2010 AMC 10B Problem 12

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 bounding to limit cases

Difficulty rating: 1420

Small hint:

Think of

100

students

Big hint:

Minimize or maximize the number who switch answers

Solution:

To minimize

x,

we want to have as many kids as possible maintain their answer.

We then need at least

70 - 50 = 20

percent of the students to change their answer.

To maximize

x,

we can have everybody that answered no change their answer, but some who answered yes must stay yes.

We need

70 - 50 = 20

percent of the people who said yes to stay yes, which means only

50 - 20 = 30

percent can switch.

This makes the maximum percent of students that can switch

50 + 30 = 80

percent. The difference is then

80 - 20 = 60.

Thus, D is the correct answer.