Question

Write the decimal expansion of the following rational number and state which type of decimal expansion has?

`12 5/6`

Answer 1

Given: `12 5/6`

Step-wise calculation:

1. Convert the mixed fraction to an improper fraction:

`12 5/6`

= `(12 xx 6 + 5)/6`

= `(72 + 5)/6`

= `77/6`

2. Perform the division 77 ÷ 6 to get the decimal expansion:

6 goes into 77 exactly 12 times since 6 × 12 = 72, remainder 5.

Now divide the remainder 5 by 6:

5 ÷ 6 = 0.8333...

So decimal expansion is 12 + 0.8333... = 12.8333...

3. The decimal part 0.8333... is repeating the digit 3 indefinitely, so the full decimal expansion is `12.8bar(3)`

The decimal expansion of `12 5/6` is `12.8bar(3)`, which is a non-terminating repeating decimal expansion because the denominator 6 = 2 × 3 contains the prime factor 3 other than 2 and 5, causing repetition in the decimal part.