Question

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 $\overline{ab}?$

$15$

$30$

$45$

$60$

$75$

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 $66\left(\frac{n}{99} - \frac{n}{100}\right) = 66 \cdot \frac{n}{9900} = \frac{n}{150}.$

Setting $\dfrac{n}{150} = 0.5$ gives $n = 75.$

Thus, the correct answer is E.

2021 AMC 12A Spring Problem 5

Solution:

Problem 5 in Other Years