Let the price of an apple be \( a \) dollars and the price of a mango be \( m \) dollars.
From Jace's purchase, we can set up the equation: \[ 8a + 6m = 30.50 \tag{1} \]
From Kiran's purchase, we have another equation: \[ 2a + 3m = 10.25 \tag{2} \]
We can solve the system of equations formed by (1) and (2). First, let's simplify equation (2) by multiplying it by 3: \[ 6a + 9m = 30.75 \tag{3} \]
Next, we can manipulate equation (1) so that we can align it with our new equation (3). We'll multiply equation (1) by 1: \[ 8a + 6m = 30.50 \]
Now we can eliminate \( m \) by multiplying both sides of (1) by 1.5 (to match the coefficients of \( m \) in equation (3)): \[ 12a + 9m = 45.75 \tag{4} \]
Now we have:
- From equation (3): \( 6a + 9m = 30.75 \)
- From equation (4): \( 12a + 9m = 45.75 \)
Next, we can subtract equation (3) from equation (4): \[ (12a + 9m) - (6a + 9m) = 45.75 - 30.75 \] This simplifies to: \[ 6a = 15 \] Dividing both sides by 6 gives: \[ a = 2.50 \]
Now that we have the price of an apple, we can substitute \( a \) back into equation (2) to find \( m \): \[ 2(2.50) + 3m = 10.25 \] This simplifies to: \[ 5 + 3m = 10.25 \] Subtracting 5 from both sides gives: \[ 3m = 5.25 \] Dividing both sides by 3 gives: \[ m = 1.75 \]
Thus, the prices are:
- Price of an apple: \( \boxed{2.50} \)
- Price of a mango: \( \boxed{1.75} \)
To verify, we can substitute these values back into the original equations:
- For Jace: \[ 8(2.50) + 6(1.75) = 20 + 10.5 = 30.50 \quad \text{(Correct)} \]
- For Kiran: \[ 2(2.50) + 3(1.75) = 5 + 5.25 = 10.25 \quad \text{(Correct)} \]
Therefore, the prices of the apples and mangos are indeed \( \boxed{2.50} \) and \( \boxed{1.75} \) respectively.