Five years ago, a high-school cafeteria charged $5.85 for three pieces of fruit and a chicken salad. Today, each piece of fruit costs 12% more, while a chicken salad costs 15% more. The new cost of three pieces of fruit and a chicken salad is $6.66. Determine the new prices of a piece of fruit and a chicken salad.

Let the cost of a piece of fruit 5 years ago be denoted by $ f $ and the cost of a chicken salad 5 years ago be denoted by $ s $. The problem provides the following information:

1. Five years ago, the cost of three pieces of fruit and a chicken salad was $5.85:
\[
3f + s = 5.85
\]
2. Today, the cost of each piece of fruit is 12% more, and the cost of a chicken salad is 15% more. Therefore, the new cost of three pieces of fruit and a chicken salad is $6.66:
\[
3(f \times 1.12) + (s \times 1.15) = 6.66
\]
Simplifying the second equation:
\[
3(1.12f) + 1.15s = 6.66
\]
\[
3.36f + 1.15s = 6.66
\]
Now we have a system of two equations:

\[
1) \quad 3f + s = 5.85
\]
\[
2) \quad 3.36f + 1.15s = 6.66
\]

### Solving the system:

From the first equation, solve for $ s $:
\[
s = 5.85 - 3f
\]

Substitute this expression for $ s $ into the second equation:
\[
3.36f + 1.15(5.85 - 3f) = 6.66
\]
Now, expand the equation:
\[
3.36f + 1.15 \times 5.85 - 1.15 \times 3f = 6.66
\]
\[
3.36f + 6.7275 - 3.45f = 6.66
\]
Simplify:
\[
-0.09f + 6.7275 = 6.66
\]
\[
-0.09f = 6.66 - 6.7275
\]
\[
-0.09f = -0.0675
\]
\[
f = \frac{-0.0675}{-0.09} = 0.75
\]
So, the price of a piece of fruit 5 years ago was $ f = 0.75 $.

Now, substitute $ f = 0.75 $ into the first equation:
\[
3(0.75) + s = 5.85
\]
\[
2.25 + s = 5.85
\]
\[
s = 5.85 - 2.25 = 3.60
\]

Thus, the price of a chicken salad 5 years ago was $ s = 3.60 $.

### New Prices:
- The new price of a piece of fruit is $ 0.75 \times 1.12 = 0.84 $.
- The new price of a chicken salad is $ 3.60 \times 1.15 = 4.14 $.

Therefore, the new prices are:
- **A piece of fruit costs $0.84**.
- **A chicken salad costs $4.14**.

Would you like further details or have any questions?

Here are 5 related questions:
1. What would the total cost be if you purchased 5 pieces of fruit and 2 chicken salads today?
2. How much did the price of a single chicken salad increase over the 5-year period?
3. What is the percentage increase in the total price from $5.85 to $6.66?
4. If fruit prices continue to increase by 12% every 5 years, what will a piece of fruit cost in another 5 years?
5. How would the system change if the fruit price increase was 10% instead of 12%?

**Tip:** When dealing with percentage changes, multiplying the original value by $1 + \text{percentage change}$ (in decimal form) is a quick way to find the new value.

Learn how to determine the new prices of a piece of fruit and a chicken salad using algebraic equations and percentage increase. This real-world problem involves solving a system of linear equations and understanding price changes over time. Suitable for students in Grades 9-11.

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