Question

wesley does this work to find the decimal equivalent of \\(\frac{23}{99}\\). what does his work tell you about the decimal equivalent?
\\(\

$$\begin{array}{r}0.23 \\\\ 99 \\overline{)23.000} \\\\ -198 \\\\ \\hline 320 \\\\ -297 \\\\ \\hline 230 \\\\ \\end{array}$$

\\)
options:

• he needs to keep going to find more digits.
• it is \\(0. \overline{23}\\). the decimal repeats when he divides into the same number twice.
• it is \\(0. \overline{23}\\). the decimal repeats when he divides into the same number twice.
• it is \\(0. 23\\). the decimal stops when he divides into the same number twice.

Explanation:

Step1: Analyze the division process

In the long division of \(23\div99\), after the first two steps, we get a remainder of \(230\) (from \(320 - 297 = 23\), then we bring down a \(0\) to get \(230\)). Notice that the original dividend was \(23\) (or \(23.000\)), and now we have a remainder that leads to \(230\), which is similar to the initial dividend part (since \(230\) is just \(23\times10\)). This means the division will start repeating the same steps as before, leading to a repeating decimal.

Step2: Evaluate the options

• Option 1: "He needs to keep going..." is incorrect because we can see the repetition starting (remainder leads back to a number related to the original dividend), so we don't need to keep going infinitely to know it repeats.
• Option 2: "It is \(0.\overline{23}\). The decimal repeats when he divides into the same number twice." This is correct. The remainder after two steps brings us back to a situation where we are dividing into a number (230, related to 23) that will cause the digits \(23\) to repeat.
• Option 3: The notation here is unclear and misrepresents the repeating decimal.
• Option 4: "The decimal stops..." is incorrect because \(99\) and \(23\) are coprime (GCD(23,99)=1) and \(99 = 9\times11\), so the decimal should be repeating, not terminating.

Answer:

It is \(0.\overline{23}\). The decimal repeats when he divides into the same number twice. (The option with this text, assuming it's one of the radio - button options, e.g., if it's the second option in the layout, we identify it as that option)