Explanation:
a 68% mixture means it has a 68/100 fraction of alcohol :
68 ml of alcohol as part of a 100 ml mixture.
I am not sure how to understand the problem description, so I solve for 2 different problems.
1. create a new 600 ml mixture with 68% alcohol.
since we want to have a 600 ml mixture, we need to multiply the denominator of the fraction by 6.
to keep the value of the fraction unchanged, we need then also to multiply the numerator by 6 :
68/100 × 6/6 = 408/600
that means they need to include 408 ml of 100% alcohol with 192 ml of the rest of the ingredients to create 600 ml of a 68% mixture.
2. we have 600 ml of 50% mixture and are adding 100% alcohol to get a 68% mixture.
the current mixture has a 50/100 fraction of alcohol.
so, 300 ml of the 600 ml mixture are 100% alcohol.
we need to add x ml of 100% alcohol to make it a 68% mixture of 600 + x ml.
that means the sum of the existing 600 ml plus the added 100% alcohol must reach the same (68/100) fraction of alcohol.
(300 + x) / (600 + x) = 68/100
100(300 + x) = 68(600 + x)
30000 + 100x = 40800 + 68x
30000 + 32x = 40800
32x = 10800
2x = 675
x = 337.5 ml
we need to add 337.5 ml of 100% alcohol to the existing 600 ml mixture to get a a mixture of 68% alcohol in the in total 937.5 ml.