2008 AIME I Problems/Problem 1
Problem
Of the students attending a school party, of the students are girls, and
of the students like to dance. After these students are joined by
more boy students, all of whom like to dance, the party is now
girls. How many students now at the party like to dance?
Solution
Solution 1
Say that there were girls and
boys at the party originally.
like to dance. Then, there are
girls and
boys, and
like to dance.
Thus, , solving gives
. Thus, the number of people that like to dance is
.
Solution 2
Let the number of girls be . Let the number of total people originally be
.
We know that from the problem.
We also know that from the problem.
We now have a system and we can solve.
The first equation becomes:
.
The second equation becomes:
Now we can sub in by multiplying the first equation by
. We can plug this into our second equation.
We know that there were originally people. Of those,
like to dance.
We also know that with these people, boys joined, all of whom like to dance. We just simply need to add
to get
See also
2008 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |