Problem:
When a student multiplied the number 66 by the repeating decimal,
1. ababβ=1.ab
where a and b are digits, he did not notice the notation and just multiplied 66 times 1.ab. Later he found that his answer is 0.5 less than the correct answer. What is the 2 - digit integer ab ?
Answer Choices:
A. 15
B. 30
C. 45
D. 60
E. 75
Solution:
The given condition is
66β
1ββ
aβbβ=66β
1ββ
aβaβbββ0.5,
which is equivalent to
0.5=66β
1ββ
aβbβbββ66β
1ββ
aβbβ=66(1ββ
aβaβbββ1ββ
aβbβ)=66β
0ββ
0β0βaβaβbβ.
Because
0ββ
aΛβbββ=99aββ
this is equivalent to
21β=66β
9aβbβ
from which aβbβ=75β.
The problems on this page are the property of the MAA's American Mathematics Competitions