Question

You bought 8 mangoes and 3 apples for $18 and 3 mangoes and 5 apples for $14.50. Define variables, write a system of equations, and solve it to find how much one mango and one apple cost.

Answer 1

Let's define the variables:

- Let \( m \) be the cost of one mango.
- Let \( a \) be the cost of one apple.

Now, we can translate the purchases into equations:

1. The first transaction states that 8 mangoes and 3 apples cost $18:
\[
8m + 3a = 18 \quad \text{(Equation 1)}
\]

2. The second transaction states that 3 mangoes and 5 apples cost $14.50:
\[
3m + 5a = 14.50 \quad \text{(Equation 2)}
\]

Now, we have the following system of equations:
\[
\begin{align*}
8m + 3a &= 18 \quad \text{(1)} \\
3m + 5a &= 14.5 \quad \text{(2)}
\end{align*}
\]

Next, we can solve this system by using the elimination or substitution method. We'll use the elimination method.

To eliminate \( a \), we can multiply Equation 1 by 5 and Equation 2 by 3 so that the coefficients of \( a \) in both equations are equal:

\[
\begin{align*}
5(8m + 3a) &= 5(18) \\
40m + 15a &= 90 \quad \text{(Equation 3)}
\end{align*}
\]

\[
\begin{align*}
3(3m + 5a) &= 3(14.5) \\
9m + 15a &= 43.5 \quad \text{(Equation 4)}
\end{align*}
\]

Now we can subtract Equation 4 from Equation 3:
\[
(40m + 15a) - (9m + 15a) = 90 - 43.5
\]

This simplifies to:
\[
31m = 46.5
\]

Now, we can solve for \( m \):
\[
m = \frac{46.5}{31} = 1.5
\]

Next, we substitute \( m \) back into one of the original equations to find \( a \). We can use Equation 1:
\[
8(1.5) + 3a = 18
\]

Calculating \( 8(1.5) \):
\[
12 + 3a = 18
\]

Now, solve for \( a \):
\[
3a = 18 - 12 \\
3a = 6 \\
a = 2
\]

Thus, the cost of one mango and one apple are:
\[
\boxed{1.5} \text{ for a mango and } \boxed{2} \text{ for an apple.}
\]