KA-0165Mixtures and weighted averages
4 points, difficulty 2 of 3Level 9–10about 120sFour litres of a solution that is 25 percent acid are mixed with six litres of a solution that is 50 percent acid. What percentage of the mixture is acid?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Percentages only average when the amounts match.
2The strategy
Work out how many litres of pure acid are in each solution, then add them.
3The full solution
0.25 x 4 = 1 litre and 0.50 x 6 = 3 litres, so 4 litres of acid in 10 litres of mixture, which is 40 percent.
Solution
The reliable way
Track the acid itself rather than the percentages. The first solution contributes 0.25 x 4 = 1 litre of acid; the second contributes 0.50 x 6 = 3 litres. Together that is 4 litres of acid in 4 + 6 = 10 litres of mixture, so the concentration is 4/10 = 40 percent. The transferable idea: a weighted average always lands nearer the ingredient you used more of - here nearer 50 than 25, which a plain average of 37.5 would miss.
The elegant way
There is more of the strong solution, so the answer must exceed the midpoint of 37.5, and only 40 does.
Why this is on the test: The plain average is the trap, and noticing which side the answer must fall on eliminates it before any arithmetic.