Many grocery stores have machines.

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:

  1. 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=0.119x

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%).

  1. 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+0.05x

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

  1. Now, let’s calculate the fees for $13.65 in coins at both machines:
  2. Coinstary=0.119(13.65)

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

  1. 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)

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

f+0.05x<0.119x

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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