Question

Is 2.74 74 repeated
a rational number?

Answer 1

Yes, the number \( 2.74 \overline{74} \) (where \( 74 \) is repeated) is a rational number. A rational number is defined as a number that can be expressed as the quotient of two integers, or as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \) is not zero.

The repeating decimal \( 2.74 \overline{74} \) can be converted to a fraction. To do this, you can set \( x \) equal to the number:

\[
x = 2.74747474\ldots
\]

Next, to eliminate the repeating part, you can multiply both sides by \( 100 \) (because the repeating part has two digits):

\[
100x = 274.74747474\ldots
\]

Now, you have two equations:

1. \( x = 2.74747474\ldots \)
2. \( 100x = 274.74747474\ldots \)

Next, you subtract the first equation from the second:

\[
100x - x = 274.74747474\ldots - 2.74747474\ldots
\]

This simplifies to:

\[
99x = 272
\]

Now, solving for \( x \):

\[
x = \frac{272}{99}
\]

Since \( 272 \) and \( 99 \) are both integers, \( x \) is indeed a rational number. Therefore, \( 2.74 \overline{74} \) is a rational number.