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A radiator holds 40 liters of pure antifreeze. A mechanic drains 8 liters, replaces them with water, and mixes thoroughly, repeating this drain-and-replace step with 8 liters of the current mixture each time. After how many steps does the antifreeze concentration first drop below 50%?
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Correct answer
B
The engine here is multiplicative decay; the scoring step is reading the boundary honestly at step 3 versus step 4. Each time the mechanic drains 8 of the 40 liters, one fifth of the current mixture leaves, so each step keeps 4/5 of whatever antifreeze is present. After k steps, the concentration is (4/5)ᵏ.
Track it one step at a time: after step 1 it is 0.8, after step 2 it is 0.64, after step 3 it is 0.512, and after step 4 it is 0.4096. The tempting call is to read 0.512 as roughly a half and stop at step 3, but 0.512 is still above 50%. The concentration first drops below 50% at step 4, where it reaches 0.4096.
The antifreeze concentration first drops below 50% after 4 steps.
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