how to do weighted averages

Question

how many grams of sugar must be added to 60 grams of a solution that is 32% sugar to obtain a solution that is 50% sugar.

Answer 1

Lala,

 

Let s = sugar and w = water.  Then s+w = 60 g and s/(s + w) = 0.32.  Replace (s + w) with 60 g

 

s/60 g = 0.32, then s = 0.32·(60 g) = 19.2 g.  This is the mass of sugar in a 32 % solution.

 

To find how much sugar to add to get a 50% solution we need the total mass of the sugar divided by the total mass of the solution (sugar + water) and set = 0.50

 

So:  (19.2 g + x)/(60 g + x) = 0.50

 

multiply both sides by the denominator:  (19.2 g + x) = 0.50·(60 g + x) = 30 g + 0.5x

 

collecting terms with x on the left:  0.5·x = (30 - 19.2) g = 10.8 g

 

Divide both sides by 0.5:  x = (10.8/0.5) g = 21.6 g (amount of sugar that must be added to get 50% solution)

 

Check:  total sugar = 19.2 g + 21.6 g =40.8 g

Total mass of solution = 60 g + 21.6 g = 81.6 g

 

total sugar ÷ total mass = 40.8/81.8 = 0.5 = 50%