# Copper that was 95% pure was melted together with copper that was 75% pure to make 15 kilograms of an alloy that was 83% pure. How many kilograms of each kind were used?

Aug 6, 2015

Mass of 95% pure copper: $\text{7.8 kg}$
Mass of 75% pure copper: $\text{7.2 kg}$

#### Explanation:

The first piece of information that you're going to use is the fact that you know how much alloy you have and its purity, which means that you know how much copper it contains.

More precisely, the alloy contains

15 color(red)cancelcolor(black)("kg alloy") * "83 kg copper"/(100color(red)cancelcolor(black)("kg alloy")) = "12.45 kg copper"

If $x$ is the mass of 95% pure copper and $y$ is the mass of 75% pure copper, then you can say that

$x + y = 15$

At the same time, the masses of actual copper must add up to give 12.45 kg. This means that you can write

$x \cdot \frac{95}{100} + y \cdot \frac{75}{100} = 12.45$

Use the first equation to write $x$ as a function of $y$, then use this expression to find $x$

$x = 15 - y$

$\left(15 - y\right) \cdot \frac{95}{100} + y \cdot \frac{75}{100} = 12.45$

$14.25 - 0.95 y + 0.75 y = 12.45$

$- 0.25 y = - 1.8 \implies y = \frac{1.8}{0.25} = \textcolor{g r e e n}{\text{7.2 kg}}$

This means that $x$ will be equal to

$x = 15 - 7.2 = \textcolor{g r e e n}{\text{7.8 kg}}$

So, if you mix 7.8 kg of 95% pure copper with 7.2 kg of 75% pure copper you get a 15 kg of 83% pure copper alloy.