Problem:
When a student multiplied the number 66 by the repeating decimal,
1β.aβ bβ aβ bββ¦=1β.aβ bβΛβ,
where a and b are digits, he did not notice the notation and just multiplied 66 times 1β.aβ bβ. Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit number aβ bβ?
Answer Choices:
A. 15
B. 30
C. 45
D. 60
E. 75
Solution:
The given condition is
66β
1ββ
abβ=66β
1ββ
abββ0.5.
which is equivalent to
0.5=66β
1ββ
aabββ66β
1ββ
abβ=66(1ββ
abββ1ββ
abβ)=66β
0ββ
00abΛβ.
Because
0ββ
aΛβbβ=9aβbβ.
this is equivalent to
21β=66β
990aβbβ.
from which abβ=(E)75β.
The problems on this page are the property of the MAA's American Mathematics Competitions