### Video Transcript

Which of the following is a
fraction that can be expressed as a repeating decimal with two alternating
digits? (A) Two-thirds, (B) four
seventeenths, (C) sixteen nineteenths, (D) one thirty-third, (E) one
thirty-fourth.

In this question, we need to
identify which of the five given fractions can be written as a repeating decimal
with two different alternating digits. In order to do this, we can use the
fraction button on the calculator or simply divide the numerator by the
denominator. In this case, we will divide the
numerators by the denominators for each fraction as shown. Note that after pressing the equals
button, we need to press the SD button to convert our answer from standard form to
decimal form.

Beginning with option (A), we
calculate that two-thirds is equal to 0.6666 and so on, which is equal to 0.6
recurring. We can place a dot or bar above any
repeating digits to denote this. Since this has only one repeating
digit, option (A) is not the correct answer.

Four divided by 17 is equal to
0.2352941 and so on. As this clearly does not have two
repeating alternating digits, option (B) is also not the correct answer.

Moving on to option (C), 16 divided
by 19 is equal to 0.8421052 and so on. Sixteen nineteenths is therefore
not the correct answer.

One divided by 33 is equal to
0.030303 and so on. This can be written as 0.03
recurring with dots above the two repeating digits. Alternatively, we can draw a bar
above the two repeating digits. Since one thirty-third written as a
repeating decimal does have two alternating digits, it appears that this is the
correct answer.

Let’s quickly check option (E)
first though. One divided by 34 is equal to
0.0294117 and so on. So this fraction cannot be
expressed as a repeating decimal with two different alternating digits.

We can therefore conclude that the
correct answer is option (D). One thirty-third can be expressed
as the decimal 0.03 recurring.