2024 AMC 12A Problem 5

Attempt Problem 5 of the 2024 AMC 12A below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

5.

A data set containing 2020 numbers, some of which are 6,6, has mean 45.45. When all the 66s are removed, the data set has mean 66.66. How many 66s were in the original data set?
44
55
66
77
88
Answer: D
Concepts:meanlinear equation
Small hint:
The 2020 numbers sum to 20⋅45=90020\cdot45=900
Big hint:
If there are kk sixes, the remaining 20−k20-k numbers sum to 900−6k=66(20−k)900-6k=66(20-k)
Solution:
The full set sums to 20⋅45=900.20\cdot45=900. Removing kk sixes leaves 20−k20-k numbers summing to 900−6k,900-6k, with mean 66,66, so 900−6k=66(20−k)900-6k=66(20-k) =1320−66k.=1320-66k. Then 60k=420,60k=420, giving k=7.k=7. Thus, the correct answer is D.
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