Let's define the cost of one apple as $a$ and the cost of one pear as $p$. From the problem, we are given two equations:

1. The cost of 2 apples and 1 pear is $1.80: $2a + p = 1.80 \quad \text{(Equation 1)}$

2. The cost of 3 apples and 5 pears is $4.10: $3a + 5p = 4.10 \quad \text{(Equation 2)}$

Step 1: Solve the system of equations

We will solve these two equations simultaneously to find the values of $a$ and $p$.

Multiply Equation 1 by 5:

$5(2a + p) = 5(1.80)$ $10a + 5p = 9.00 \quad \text{(Equation 3)}$

Subtract Equation 2 from Equation 3:

$(10a + 5p) - (3a + 5p) = 9.00 - 4.10$ $7a = 4.90$ $a = \frac{4.90}{7} = 0.70$

Now that we know $a = 0.70$, substitute this value into Equation 1 to find $p$.

Substitute $a = 0.70$ into Equation 1:

$2(0.70) + p = 1.80$ $1.40 + p = 1.80$ $p = 1.80 - 1.40 = 0.40$

So, the cost of one apple is $a = 0.70$ dollars, and the cost of one pear is $p = 0.40$ dollars.

Step 2: Find the total cost of 4 pears and 5 apples

Now, we want to find the total cost of 4 pears and 5 apples.

$\text{Total cost} = 5a + 4p = 5(0.70) + 4(0.40) = 3.50 + 1.60 = 5.10$

Step 3: Find the average cost per item

The total number of items is 4 pears and 5 apples, which is 9 items in total. The average cost per item is:

$\text{Average cost per item} = \frac{5.10}{9} = 0.5667 \approx 0.57 \, \text{dollars}$

Final Answer:

The average cost per item of 4 pears and 5 apples is approximately $0.57.

---

Do you want more details or have any questions?

Here are 5 related questions:

1. How would the solution change if the cost of the pear was higher?
2. Can you solve the same system of equations using substitution instead of elimination?
3. What if 2 apples and 3 pears cost $2.30? How would the system change?
4. What is the total cost of 10 apples and 8 pears with the same prices?
5. How do you interpret the difference between the price of apples and pears?

Tip: Always double-check your calculations when working with simultaneous equations to avoid errors when manipulating the equations!