Problem:
When a student multiplied the number 66 by the repeating decimal,
1​.a​ b​ a​ b​…=1​.a​ b​ˉ​,
where a and b are digits, he did not notice the notation and just multiplied 66 times 1​.a​ b​. Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit number a​ b​?
Answer Choices:
A. 15
B. 30
C. 45
D. 60
E. 75
Solution:
The given condition is
66⋅1​⋅ab​=66⋅1​⋅ab​−0.5.
which is equivalent to
0.5=66⋅1​⋅aab​−66⋅1​⋅ab​=66(1​⋅ab​−1​⋅ab​)=66⋅0​⋅00abˉ​.
Because
0​⋅aˉ​b​=9a​b​.
this is equivalent to
21​=66⋅990a​b​.
from which ab​=(E)75​.
The problems on this page are the property of the MAA's American Mathematics Competitions