Multiple choice

A jar contains some quantity of apple juice. Kyla draws 8 ounces of juice from it and then fills the jar with water. She performs this activity three more times. The ratio of the quantity of apple juice now left in the jar to that of water is 16 : 65. How much apple juice did the jar originally hold?

  1. 18 ounces

  2. 22 ounces

  3. 24 ounces

  4. 26 ounces

  5. 30 ounces
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let x be the initial amount of apple juice. After 4 draws of 8 ounces, the fraction of apple juice remaining is (1 - 8/x)^4. Given the ratio of juice to water is 16:65, the fraction of juice remaining is 16/(16+65) = 16/81. Setting (1 - 8/x)^4 = 16/81, we get 1 - 8/x = 2/3, so 8/x = 1/3, which means x = 24.

AI explanation

Using the formula for repeated replacement, the ratio of remaining pure apple juice to the initial quantity is the initial volume minus 8, divided by the initial volume, raised to the power of 4. This ratio is provided by the given ratio of remaining apple juice to water, where the remaining juice corresponds to 16 parts out of a total of 81 parts, resulting in the fraction 16 divided by 81. Equating this fraction to the replacement ratio gives the initial volume minus 8, divided by the initial volume, equaling the fourth root of 16 divided by 81, which simplifies to 2 divided by 3. Solving this equation gives the initial volume as 24 ounces.

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