Final answer:
Decimals like 0.1, 0.13, and 0.157 are identical to their fractions form. We can conjecture that repeating decimals can be expressed as a fraction with the numerator being the repeating part and the denominator being 9, 99, and so on depending on the number of repeating digits.
Step-by-step explanation:
The task here is to Compare 0.1 and 0.1, 0.13 and 0.13, and 0.157 and 0.157 when written as fractions. It is important to note that repeating decimals are always expressed as fractions. To show this, we convert each of these decimals to fractions: 0.1 is the same as 1/10, 0.13 is the same as 13/100, and 0.157 is the same as 157/1000. Thus, the fraction forms of these decimals are identical to the decimal forms. From this observation, we can make the conjecture that expressing repeating decimals like these as fractions will produce a fraction whose numerator is the repeating part and the denominator is the number 9 (for one repeating digit), 99 (for two repeating digits), and so on, depending on the number of digits repeating. This rule applies because the fractional representation of a repeating decimal provides an exact value of the decimal, which cannot be achieved by the decimal itself.
Learn more about Fractions and Decimals