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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 percent?
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Correct answer
B
The engine is multiplicative decay; the scoring step is reading the boundary honestly at step 3 versus step 4. Each drain removes 8 of the 40 liters, one fifth of the mixture, so each step keeps 4/5 of whatever antifreeze is present, and after k steps the concentration is (4/5)ᵏ.
Track it one step at a time: 0.8, then 0.64, then 0.512, then 0.4096. The tempting call, seated as choice A, is reading 0.512 as roughly a half and stopping at step 3; but 0.512 sits above 50 percent, so the concentration first falls below the line at step 4, at 0.4096.
Choice C is the off-by-one that labels that same 0.4096 as step 5; D decays by one tenth per step from misreading 8 out of 40; E races below 20 percent, the fraction drained, instead of 50. Dilution multiplies, and 0.512 has not crossed the line: the answer is choice B.
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