Question

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?

Answer 1

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 `"100 g"` will contain `"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 `"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

`35/100x + 3 = 69/100y`

Use the first equation to get

`x = y - 3`

Plug this into the second equation to get

`35/100 * (y - 3) + 3 = 69/100y`

`35/100y - 105/100 + 3 = 69/100y`

This is equivalent to

`(35 - 69)/100 * y = -195/100`

`34y = 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 `"2.74 g"`, and mix it with a `"3-g"`, `100%` gold coin, the resulting coin will have a mass of `"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.