A drum contains a blend of three chemicals: P, Q, and R, in the ratio of 3:5:2. 30 litres of this blend are removed, and then 12 litres of chemical P and 8 litres of chemical Q are poured into the drum. If the resultant quantity of chemical Q is 20 litres more than the resultant quantity of chemical P, what was the initial total quantity of the mixture in the drum (in litres)?

Correct — C. Let the drum hold T litres at the start. Taking out part of a blend leaves its ratio at 3:5:2, so split what is left.

Remove 30 L: T − 30 litres left, still 3:5:2
P left = 0.3(T − 30), Q left = 0.5(T − 30)
Add 12 L to P: P = 0.3(T − 30) + 12
Add 8 L to Q: Q = 0.5(T − 30) + 8

Subtract: Q − P = 0.2(T − 30) − 4
Set Q − P = 20: 0.2(T − 30) = 20 + 4 = 24
Divide by 0.2: T − 30 = 120
Add back the 30: T = 150 → option (c)

Why the others are wrong

• (a)
  100 — 100 L at the start leaves 70 L: P = 21 + 12 = 33 and Q = 35 + 8 = 43. The gap is 10 L, not the 20 L the question states.
• (b)
  120 — 120 L is what remains after the removal (T − 30 = 120), not the starting quantity. Taken as the start, it leaves 90 L and a Q − P gap of only 14 L.
• (d)
  180 — 180 L at the start leaves 150 L: P = 45 + 12 = 57 and Q = 75 + 8 = 83. The gap is 26 L, overshooting the 20 L given.

Taking litres out of a well-mixed blend removes each chemical in the blend's own ratio, so the drum still holds P, Q and R at 3:5:2 afterwards. Only the litres poured in later change the ratio.

So split the remainder (T − 30) by 3:5:2 first, then add the new P and Q on top. The stated gap, Q exceeding P by 20 L, gives one equation in T.

Key facts

• Removing part of a uniform mixture leaves the ratio of its components unchanged.
• In a 3:5:2 blend, P is 3⁄10 and Q is 5⁄10 of any amount taken out or left behind.
• Here the drum starts at 150 L, drops to 120 L, and ends with P = 48 L and Q = 68 L, a gap of 20 L.

Common traps

• Stopping at T − 30 = 120: that is the quantity left after the removal, and it sits in the options as (b).
• Splitting the whole drum instead of the remainder: 0.2T − 4 = 20 also lands on 120.
• Adding the 12 L and 8 L before splitting by 3:5:2, which shares new chemical into R.

No directly related past PYQ was found.