cups of the 10% sugar solution
cups of the 30% sugar solution
============================================================
Work Shown:
x = number of cups of the 10% solution
y = number of cups of the 30% solution
The two mixes add together to get 8 cups since this is the total amount we want. Therefore, the first equation is simply x+y = 8
Solve for y to get y = 8-x. We'll use this equation later.
-----------------------
The second equation is a bit more tricky.
0.10x = amount of pure sugar from just the 10% batch
0.30y = amount of pure sugar from just the 30% batch
0.10x+0.30y = total amount of pure sugar from both batches
25% of 8 = 0.25*8 = 2 = total amount of pure sugar required
0.10x+0.30y = 2 is the second equation
-----------------------
Apply substitution to solve for x and y
0.10x+0.30y = 2 ..... start with the second equation
0.10x+0.30(8-x) = 2 ....... plug in y = 8-x
0.10x+2.4-0.30x = 2 ..... distribute
-0.20x + 2.4 = 2
-0.20x = 2-2.4 ....... subtract 2.4 from both sides
-0.20x = -0.4
x = -0.4/(-0.20) ..... divide both sides by -0.20
x = 2
Now that we know the value of x, we can find y.
y = 8-x
y = 8-2 ... plug in x = 2
y = 6
So we need 2 cups of the 10% solution and 6 cups of the 30% solution.
-----------------------
As a check,
x+y = 2+6 = 8 works out
0.10*x+0.30*y = 0.10*2+0.30*6 = 0.2+1.8 = 2 works out as well