Question

0.14189189189.... can be expressed as a rational number

• A. $\frac{7}{3700}$
• B. $\frac{7}{50}$
• C. $\frac{525}{111}$
• D. $\frac{21}{148}$

Answer 1

Answer: (D)

$\frac{21}{148}$

Answer 2

Let $x = 0.14\overline{189}$. To convert the repeating decimal to a fraction, we can follow these steps:

1. Multiply $x$ by 100 to move the decimal point two places to the right: $100x = 14.\overline{189}$
2. Multiply $100x$ by 1000 to move the decimal point three more places to the right (covering one repeating block): $100000x = 14189.\overline{189}$
3. Subtract the first equation from the second to eliminate the repeating part: $100000x - 100x = 14189.\overline{189} - 14.\overline{189}$ which simplifies to $99900x = 14175$
4. Solve for $x$: $x = \frac{14175}{99900}$
5. Simplify the fraction by dividing both numerator and denominator by their greatest common divisor.

$\frac{14175}{99900} = \frac{2835}{19980} = \frac{567}{3996} = \frac{63}{444} = \frac{21}{148}$

Therefore, $0.14189189189... = \frac{21}{148}$