Question

using division to write fractions as decimals - item 52309
perry wants to find the decimal equivalent of \\(\frac{19}{3}\\), so he divides. study perrys work shown here, and then answer the questions below.
\\(\

$$\begin{array}{r}6.3\\\\3\\overline{)19.0000}\\\\ -18\\\\\\hline10\\\\ -9\\\\\\hline1\\end{array}$$

\\)

continue perrys work to find the next three digits in the decimal equivalent of \\(\frac{19}{3}\\).
6.3\square\square\square

what do you notice about this work and the decimal equivalent of \\(\frac{19}{3}\\)?
each of the digits after the decimal point will be \\(\square\\) because this number is \\(\square\\)
the difference between 10 and the product of these digits and 3 is always \\(\square\\)

Explanation:

Step1: Analyze the division process

We have the remainder 1 from the previous step. To continue the division, we add a zero to make it 10 (since we are dealing with decimals, we can add zeros after the decimal point). Now, we divide 10 by 3.
$10 \div 3 = 3$ with a remainder of $10 - 3\times3 = 1$.

Step2: Find the next digit

The quotient digit here is 3. So the next digit after 6.3 is 3.

Step3: Repeat the process for the second next digit

Again, we have a remainder of 1, add a zero to make it 10, divide by 3, we get 3 as the quotient digit (remainder 1 again). So the second next digit is 3.

Step4: Repeat for the third next digit

Similarly, remainder 1, add zero to get 10, divide by 3, quotient digit 3 (remainder 1). So the third next digit is 3.

Answer:

The next three digits in the decimal equivalent of $\frac{19}{3}$ (wait, actually it's $\frac{19}{3}$? Wait, the original fraction Perry is working on—wait, the division is 19 divided by 3? Wait, the fraction is $\frac{19}{3}$? Wait, the problem says "the decimal equivalent of $\frac{19}{3}$"? Wait, no, the first part says "Perry wants to find the decimal equivalent of $\frac{19}{3}$". So continuing the division:

After 6.3, the next three digits are 3, 3, 3.

For the second part, each of the digits after the decimal point will be 3 because this number is a repeating decimal (since the remainder keeps repeating as 1, leading to the same quotient digit 3 each time), and the difference between 10 and the product of these digits (3) and 3 is always 1 (since $10 - 3\times3 = 1$).

But for the first sub - question (finding next three digits), the answer is 333.