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?

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.