Video Transcript
In this lesson, we will learn how
to convert between fractions and recurring decimals using a calculator. We will see how to do this for
mixed fractions and finite decimal expansions. This will allow us to easily
compare the sizes of mixed and nonmixed fractions, finite decimals, and recurring
decimals.
Before we begin with recurring
decimal expansions, it is worth recalling that there are many different ways of
representing the same number. For example, we can recall that we
can rewrite the improper fraction nine-eighths as the mixed number one and
one-eighth, or we can rewrite this as a decimal 1.125. All of these numbers are
equivalent. If we multiply any of these numbers
by eight, we will get nine.
We can calculate each of these
conversions by hand. However, it is much easier to use a
calculator. For instance, we can type
nine-eighths in the calculator using the fraction button and then pressing equals or
by using the divide button. We then press the S to D button to
convert the answer into a decimal.
We can follow a very similar
process for the conversion of the other forms of this number. For example, we can use the mixed
number button on the calculator to input one and one-eighth and then use the
conversion button to convert between a fraction and a decimal. If we clear the calculator and try
this process for other fractions, we can notice something interesting that happens
for some fractions. For instance, if we input one-third
into the calculator and press the conversion button to find this decimal expansion,
we can see that it contains a repeating expansion of three. This value of three repeats
infinitely. So we can say that one-third is
equal to 0.333, and this expansion repeats indefinitely. We can write this as an ellipsis at
the end of the decimal.
However, the ellipses at the end of
the expansion usually refers to any expansion that continues. It does not have to repeat. Instead, we usually use a dot or a
bar over the repeating digits to show that they are the digits that repeat in the
expansion. It is important to note that we put
a dot or a bar only over the digits that repeat. For instance, consider the
following decimal number. We can see that in the expansion of
this number, only the digits of two and three are repeating. After identifying the repeating
digits, we can put a dot over the first and last repeating digit to show the
repeating digits. This gives us the following.
Finally, we only put the dots over
the first and last digit that is repeating. For instance, in this decimal, we
note that three of the digits — seven, eight, and nine — are repeating. In this case, we only put a dot
over the seven and nine as shown. Another useful fact worth noting is
that the quotient of any two integers, where we do not divide by zero, will have a
finite or repeating decimal expansion.
This final example also highlights
a key difference between the dot and bar notations. When using the bar notation, we
draw the bar over all of the repeating digits. However, with the dot notation, we
only draw the dot over the first and last repeating digit.
Let’s now see an example of using a
calculator to identify which of a list of fractions has a repeating decimal
expansion.
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.
In our next example, we will
convert a fraction into a decimal with a repeating expansion by using a
calculator.
Write eight over three as a
recurring decimal.
In this question, we are given
a fraction and asked to write this as a recurring decimal. We can do this by using a
calculator. First, we need to input eight
over three into the calculator. We can do this by using the
fraction button and inputting the numerator and denominator. Then, we press the equals
button. Finally, we need to click the
button on the calculator to convert into a decimal. This is usually denoted with an
S and a D. We get the following output on
our calculator.
We can now note that this is a
repeating expansion of the digit six, where the final digit on the calculator’s
display is rounded up. We can then write this as a
recurring decimal expansion by putting a dot or a bar over the repeating
digit. We obtain that eight over three
is equal to 2.6 recurring.
In our next example, we will see
how to apply this process to a mixed number.
Write 11 and five-sixths as a
recurring decimal.
In this question, we are given
a mixed fraction and asked to find its recurring decimal expansion. We can do this by using a
calculator. We need to start by inputting
the number into a calculator. We can do this by using the
addition and division button or by using the mixed number button. Once we have entered the mixed
number, we press the equals button. Now, we need to click the
button to convert the output of the calculator to a decimal. This is usually the S to D
button. We get an output of the
calculator like the following.
We can see that in the decimal
expansion, the digit three is the only one that is repeating. So, when writing this in
recurring notation, we only need to put a dot over this digit. Hence, we have that 11 and
five-sixths is equal to 11.83 recurring.
