Mr. Smith goes to the farmers market twice each month to buy fresh produce The first time he went, he purchased six peaches and four apples from one of the vendors for $13.50 the next time he won he purchased five apples and four peaches for $12.50. How much did the apples and peaches cost individually.

Answer 1

Let \( p \) represent the price of a peach, and \( a \) represent the price of an apple.

From the information given, we can set up the following two equations based on Mr. Smith's purchases:

1. For the first purchase: \[ 6p + 4a = 13.50 \quad \text{(1)} \]

2. For the second purchase: \[ 4p + 5a = 12.50 \quad \text{(2)} \]

Now, we can solve this system of equations.

First, let's multiply both equations by 2 to eliminate the decimals:

1. Multiply equation (1) by 2: \[ 12p + 8a = 27 \quad \text{(3)} \]

2. Multiply equation (2) by 2: \[ 8p + 10a = 25 \quad \text{(4)} \]

Next, we can eliminate one of the variables. Let's eliminate \( a \) by making the coefficients of \( a \) equal in both equations.

To do this, we can multiply equation (3) by 5 and equation (4) by 4:

1. From equation (3): \[ 60p + 40a = 135 \quad \text{(5)} \]

2. From equation (4): \[ 32p + 40a = 100 \quad \text{(6)} \]

Now we can subtract equation (6) from equation (5): \[ (60p + 40a) - (32p + 40a) = 135 - 100 \] \[ 28p = 35 \] \[ p = \frac{35}{28} = \frac{5}{4} = 1.25 \]

Now that we have the price of a peach \( p = 1.25 \), we can substitute this value back into one of the original equations to find the price of an apple. Let's use equation (1):

\[ 6(1.25) + 4a = 13.50 \] \[ 7.50 + 4a = 13.50 \] \[ 4a = 13.50 - 7.50 \] \[ 4a = 6 \] \[ a = \frac{6}{4} = \frac{3}{2} = 1.50 \]

So, the final prices are:

• The price of a peach ( \( p \) ) is \( $1.25 \).
• The price of an apple ( \( a \) ) is \( $1.50 \).