Question

for each number, determine if it can be written as a fraction. then state the reason.
(a) (0.overline{65})
can this be written as a fraction?
(\bigcirc)yes (\bigcirc)no
give the reason.
(\bigcirc) this is a terminating decimal.
(\bigcirc) this is a repeating decimal.
(\bigcirc) this is a decimal that neither terminates nor repeats.
(b) (0.35555ldots)
can this be written as a fraction?
(\bigcirc)yes (\bigcirc)no

Explanation:

Part (a)

Step 1: Identify Decimal Type

The number \(0.\overline{65}\) has a repeating bar over "65", meaning the digits "65" repeat indefinitely. Repeating decimals are rational numbers, and all rational numbers can be written as fractions.

Step 2: Choose Reason

Since \(0.\overline{65}\) is a repeating decimal (the "65" repeats), it is a rational number and can be expressed as a fraction. A terminating decimal has a finite number of digits after the decimal point, and a non - repeating, non - terminating decimal is irrational (cannot be a fraction). So the reason is that it is a repeating decimal.

Step 1: Identify Decimal Type

The number \(0.35555\ldots\) has the digit "5" repeating indefinitely. Repeating decimals are rational numbers, and all rational numbers can be written as fractions.

Step 2: Conclusion on Fraction Representation

Since \(0.35555\ldots\) is a repeating decimal (the "5" repeats), it is a rational number and thus can be written as a fraction.

Answer:

Can this be written as a fraction? \(\boldsymbol{\text{Yes}}\)

Give the reason. \(\boldsymbol{\text{This is a repeating decimal.}}\)

Part (b)