Half litre of a solution contains 15% of alcohol. To change the alcohol concentration to 50%, find the quantity of alcohol to be mixed.

Correct — B. Half a litre is 500 ml, of which 15% is alcohol.

Alcohol = 15% of 500 = 75 ml, so the rest is 500 − 75 = 425 ml

Pouring in pure alcohol leaves that 425 ml untouched, and it has to end up as the other half of the mixture:
Final volume = 425 × 2 = 850 ml

Alcohol to be added = 850 − 500 = 350 ml → option (b)

Why the others are wrong

• (a)
  400 ml — 400 ml overshoots: 75 + 400 = 475 ml of alcohol in 900 ml of mixture is 52.8%, past the 50% asked for.
• (c)
  175 ml — 175 ml is 250 − 75 — the answer if the alcohol you pour in did not also enlarge the total. Added, it gives 250 ml in 675 ml, only 37%.
• (d)
  250 ml — 250 ml is 50% of the original 500 ml, the alcohol the mixture would need if its volume stayed at half a litre. Poured in, it gives 325 ml in 750 ml, or 43.3%.

When you add one pure ingredient, the other ingredient's quantity never moves. That is the whole trick.

Here 425 ml is not alcohol and stays 425 ml. If alcohol is to be half the final mixture, the non-alcohol is the other half, so the final volume is 2 × 425 = 850 ml and the top-up is 850 − 500 = 350 ml.

Algebra lands in the same place: (75 + x)⁄(500 + x) = 1⁄2 gives 150 + 2x = 500 + x, so x = 350.

Key facts

• Half a litre is 500 ml, and 15% of it is 75 ml of alcohol.
• Adding pure alcohol leaves the 425 ml of non-alcohol unchanged.
• For alcohol to reach 50%, the non-alcohol must be the other 50%, fixing the final volume at 850 ml.
• Check the answer: 75 + 350 = 425 ml of alcohol in 850 ml is exactly half.

Common traps

• Treating 15% + x% = 50% as if percentages of different totals could be added.
• Forgetting that the alcohol poured in also enlarges the denominator.
• Mixing litres and millilitres in one line, so 0.5 and 500 collide.

No directly related past PYQ was found.