Orange​ juice, a raisin​ bagel, and a cup of coffee from​ Billy's Breakfast Bar cost a total of ​$3.60. Billy posts a notice announcing​ that, effective the following​ week, the price of orange juice will increase​ 50% and the price of bagels will increase​ 20%. After the​ increase, the same purchase will cost a total of ​$4.50​, and the orange juice will cost twice as much as coffee. Find the price of each item before the increase.
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Part 1
What was the cost of a glass of orange juice before the​ increase? ​$
  
enter your response here
What was the cost of a raisin bagel before the​ increase? ​$
  
enter your response here
What was the cost of a cup of coffee before the​ increase? ​$
  
enter your response here

Let's assign variables to represent the prices of the three items before the price increase:

- Let $ x $ be the price of a glass of orange juice.
- Let $ y $ be the price of a raisin bagel.
- Let $ z $ be the price of a cup of coffee.

From the information provided, we can set up the following system of equations:

### Equation 1 (based on the original total price):
\[
x + y + z = 3.60
\]
### Equation 2 (after the price increase):
The price of orange juice increases by 50%, so the new price of orange juice is $ 1.5x $.
The price of a raisin bagel increases by 20%, so the new price of the bagel is $ 1.2y $.
The price of coffee remains the same at $ z $.

Thus, the total price after the increase is:
\[
1.5x + 1.2y + z = 4.50
\]
### Equation 3 (orange juice is twice the price of coffee after the increase):
\[
1.5x = 2z
\]

Now, let's solve this system of equations.

#### Substituting $ z $ from the third equation:
\[
z = \frac{1.5x}{2} = 0.75x
\]

Now, substitute $ z = 0.75x $ into both Equation 1 and Equation 2.

### Equation 1:
\[
x + y + 0.75x = 3.60 \quad \text{which simplifies to} \quad 1.75x + y = 3.60
\]

### Equation 2:
\[
1.5x + 1.2y + 0.75x = 4.50 \quad \text{which simplifies to} \quad 2.25x + 1.2y = 4.50
\]

We now have the system of two equations:
1. $ 1.75x + y = 3.60 $
2. $ 2.25x + 1.2y = 4.50 $

We will solve these equations step-by-step.The solution to the system of equations is:

- The price of a glass of orange juice before the increase was $ \$1.20 $.
- The price of a raisin bagel before the increase was $ \$1.50 $.

Now, let's find the price of a cup of coffee before the increase using the relation $ z = 0.75x $:

\[
z = 0.75 \times 1.20 = 0.90
\]

- The price of a cup of coffee before the increase was $ \$0.90 $.

Would you like any further details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the total cost change if coffee prices increased by 10% as well?
2. What is the percentage increase in the total price after the adjustments?
3. Can you calculate the total price if the cost of coffee decreased by 20% instead?
4. How much more would a person spend on coffee if they bought two cups instead of one?
5. How would you model this scenario with three unknown price increases?

**Tip**: When working with price changes, it's helpful to represent the changes as a percentage multiplier to simplify calculations.

Orange juice, a raisin bagel, and a cup of coffee from Billy's Breakfast Bar cost a total of $3.60. Billy posts a notice announcing that, effective the following week, the price of orange juice will increase 50% and the price of bagels will increase 20%. After the increase, the same purchase will cost a total of $4.50, and the orange juice will cost twice as much as coffee. Find the price of each item before the increase. What was the cost of a glass of orange juice before the increase? What was the cost of a raisin bagel before the increase? What was the cost of a cup of coffee before the increase?

Learn how to solve a system of equations to find the original prices of orange juice, raisin bagel, and coffee before a price increase at Billy's Breakfast Bar. Understand the relationship between percentages and price changes. Suitable for Grades 9-12 students studying algebra.

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