Lesson Video: Converting between Fractions and Recurring Decimals Mathematics • First Year of Preparatory School

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.