In our next example, we will covert
a fraction into its equivalent recurring decimal expansion to compare its size to a
given decimal.
Benjamin was told that he could
have one-third or 0.34 of the total jam inside a jar. Convert one-third to a decimal,
and determine which option would give Benjamin more jam.
In this question, we are given
two possible proportions for the amount of jam that Benjamin can have. We want to compare the sizes of
these two proportions. We could do this by rewriting
both numbers as fractions with the same denominator. However, we are told to do this
by first converting one-third into a decimal. We do this by typing one-third
into the calculator by using the fraction button, then pressing equals. Finally, we press the S to D
button to convert the output to a decimal to obtain the shown output.
We can write this as 0.3
recurring with the dot over the three since the three digit is repeating. However, it is not necessary to
do this to compare the sizes of the portions. Instead, we can note that both
decimals have the same integer part and the same first decimal digit. However, the second digit of
one-third is lower than that of 0.34. So, it is the larger
portion. Hence, one-third is equal to
0.3 recurring, and 0.34 is the option to give Benjamin more jam.
Let’s now see an example of
converting a decimal with a recurring expansion into a fraction.
Use a calculator to convert 3.1
recurring into a fraction.
In this question, we are asked
to convert a recurring decimal into a fraction by using a calculator. To do this, we first recall
that the dot above the digit one tells us that this digit repeats indefinitely
in the decimal expansion of this number. We can keep adding more and
more digits to get closer to the actual value of this number. We can use this idea and a
calculator to find the fraction that is equivalent to this expansion. We continue to add more and
more of the repeated digits of the expansion into the calculator and then press
equals.
If we input enough repeating
decimals, we will get a fractional output of 28 over nine. It is worth noting if we do not
input enough digits, we will get a decimal answer. So we should make sure that our
answer is a fraction. We can check that this fraction
is the correct answer by converting 28 over nine into a recurring decimal to
show that it is equal to 3.1 recurring. Hence, we showed that 3.1
recurring is equal to 28 over nine.
In our final example, we will solve
a real-world problem by converting fractions into decimals.
Emma can dig 134 over 99 holes
in an hour, and Charlotte can dig one and five over 18 holes in an hour. By converting both fractions
into decimals, determine who is faster at digging holes.
In this question, we are given
the number of holes that two different people can dig in one hour. We know that the larger of
these values tells us who is faster at digging holes, so we need to determine
which of these numbers are larger. We are told to do this by
converting each number into a decimal. Let’s start with the first
fraction. We input the fraction into the
calculator, press equals, and then press the conversion button to obtain the
shown result. We can see that in the decimal
expansion, the digits three and five are repeating, where the last digit in the
calculator’s display is rounded up. Therefore, Emma can dig 1.35
repeating holes in an hour.
We can now clear the
calculator’s memory and follow a similar process for the second number. We use the mixed number button,
input the number, press equals, and then convert the output into a decimal. We see that the digit seven is
repeating, so we have shown that Charlotte can dig 1.27 recurring holes in an
hour. We then see that the integer
part of the two numbers are the same. However, the first decimal
digit in the number of holes Emma can dig is larger. This means that 1.35 recurring
is larger than 1.27 recurring. So we can say that Emma is
faster at digging holes.
Let’s now go over the key points
from this lesson. We saw that we can use dots or
sometimes a horizontal bar over the digits of a decimal to show that they repeat
indefinitely. For instance, we can write the
decimal 0.123, where the two and three digits repeat indefinitely in the following
different ways. We also saw that the quotient of
any two integers, where the denominator is nonzero, will have either a finite or
repeating decimal expansion.
We also saw that we can convert
between a fraction and a decimal by using the S to D button on a calculator. This is useful for finding the
repeating or finite decimal expansion of a fraction. We can also convert a recurring
decimal into a fraction using a calculator by inputting enough of the repeating
digits so that the calculator outputs a fraction. Finally, we saw that we can compare
the size of fractions by converting them into decimals and comparing the size of
their expansions.