How do you multiply repeating decimals?

Hint: The above question is based on the concept of repeating decimals. The main approach towards solving the question is to take two repeating decimal numbers and multiplying them in such a way that by rounding off the infinite number to a finite number and then multiply both of them accordingly.

Complete step-by-step solution:
A repeated decimal number is also named as a recurring decimal number whose decimal representation is periodic i.e., the same pattern or sequence of numbers is repeated infinitely. We can also represent a repeating digit by drawing a bar over them. For example, if the number is 0.47474747…. then it can write it in the following way.
$0.\overline {47} $
The repeating pattern can also be in a single number. For example, the number 0.47777……. Here 7 is the repeating number therefore the bar will be on 7.
\[0.4\overline 7 \]
So now we need to see how to multiply two repeating decimals. So will take two repeating decimal numbers i.e.,0.6666666666 and 0.11111111111.
So now to multiply these numbers we first need to round-of both the numbers into finite numbers. The first number can be written as 0.667 and second can be written as 0.111.
On multiplying we get,
\[0.667 \times 0.111 = 0.074\]

Note: An important thing to note is that another method to solve the repeating decimal numbers.We can do this by converting into fractions \[0.\overline 6 = \dfrac{2}{3}\] and \[0.\overline 1 = \dfrac{1}{9}\] and then on multiplying both of these fractions together we get, \[\dfrac{2}{{27}}\] which is then converted into decimal format and can be written as 0.074.