Lesson Explainer: Converting between Fractions and Recurring Decimals | Nagwa Lesson Explainer: Converting between Fractions and Recurring Decimals | Nagwa

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

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In this explainer, we will learn how to convert fractions to recurring decimals and recurring decimals to fractions using a calculator.

There are many different ways of representing fractions. For example, we can write them as proper or improper fractions, mixed numbers, or decimals. It can be quite difficult to convert a fraction into a decimal and vice versa, so these calculations are often done on a calculator.

For example, we can convert into a decimal by typing into a calculator, either by typing or by using the fraction button, , and pressing the equals button. We can then press the button to convert from the standard output form to a decimal, . The output of the calculator will then look as follows.

This tells us that the decimal form of is 0.125. We can follow this process for any fraction. However, we will notice something interesting for certain fractions.

Let’s clear the calculator and then apply this process to . We input into the calculator, press the equals button, and then press the conversion button, . This gives us the following.

We see that the decimal expansion of contains a repeating expansion of the digit 3. We can write this using the repeating expansion notation; we write a dot over the repeating digit: .

In fact, all fractions (which are the quotients of integers) have either a finite decimal expansion or a repeating decimal expansion. Thus, to represent these repeating expansions, we need to formally introduce the following notation.

Definition: Recurring Decimal Digits

We use dots to represent all of the recurring decimal digits of a number. Only the digits that repeat will have a dot over them.

For example,

It is also worth noting that the repeating digits are sometimes represented by lines over the first and last repeating digit. So,

In our first example, we will determine which of five given fractions has a recurring decimal expansion.

Example 1: Identifying Fractions That Can Be Expressed as Recurring Decimals

Which of the following fractions has a recurring decimal expansion?

To answer this question, we need to convert all of the given options into decimals. We also recall that recurring digits in a decimal expansion are represented by a dot over the repeating digits.

To convert a fraction into a decimal using a calculator, we first need to input the fraction into the calculator. We can do this by either using the button or the button, which allows us to separately input the numerator and denominator. Once we have input the fraction, we press the equals button and then the button to convert from the standard output form to a decimal, .

We can follow this process for each of the options separately. First, we input into the calculator, press the equals button, and then press the button. The calculator will then display the following.

Since the output of 0.375 does not have any repeating digits, this fraction does not need recurring notation in its expansion. We can now clear the calculator screen, either by using the clear button or by pressing the on button, and repeat this process for the remaining options.

We have the following.

So, and it has a finite decimal expansion.

Next, we get the following.

Thus, and it has a finite decimal expansion.

Then, we get the following.

We can note that the section of digits 615384 is repeating:

The calculator rounds up the last digit, which it can show on the screen in this case. We represent this using repeating expansion notation; we put dots over the beginning and end of the repeating digits:

We can also check the final option in the same way.

We see that and it has a finite decimal expansion.

Hence, only option D, , has a recurring decimal expansion.

An interesting point about the previous example is that there is a method of determining whether a fraction will have a recurring expansion without a calculator. Although it is beyond the scope of the explainer to prove this result, it is true that only fractions with the simplified form for natural numbers and will have a finite decimal expansion. In other words, the fractions with prime factors that are not 2 or 5 in the denominator will have a recurring expansion.

Of course, this is not necessary to answer the question since we can just evaluate the fractions using a calculator, but it is an interesting fact.

Let’s now see some examples of converting fractions into decimals with repeating decimal digits.

Example 2: Converting an Improper Fraction into a Decimal

Write as a recurring decimal.

To convert a fraction into a decimal using a calculator, we first need to input the fraction into the calculator. We do this by using either the button or the button, which allows us to input the numerator and denominator. Once we have done this, we press the equals button and then we need to use the button on the calculator to convert the standard form into a decimal. This is often labeled .

The calculator display will then look as follows.

This is a repeating expansion of the digit 6, where we note the last digit on the display is rounded up.

We represent this using repeating expansion notation; we put a dot over the repeating digit:

In our next example, we will convert a mixed number into a decimal with repeating decimal digits.

Example 3: Converting a Mixed Number to a Decimal

Write as a recurring decimal.

Some calculators have a button that allows us to directly convert mixed numbers into fractions. This button will look like this: . If we press this button, we input the integer part into the left box and the fractional parts into the right boxes. If we then press the equals button, we get

Of course, we can find this directly by noting that , so

To convert a fraction into a decimal, we need to press the button on the calculator for converting fractions into decimals, . The calculator screen will then look as follows.

