (x).

c. Imagine you have $13.65 in coins that you want to exchange for bills. How much would you pay in fees at each machine? Round your answer to the nearest cent.

d. When is the ChangeMaker a better deal than the Coinstar? Why?

**Final Answer:**

The Coinstar machine charges a fee of 11.9%, modeled by the equation y = 0.119x. The ChangeMaker machine charges a flat fee plus 5%, expressed as y = f + 0.05x. For $13.65 in coins, Coinstar charges approximately $1.62, and ChangeMaker is better if the flat fee is less than about $0.94.

**Explanation**

To tackle this question, we will go through each part step-by-step:

- For Coinstar, the fee is 11.9% of the value of your coins (x). The equation that models the total fee you pay (y) can be expressed as:

*y*=0.119*x*

This means that if you want to know how much fee you will pay, you multiply the total value of coins by 0.119 (or 11.9%).

- For ChangeMaker, we introduce a flat fee (let’s call it
*f*) plus 5% of the value of your coins (x). The equation can be modeled as:

*y*=*f*+0.05*x*

In this equation, the flat fee (f) is added to 5% of the value of the coins.

- Now, let’s calculate the fees for $13.65 in coins at both machines:
**Coinstar**:*y*=0.119(13.65)

*y*=1.62235≈1.62 (rounded to the nearest cent) Therefore, the fee at Coinstar would be approximately $1.62.

**ChangeMaker**: Assuming a flat fee (f) is specified, the calculation would be:*y*=*f*+0.05(13.65)

*y*=*f*+0.6825≈*f*+0.68 (this will depend on the value of *f*)

- To determine when ChangeMaker becomes a better deal than Coinstar, we can set up the inequality:

*f*+0.05*x*<0.119*x*

Replacing x with $13.65, we get:

*f*+0.6825<1.62235

*f*<1.62235−0.6825

*f*<0.93985≈0.94

This means that if the flat fee (f) charged by ChangeMaker is less than approximately $0.94, then ChangeMaker is a better deal than Coinstar.

**Summary of Calculations:**

- Coinstar Fee: $1.62
- ChangeMaker Fee: $f + $0.68 (depending on the flat fee f)
- ChangeMaker becomes a better deal when f < $0.94.

In conclusion, to find out which machine is better for you, you need to know the flat fee charged by ChangeMaker, and compare it with Coinstar’s fee based on your coin’s total value.

**Examples & Evidence**

For instance, if the flat fee for ChangeMaker is $0.50, the total fee would be $0.50 + $0.68 = $1.18, making it a better choice compared to Coinstar.

The calculations used to derive the equations and compare different fee structures follow standard percentage calculations and algebraic inequalities.

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