Video Transcript
Which of the following
fractions has a recurring decimal expansion? Option (A) three-eights. Option (B) one-half. Option (C) three-quarters. Option (D) three-fifths. Or is it option (E) eight
thirteenths?
In this question, we are given
a list of five fractions and asked to determine which of these has a recurring
decimal expansion. We can find the decimal
expansion of each of the numbers by using a calculator to convert each into a
decimal and then use this to see which has a recurring decimal expansion.
We do this in steps. First, we want to write the
fraction into the calculator. We can do this by using the
fraction button or the division button. Next, we type the values of the
numerator and denominator into the fraction. Then, we press equals. Finally, we press the S to D
button to convert the answer into a decimal. If we apply this process to
option (A), we see that three-eighths is equal to 0.375. This expansion is finite. We can see that it has no
recurring digits, so this is not the correct answer. It is not necessary. However, we can write this next
to the value to keep track of our calculations.
Now, we clear our calculator
and apply the same process for option (B). Doing this gives us that
one-half is equal to 0.5. Once again, this is not a
recurring decimal expansion.
Clearing our calculator and
then applying this process to option (C) shows us that three-quarters is equal
to 0.75. Once again, this is not a
recurring decimal expansion.
Clearing our calculator and
then applying this process to option (D) shows us that three-fifths is equal to
0.6. Once again, this is not a
recurring decimal expansion. Finally, we can apply this
process to option (E), and we notice something interesting. At first, it may appear that
there is not a repeating expansion. However, the final digit of the
calculator’s display is rounded up.
In fact, the first six digits
of the expansion are repeating. We can represent this using the
dot notation for repeating expansions. We place a dot over the first
and last repeating digits. In either case, we see that
only option (E), eight thirteenths, has a repeating decimal expansion.
There is an interesting fact
that is beyond the scope of this video to prove about which fractions have
finite and which have repeating expansions. It is possible to show that any
fraction in the form 𝑎 over two to the 𝑛th power times five to the 𝑛th power
for any integer 𝑎 and natural numbers 𝑛 and 𝑚 has a finite decimal
expansion. All other fractions have a
recurring decimal expansion. Of course, we do not need to
use this fact to answer questions like this since we can convert using a
calculator.
