Multiple choice

A chef has a 80-liter vat filled with concentrated fruit syrup. He draws out 8 liters of the syrup and replaces it with 8 liters of plain water. This procedure — removing 8 liters of the current mixture and adding 8 liters of water — is repeated multiple times. What is the minimum number of such repetitions needed so that the quantity of syrup remaining in the vat becomes less than the quantity of water?

  1. 4

  2. 5

  3. 7

  4. 8
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

This is a mixture problem where the amount of syrup remaining after n replacements is 80 * (1 - 8/80)^n = 80 * (0.9)^n. We want 80 * (0.9)^n < 40, or (0.9)^n < 0.5. For n=6, 0.9^6 = 0.531. For n=7, 0.9^7 = 0.478. Thus 7 repetitions are needed.

AI explanation

Using the replacement formula, Final = Initial * (1 - Replaced/Total)^n, we need the syrup to be less than 50% of the 80 liter vat. Thus 80 * (1 - 8/80)^n < 40, which simplifies to (0.9)^n < 0.5. Testing values shows n=6 yields 0.531, while n=7 yields 0.478, so the minimum repetitions needed is 7.

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