How many students now at the party like to dance?

Question

You're hosting a party. 75% of the students are girls. 30% of the students like to dance. Some more friends arrive: a group of 12 boys. All 12 boys like to dance! Now, the party is 70% girls.

How many students now at the party like to dance?

Answer 1

This problem isn't worded very clearly. The only way I can make it work out without ending up with fractional people (!!!) is to assume that the 30% that like to dance is constant. That is, it doesn't refer just to the party goers before the arrival of the 12 extra boys.

 

So..... if at the beginning, 75% are girls, then 25% are boys. After the 12 extra boys arrive, the new number of boys makes up 30% of the total, because the girls' percentage has fallen to 70%.

We can write an equation that expresses this, where T = total guests before the extra 12 arrive.

 

(%Boy)(total before) + (12 more) = (Boy % after)*(new total)

.25(T) + 12 = .30(T + 12)

.25(T) + 12 = .30T + 3.6

             8.4 =.05(T)

             168 = T

             180 = T +12 boys

168 is the number of guests before the extra 12 boys arrive. After they arrive, there are 180 guests. IF 70% are girls, there are 126 girls. 30% Boys = 54 boys.

 

If the %-age of dancers is still 30% then there are 54 dancers. (.30) * 180 = 54 (but we don't know how many of these are boys vs. girls)

 

If the 30% applies to the total before the new arrivals: .30(168) = 50.4 After the 12 more arrive; 50.4 + 12 = 62.4 dancers - but this is now more than 30%. (Don’t know if I’d like to be that 0.4 person!)

Answer 2

Cashlin, these kind of problems involve converting the ratio or percentage into a number of people.  Check your numbers though, my math is giving me fractional people, which is always suspect in these kinds of problems).  There are often faster ways to do these kinds of problems for standardized tests, but this method will always work even if you miss a short-cut.

 

We start with defining some variables.  I recommend starting with actual numbers of people with my variable definitions.  So

"N" is the number of students originally at the dance.

G=# of girls originally, so B=# of boys originally=N-G

D=# who like to dance originally, N-D=Number who didn't like to dance originally.

 

Now use the given information to create equations in terms of these variables

Equation (1)

75% of the students are girls.

.75=G/N or .75N=G

Equation (2)

30% of the students like to dance

.3=D/N or .3N=D

 

Some more friends arrive: a group of 12 boys.

Our N has changed, the new number of people at the party is N+12

Number of Girls is still G.  (Boys is now B+12, but we won't need this.)

 

Equation (3)

All 12 boys like to dance!

The number of people who now like to dance = D+12 (this will be our answer).

 

Equation (4)

Now, the party is 70% girls.

.7 = G/(N+12)

G= .7(N+12)

 

We can now solve for D

Equation (1) and (4) are both G, so we can solve for N

.75 N = .7(N+12)

75 N = 70(N+12)

75 N = 70N + 12*70

5 N = 12*70

N = 12*14 = 168

 

Now plug this into equation 2

.3(168)=50.4 Students who originally liked to dance

 

But don't forget the 12 new students (D+12) or 50.4 + 12 = 62.4 students who like to dance.