Question

Change the decimal to a fraction: 0.1213(1 point)
Responses

This decimal is Irrational and cannot be written as a fraction
This decimal is Irrational and cannot be written as a fraction

121310
12 and 13 tenths

12131,000
1 the fraction with numerator 213 and denominator 1 comma 000

1213999
1213 over 999

121310,000

Answer 1

To convert the repeating decimal \( 0.1213\overline{3} \) into a fraction, let's denote it as \( x \):

\[
x = 0.1213\overline{3}
\]

Next, we can express the repeating part separately. First, it's useful to separate \( 0.1213 \) from the repeating decimal \( 0.0003\overline{3} \):

1. **Convert the non-repeating part:**
\[
0.1213 = \frac{1213}{10000}
\]

2. **Convert the repeating part \( 0.0003\overline{3} \):**
Let \( y = 0.0003\overline{3} \):
- Multiply \( y \) by 10 to move the decimal point one place right:
\[
10y = 0.003333\ldots
\]
- Now subtract \( y \) from \( 10y \):
\[
10y - y = 0.003333\ldots - 0.0003333\ldots \implies 9y = 0.003
\]
- Thus,
\[
y = \frac{0.003}{9} = \frac{3}{9000} = \frac{1}{3000}
\]

3. **Combine both parts:**
\[
x = 0.1213 + 0.0003\overline{3} = \frac{1213}{10000} + \frac{1}{3000}
\]

To add these fractions, we need a common denominator. The least common multiple of 10000 and 3000 is 30000.

- Convert \( \frac{1213}{10000} \):
\[
\frac{1213 \times 3}{10000 \times 3} = \frac{3639}{30000}
\]

- Convert \( \frac{1}{3000} \):
\[
\frac{1 \times 10}{3000 \times 10} = \frac{10}{30000}
\]

Now, we sum them up:
\[
x = \frac{3639}{30000} + \frac{10}{30000} = \frac{3649}{30000}
\]

Thus, \( 0.1213\overline{3} \) as a fraction is \( \frac{3649}{30000} \).

From the given options, **none of them represent the correct fraction for \( 0.1213\overline{3} \)**. The correct conversion yields \( \frac{3649}{30000} \).

Answer 2

Which answer is correct