Question

Your first mate is a chemist (of course) so you have him mixing gunpowder back at the ship. He uses a mixture consisting of 14% potassium nitrate mineral ( x \rightarrow pounds) with a 24% potassium nitrate powder (y) to make 100 pounds of gunpowder containing 18% potassium nitrate.

Which systems of linear equations models this situation?

Answer 1

To model this situation, we will set up a system of linear equations based on the information given. We are trying to mix two types of potassium nitrate to create a compound gunpowder.

Let:
- \( x \) = amount (in pounds) of the 14% potassium nitrate mineral.
- \( y \) = amount (in pounds) of the 24% potassium nitrate powder.

From the problem statement, we have two key pieces of information:

1. The total weight of the mixture should be 100 pounds:
\[
x + y = 100
\]

2. The total amount of potassium nitrate in the mixture should provide 18% of the total weight (which is 100 pounds) as potassium nitrate. This means the total potassium nitrate from both components should equal 18 pounds (since \( 0.18 \times 100 = 18 \)):

The amount of potassium nitrate contributed by the 14% mineral is \( 0.14x \) and from the 24% powder is \( 0.24y \). So, we can set up the second equation as follows:
\[
0.14x + 0.24y = 18
\]

Thus, the system of linear equations modeling this situation is:
\[
\begin{cases}
x + y = 100 \\
0.14x + 0.24y = 18
\end{cases}
\]