2 apples and 1 pear cost $1.80. 3 such apples and 5 such pears cost $4.10. Find the average cost per item of 4 such pears and 5 such apples

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**.

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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!

2 apples and 1 pear cost $1.80. 3 such apples and 5 such pears cost $4.10. Find the average cost per item of 4 such pears and 5 such apples.

Solve the system of linear equations to find the cost of apples and pears, and calculate the average cost per item for 4 pears and 5 apples. Learn key algebraic concepts such as simultaneous equations and averages. Suitable for Grades 8-10.

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