Question

What is the rational number equivalent to 3 point 12 with a bar over 12?

Group of answer choices

3 and 4 over 33

3 and 8 over 33

3 and 10 over 39

3 and 5 over 39

Answer 1

To convert the repeating decimal \(3.12\overline{12}\) into a rational number, we can break it down as follows:

Let \(x = 3.121212...\)

Now, to eliminate the repeating part, we can multiply \(x\) by \(100\) (which shifts the decimal two places to the right):

\[
100x = 312.121212...
\]

Next, we can set up an equation by subtracting the original \(x\) from this new equation:

\[
100x - x = 312.121212... - 3.121212...
\]
\[
99x = 312 - 3
\]
\[
99x = 309
\]

Now, we solve for \(x\):

\[
x = \frac{309}{99}
\]

To simplify \(\frac{309}{99}\), we find the greatest common divisor (GCD) of 309 and 99. Both numbers can be divided by 3:

\[
309 \div 3 = 103
\]
\[
99 \div 3 = 33
\]

Thus, the simplified form is:

\[
x = \frac{103}{33}
\]

Now, we need to rewrite this in the form of a mixed number. We divide \(103\) by \(33\):

\[
103 \div 33 = 3 \quad \text{(with a remainder of } 4\text{)}
\]

Now we write it as:

\[
\frac{103}{33} = 3 + \frac{4}{33}
\]

So, the rational number equivalent to \(3.12\overline{12}\) is:

\[
3 \frac{4}{33}
\]

However, looking at your provided answer choices, the equivalent that fits closest is:

**3 and 4 over 33**

This matches our calculation.