Question

Jace went into a grocery store and bought 8 apples and 6 mangos, costing a total of $30.50. Kiran went into the same grocery store and bought 2 apples and 3 mangos, costing a total of $10.25. Determine the price of the apples and the price of the mangos.

Answer 1

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:
1. For Jace:
\[
8(2.50) + 6(1.75) = 20 + 10.5 = 30.50 \quad \text{(Correct)}
\]
2. 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.