Question

for each number, determine if it can be written as a fraction, then state the reason.
(a) 0.87
can this be written as a fraction?
○yes ○no
give the reason.
○ this is a terminating decimal.
○ this is a repeating decimal.
○ this is a decimal that neither terminates nor repeats.
(b) $0.\overline{3}$
can this be written as a fraction?
○yes ○no
give the reason.
○ this is a terminating decimal.
○ this is a repeating decimal.
○ this is a decimal that neither terminates nor repeats.

Explanation:

Step1: Analyze 0.87 (terminating decimal)

A terminating decimal can be written as a fraction with a denominator that is a power of 10.
$0.87 = \frac{87}{100}$

Step2: Classify 0.87's decimal type

0.87 ends after two decimal places, so it is a terminating decimal, which is rational (can be written as a fraction).

Step3: Analyze $0.\overline{3}$ (repeating decimal)

A repeating decimal is a rational number, so it can be converted to a fraction.
Let $x = 0.\overline{3} = 0.333...$
Multiply both sides by 10: $10x = 3.333...$
Subtract the original equation: $10x - x = 3.333... - 0.333...$
$9x = 3$
$x = \frac{3}{9} = \frac{1}{3}$

Step4: Classify $0.\overline{3}$'s decimal type

$0.\overline{3}$ has a repeating digit, so it is a repeating decimal, which is rational.

Answer:

(a)
Can this be written as a fraction? Yes
Give the reason. This is a terminating decimal.

(b)
Can this be written as a fraction? Yes
Give the reason. This is a repeating decimal.