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B
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Difficulty:
65%
(hard)
Question Stats:
65% (02:58) correct
35%
(02:22)
wrong
based on 65
sessions
History
Date
Time
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Not Attempted Yet
Calogero has two solutions: solution A contains 90% alcohol and comes in 0.4-liter bottles, while solution B contains 20% alcohol and comes in 0.8-liter bottles. He combines whole bottles of each to obtain a mixture that is 60% alcohol.
What is the smallest possible volume of the mixture Calogero obtained ?
(A) 4.4 liters
(B) 5.6 liters
(C) 6.6 liters
(D) 8.8 liters
(E) 11 liters
What is the smallest possible volume of the mixture Calogero obtained ?
(A) 4.4 liters
(B) 5.6 liters
(C) 6.6 liters
(D) 8.8 liters
(E) 11 liters
16 Apr 2026, 00:53
Kudos
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This is a mixture problem with a twist: whole bottles only. That constraint is where most people lose the point.
1. Set up the mixing equation. Let a = number of A bottles, b = number of B bottles.
Alcohol from A: 0.9 x 0.4a = 0.36a
Alcohol from B: 0.2 x 0.8b = 0.16b
Total volume: 0.4a + 0.8b
For a 60% mixture: 0.36a + 0.16b = 0.6(0.4a + 0.8b)
2. Expand and simplify.
0.36a + 0.16b = 0.24a + 0.48b
0.12a = 0.32b
3a = 8b
So the ratio a:b = 8:3.
3. Here's the trap. A lot of people stop at the ratio and then calculate volume using a = 8 and b = 3 without thinking about the bottle sizes. That actually works here, but the key insight is: the whole-bottle constraint means a and b must be positive integers. The smallest integer solution for 3a = 8b is a = 8, b = 3.
4. Calculate the minimum volume.
0.4 x 8 + 0.8 x 3 = 3.2 + 2.4 = 5.6 liters
Answer: B.
Any multiple works too (a = 16, b = 6 gives 11.2 liters, etc.), but 5.6 is the minimum. Whenever you see "whole units," your ratio gives you the building block, not the final answer.
1. Set up the mixing equation. Let a = number of A bottles, b = number of B bottles.
Alcohol from A: 0.9 x 0.4a = 0.36a
Alcohol from B: 0.2 x 0.8b = 0.16b
Total volume: 0.4a + 0.8b
For a 60% mixture: 0.36a + 0.16b = 0.6(0.4a + 0.8b)
2. Expand and simplify.
0.36a + 0.16b = 0.24a + 0.48b
0.12a = 0.32b
3a = 8b
So the ratio a:b = 8:3.
3. Here's the trap. A lot of people stop at the ratio and then calculate volume using a = 8 and b = 3 without thinking about the bottle sizes. That actually works here, but the key insight is: the whole-bottle constraint means a and b must be positive integers. The smallest integer solution for 3a = 8b is a = 8, b = 3.
4. Calculate the minimum volume.
0.4 x 8 + 0.8 x 3 = 3.2 + 2.4 = 5.6 liters
Answer: B.
Any multiple works too (a = 16, b = 6 gives 11.2 liters, etc.), but 5.6 is the minimum. Whenever you see "whole units," your ratio gives you the building block, not the final answer.
16 Apr 2026, 11:23
Kudos
Bookmarks
given A 90% alcohol in 0.4 ltr bottle let x bottles
and B 20% alcohol in 0.8 ltr bottle let y bottles
when whole bottles combined to get the solution of 60% alcohol concentration.
minimum(0.4x+0.8y)=?
90*0.4x+20*0.8y=60(0.4x+0.8y)
36x+16y=24x+48y
3x=8y
i.e. for whole bottles y should be a multiple of 3 and x should be a multiple of 8.
so minimum amount of liquid =8*0.4+3*0.8=3.2+2.4=5.6 ltrs
B
and B 20% alcohol in 0.8 ltr bottle let y bottles
when whole bottles combined to get the solution of 60% alcohol concentration.
minimum(0.4x+0.8y)=?
90*0.4x+20*0.8y=60(0.4x+0.8y)
36x+16y=24x+48y
3x=8y
i.e. for whole bottles y should be a multiple of 3 and x should be a multiple of 8.
so minimum amount of liquid =8*0.4+3*0.8=3.2+2.4=5.6 ltrs
B
