2024 AMC 12A Problem 5

Below is the professionally curated solution for Problem 5 of the 2024 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2024 AMC 12A solutions, or check the answer key.

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Concepts:mean linear equation

Difficulty rating: 1230

A data set containing $20$ numbers, some of which are $6,$ has mean $45.$ When all the $6$ s are removed, the data set has mean $66.$ How many $6$ s were in the original data set?

$4$

$5$

$6$

$7$

$8$

Solution:

The full set sums to $20\cdot45=900.$ Removing $k$ sixes leaves $20-k$ numbers summing to $900-6k,$ with mean $66,$ so $900-6k=66(20-k)=1320-66k.$ Then $60k=420,$ giving $k=7.$ Thus, the correct answer is D.

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