# How many grams of gold should a coin of 35% gold be if when combined with a 3 grams pure gold necklace, it forms a metal that is 69 % gold?

Dec 14, 2015

Here's what I got.

#### Explanation:

Since I'm not sure if you're asking for the mass of the 35% gold coin or for the mass of gold it must contain, I'll show you how to find both values.

So, let's say that the coin that is 35% gold has a total mass of $x$ grams. For this coin, every $\text{100 g}$ will contain $\text{35 g}$ of gold, which means that your coin will contain

x color(red)(cancel(color(black)("g coin"))) * "35 g gold"/(100color(red)(cancel(color(black)("g coin")))) = 35/100x " g gold"

Now let's say that the mass of the 69% gold coin is equal to $y$ grams. This coin must contain

y color(red)(cancel(color(black)("g coin"))) * "69 g gold"/(100color(red)(cancel(color(black)("g coin")))) = 69/100y" g gold"

Now, use the $\text{3-g}$ pure gold coin to write two equations with two unknowns, the mass of gold in the second coin and the total mass of the second coin

$x + 3 = y$

and

$\frac{35}{100} x + 3 = \frac{69}{100} y$

Use the first equation to get

$x = y - 3$

Plug this into the second equation to get

$\frac{35}{100} \cdot \left(y - 3\right) + 3 = \frac{69}{100} y$

$\frac{35}{100} y - \frac{105}{100} + 3 = \frac{69}{100} y$

This is equivalent to

$\frac{35 - 69}{100} \cdot y = - \frac{195}{100}$

$34 y = 195 \implies y = 5.74$

This means that $x$ will be equal to

$x = 5.74 - 3 = 2.74$

So, if you start with a 35% gold coin that has a total mass of $\text{2.74 g}$, and mix it with a $\text{3-g}$, 100% gold coin, the resulting coin will have a mass of $\text{5.74 g}$ and be 68% gold.

The first coin will contain

2.74 color(red)(cancel(color(black)("g coin"))) * "35 g gold"/(100color(red)(cancel(color(black)("g coin")))) = "0.959 g gold"

I'll leave the values rounded to three sig figs, despite the fact that your values would only justify one sig fig.