1995 AMC 12 Problem 16

Question

Anita attends a baseball game in Atlanta and estimates that there are

50{,}000

fans in attendance. Bob attends a baseball game in Boston and estimates that there are

60{,}000

fans in attendance. A league official who knows the actual numbers attending the two games notes that:

i. The actual attendance in Atlanta is within

10\%

of Anita’s estimate.

ii. Bob’s estimate is within

10\%

of the actual attendance in Boston.

To the nearest

1{,}000,

the largest possible difference between the numbers attending the two games is

10{,}000

11{,}000

20{,}000

21{,}000

22{,}000

In the two statements, the

10\%

is taken of different quantities

Find both attendance intervals, then maximize the larger endpoint minus the smaller endpoint

Atlanta’s actual attendance

A

satisfies

45{,}000\le A\le55{,}000.

For Boston’s actual attendance

B,

the condition is

|60{,}000-B|\le0.1B,

so

\frac{60{,}000}{1.1}\le B\le\frac{60{,}000}{0.9}.

The largest difference is therefore

\frac{60{,}000}{0.9}-45{,}000 =21{,}666.\overline6,

which rounds to

22{,}000.

Thus the correct answer is E.

Problem 16 in other years