Question

the following division was performed to express \\(\frac{32}{15}\\) as a decimal.
\\\

$$\begin{array}{r}2.13\\\\15\\overline{)32.00}\\\\-30\\\\\\hline20\\\\-15\\\\\\hline50\\\\-45\\\\\\hline5\\end{array}$$

\\
(a) continue the division shown above to fill in the next three digits in the decimal form of \\(\frac{32}{15}\\).
\\(2.13334\\)
(b) when \\(\frac{32}{15}\\) is written as a decimal, which digit or group of digits repeat?
\\(\bigcirc\\) 1 \\(\bigcirc\\) 3 \\(\bigcirc\\) 13 \\(\bigcirc\\) 213 \\(\bigcirc\\) none of these

Explanation:

Step1: Bring down 0 to remainder 5

Current remainder: $5 \implies 50$

Step2: Divide 50 by 15

$\frac{50}{15}=3$, remainder $50-45=5$

Step3: Repeat Step1 and Step2

Remainder 5 becomes 50, divide by 15: quotient 3, remainder 5. Repeat again: quotient 3, remainder 5.

Step4: Identify repeating digit

The remainder cycles to 5 indefinitely, so the quotient digit repeats.

Answer:

(a) The next three digits are 3, 3, 3, so the decimal is 2.13333
(b) B. 3