What is the %m/m of 75.0 g of ethanol dissolved in 500.0 g of water?

By definition, a solution's percent concentration by mass, `"% m/m"`, tells you the number of grams of solute present for every `"100 g"` of the solution.

This means that if you know the number of grams of ethanol dissolved in exactly `"100 g"` of this solution, you know the solution's percent concentration by mass.

Now, you know that your solution contains `"75.0 g"` of ethanol and `"500.0 g"` of water. The total mass of the solution is equal to

`"75.0 g " + " 500.0 g" = "575.0 g"`

So you can say your `"575.0-g"` sample of this solution contains `"75.0 g"` of ethanol, the solute.

This means that `"100 g"` of this solution will contain

`100 color(red)(cancel(color(black)("g solution"))) * "75.0 g ethanol"/(575.0 color(red)(cancel(color(black)("g solution")))) = "13.043 g ethanol"`

Since this represents the number of grams of ethanol present per `"100 g"` of the solution, you can say that the solution's percent concentration by mass is equal to

`color(darkgreen)(ul(color(black)("% m/m = 13.0% ethanol")))`

The answer is rounded to three sig figs, the number of sig figs you have for the mass of ethanol.