This is a repeating expansion of the digit 3. We represent this using repeating expansion notation; we put a dot over the repeating digit:

In our next example, we will convert a fraction into a recurring decimal to allow for a comparison with another fraction. We will then use this to solve a real-world comparison problem.

Example 4: Converting Fractions to Recurring Decimals to Solve Word Problems

Karim was told he could have or 0.34 of the total amount of jam inside a jar. Convert to a decimal, and determine which option would give Karim more jam.

We want to determine which of the two options will give Karim a larger portion of jam; this means we need to compare the sizes of a fraction and a decimal. One way to do this is to convert the fraction into a decimal so that we can instead compare the sizes of the decimals.

To convert into a decimal using a calculator, we type , which is the same as , into the calculator and press the equals button. Then, we press the convert from standard form into decimal button, , to get the following.

This is a repeating expansion of the digit 3. We represent this using repeating expansion notation; we put a dot over the repeating digit:

We can now compare the decimal expansions. We note that both and 0.34 have integer parts of zero and they have the same first decimal digit; however, 0.34 has the higher second decimal digit and so is larger.

Hence, and 0.34 of the total amount of jam is the larger portion.

It is worth noting that converting both numbers into decimals is not the only way to answer the question above. We can also use a calculator to note that . We can then write and to have the same denominators by noticing that their lowest common multiple is 150. We have

Now, since these fractions have the same denominator, the larger portion is the one with the larger numerator, which is , or 0.34. Of course, this method becomes more difficult the more complex the denominators of the fractions are.

Let’s now see some examples of converting decimals with repeating digits into fractions.

Example 5: Converting a Recurring Decimal into a Fraction

Use a calculator to convert into a fraction.

We first recall that the dot over the decimal digit 1 tells us that this digit is repeating, so

We can convert this number into a fraction using a calculator.

First, we type the digit 3 and the decimal point. Second, we repeat the digit 1 enough times so that the calculator will output a fraction. For example, we can type 3.11111111111111 and then press the equals button to get the following.

We note that we need the calculator to output a fraction; if we get a decimal answer, then we did not repeat the digit enough times.

Hence,

In our next example, we will convert a decimal with multiple repeating digits into a fraction.

Example 6: Rewriting a Recurring Decimal as a Fraction

Use a calculator to write as a fraction.

We begin by recalling that the dots over the decimal digits 2 and 3 tell us that these digits are repeating. In particular, the decimal digit of 1 is not repeating. So,

We can convert this number into a fraction using a calculator. We first type the nonrepeating part of the decimal expansion: “1.” We then repeat the digits 23 enough times so the that the calculator will output a fraction. For example, we can type 1.232323232323 and then press the equals button to get the following.

We note that we need the calculator to output a fraction; if we get a decimal answer, then we did not repeat the digit enough times.

Hence,

In our final example, we will solve a real-world comparison problem by converting fractions into decimals.

Example 7: Real-World Problem Involving the Comparison of Fractions

Engy can dig holes in an hour, and Sarah can dig holes in an hour. By converting both fractions into decimals, determine who is faster at digging holes.

We want to determine which of and is larger, and we are told to do this by comparing their decimal expansions. We can find the decimal expansions of each number by using a calculator.

First, we input into the calculator either by typing and pressing the equals button or by using the button, typing the numerator and denominator in, and then pressing the equals button. In either case, we can then press the convert from standard form to decimal button, , to get the following.

This is a repeating expansion of 35 where the final digit on the display is rounded up. We can write this in repeated expansion notation by writing dots over the repeating digits:

We can follow a similar process to rewrite as a decimal. First, we clear the calculator so that we can input the next number. We can either write as or use the mixed number button on the calculator, , to input the integer and fractional parts separately. In either case, we press the equals button after inputting and then press the conversion button to get the following output on the calculator’s display.

We can see that , so Sarah is faster at digging holes.

Let’s finish by recapping some of the important points from this explainer.

Key Points

• Dots (or sometimes bars or lines) above the decimal digits of a number indicate that those digits are repeating indefinitely. For example, .
• We can convert a quotient of integers into a decimal with either a terminating or a repeating decimal expansion.
• To convert a fraction into a decimal using a calculator, we use the button .
• We can find a fractional representation of a repeating decimal using a calculator by inputting the repeating expansion enough times to make the calculator output a fraction.
• We can compare the size of fractions by rewriting them as decimals.

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