A vessel contains a mixture of 100 litres of milk and water.

Question

A vessel contains a mixture of 100 litres of milk and water. The concentration of milk is 90%. 10% of the contents of the vessel are withdrawn and replaced with an equal amount of water. The minimum number of times that this procedure must be carries out such that the concentration of milk in the vessel is less than 66.66% is

A. 1
B. 2
C. 3
D. 5
E. 6

Show Answer

Official Answer

C

A. 1
B. 2
C. 3
D. 5
E. 6

ankushsambare · Answer

Initially, the vessel contains 100 litres with 90% milk.
So milk = 90 litres.
Each time:
 
 10% of the mixture is removed 
 Then replaced with water  
So every time, 90% of the milk remains.
Milk left after n operations:
90 × (0.9)^n
We need the milk concentration to become less than 66.66%.
Since total volume always remains 100 litres, we need:
milk < 66.66 litres
So,
90 × (0.9)^n < 66.66
Divide by 90:
(0.9)^n < 0.7407
Now check powers:
 
 n = 1 → 0.9 
 n = 2 → 0.81 
 n = 3 → 0.729  
Since 0.729 < 0.7407, the condition is satisfied at n = 3.
 Answer: C. 3

ChillSpike · Answer

But this is a quant question, i.e no calculator

egmat · Answer

Hi ChillSpike,

Great instinct to be suspicious, but here the arithmetic is fully hand-friendly. The step you saw (checking powers of  0.9 ) looks calculator-heavy only because it was written as decimals. Let me show you the no-calculator route.

Track the milk in litres, not as a power.  Each procedure removes  10%  of the milk and tops up with water, so each time you just knock  10%  off the milk. Taking  10%  off a number is easy: move the decimal one place and subtract.

Start with  90  litres of milk:
- After  1  time:  90  -  9  =  81 
- After  2  times:  81  -  8.1  =  72.9 
- After  3  times:  72.9  -  7.29  =  65.61

No powers, no long multiplication - just subtract one-tenth each step.

Now the comparison, without messy decimals.  Notice that  66.66%  is just  2/3 , so the target milk amount is  66.66  litres (since the total stays  100 ). You only need to see when the milk drops below that:
-  81  and  72.9  are both above  66.66  - not yet.
-  65.61 is below 66.66  - condition met at the  3rd  time.

Therefore  the answer is  C (3) , reached with nothing but "take  10%  off" three times and one comparison.

The takeaway: when you see repeated "remove x% and replace," don't reach for exponent computations. Just multiply (or here, subtract) step by step - GMAT keeps these numbers clean on purpose, exactly because there's no calculator.

Answer: C

A vessel contains a mixture of 100 litres of milk and water.

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A vessel contains a mixture of 100 litres of milk and water.

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