2010 AMC 12B Problem 6

Question

At the beginning of the school year,

50\%

of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and

50\%

answered “No.” At the end of the school year,

70\%

answered “Yes” and

30\%

answered “No.” Altogether,

x\%

of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of

x?

0

20

40

60

80

Answer: D

Concepts:percentage extremal argument

Difficulty rating: 1410

Small hint:

Assume

100

students; the “Yes” count rises from

50

to

70

Big hint:

At least

20

must switch, and only

30

end with “No,” bounding how many can switch

Solution:

Assume

100

students. The number of “Yes” answers rises from

50

to

70,

so at least

70-50=20

students switched from “No” to “Yes”; thus

x\ge20.

Since only

30

students answer “No” at the end, at least

50-30=20

of the original “Yes” students still answer “Yes,” so at most

80

students switched; thus

x\le80.

Both extremes are achievable, so the difference is

80-20=60.

Thus, the correct answer is D.