2021 AMC 12A Problem 5

Problem 5 of 25EasierAlgebra Number Theory

When a student multiplied the number

66

by the repeating decimal

1.\overline{ab} = 1.ababab\ldots,

where

a

and

b

are digits, he did not notice the notation and just multiplied

66

by the terminating decimal

1.ab.

Later he found that his answer was

0.5

less than the correct answer. What is the two-digit integer

\overline{ab}?

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Solution

Let

n = \overline{ab}

be the two-digit integer. Then

1.\overline{ab} = 1 + \dfrac{n}{99}

while the terminating value is

1.ab = 1 + \dfrac{n}{100}.

The correct product minus the student’s product is

\begin{aligned} &66\left(\frac{n}{99} - \frac{n}{100}\right) \\ &= 66 \cdot \frac{n}{9900} = \frac{n}{150}. \end{aligned}

Setting

\dfrac{n}{150} = 0.5

gives

n = 75.

Thus, the correct answer is E.