AMC 12 A

Algebra Difficulty 3.8 AMC 10/12 Find the answer United States

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?

Pick one

Solution

Let the number of $6$ s in the original data set be $k$ . Numbers in the original data set have sum $20 \cdot 45 = 900$ , and the data set with the $6$ s removed has sum $900 - 6k$ . Thus
$\frac{900 - 6k}{20 - k} = 66.$
Solving this equation for $k$ yields $k = 7$ . For example, the data set could consist of $7$ copies of $6$ and $13$ copies of $66$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.