AMC 10 · 2017 · #10
Grade 7 rate-ratioPick an answer.
Tool #2 (Make a Systematic List): every student falls into one of four boxes — like/say-like, like/say-dislike, dislike/say-dislike, dislike/say-like — so I list all four and count each. Tool #9 (Solve an Easier Related Problem): instead of juggling percentages, I pretend there are exactly 100 students so each percent becomes a plain count. Tool #16 (Change Focus): the question only cares about students who SAY they dislike dancing, so I narrow my attention to just that group at the end.
Use 100 students
Taking 100 students turns percentages into people.
With a total of 100, every percent turns straight into a head count.
7.RP.A.3Solve An Easier Related ProblemSplit the likers
The likers split in two by what they say.
The two say-groups inside the likers must add back to all 60 likers.
The two say-groups inside the likers must add back to all the likers.
▸ Why?
Every liker says one thing or the other, so the group is exactly its two parts put together.
▸ Why?
With a total of a hundred students, each percent turns straight into a head count.
Split the dislikers
The dislikers split the same way.
The two say-groups inside the dislikers must add back to all 40 dislikers.
7.RP.A.3Make A Systematic ListGather the 'say dislike' group
Only the say-dislike students are gathered, from both groups.
Only the 'say dislike' students matter, so I pull together exactly those two boxes.
6.RP.A.3Change Focus Count The ComplementTake the fraction
Inside that group the fraction is 25 percent.
The fraction is just the wanted box over the whole 'say dislike' group.
6.RP.A.3Make A Systematic ListPretend there are 100 students, sort them into four boxes by what they feel and what they say, then take 'actually like it' over 'everyone who says dislike' — that is 12/48=25%, choice (D).
- Use 100 students
- Split the likers
- Split the dislikers
- Gather the 'say dislike' group
- Take the fraction