1958 AMC 12 Problem 14

Question

At a dance party a group of boys and girls exchange dances as follows: one boy dances with

5

girls, a second boy dances with

6

girls, and so on, the last boy dancing with all the girls. If

b

represents the number of boys and

g

the number of girls, then:

b=g

b=\dfrac g5

b=g-4

b=g-5

It is impossible to determine a relation between

b

and

g

without knowing

b+g

The numbers of partners form the sequence

5,6,7,\ldots,g

There are

b

terms, so express the last term using the arithmetic-sequence formula

The last of the

b

terms in the sequence

5,6,7,\ldots

is

5+(b-1)=b+4.

This last number equals the total number

g

of girls. Hence

g=b+4,

or

b=g-4.

Thus, the correct answer is C.

Problem 14 in other years