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.