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6. Terminating and Repeating Decimals
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Chapter 3
6. 

Terminating and Repeating Decimals

This lesson provides a deep dive into the world of decimals, specifically focusing on terminating and repeating decimals. It explains how these types of decimals can be understood through the use of long division and their relation to rational numbers. For example, if one is trying to convert a fraction to a decimal, understanding whether the decimal will terminate or repeat can be crucial. This knowledge is not just theoretical; it has practical applications in various fields such as finance for calculating interest rates, in engineering for precise measurements, and even in everyday scenarios like splitting a bill. The aim is to equip the reader with the tools needed to tackle real-world problems that involve decimals.
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10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Terminating and Repeating Decimals
Slide of 12
There are many ways to write the same rational number, including fractions, mixed numbers, and decimal numbers.

Fraction & Mixed Number & Decimal 14/5 & 2 45 & 2.8 Decimal numbers can be classified based on the digits after the decimal point. As an example, consider the decimal number corresponding to 13. Is there a repeating digit? How about the decimal equivalent of 17? This lesson explores decimal number types and develops methods for converting them into fractions.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Decimal Forms of Fractions

A rational number can be represented as a decimal number with the help of the long division method. Rewrite the fractions shown in the table as decimals.

Fraction Decimal
2/9
1/3
3/4
4/11
3/8

Consider the following questions as a guide to classify decimals.

  • Which fractions have a remainder of zero?
  • Which fractions do not have a remainder of zero?
  • Which fractions have the same operations repeated in the long division after a certain point?

Extra

 Applet for Calculating the Decimal Numbers
The applet below will help convert the fractions to decimal numbers. It is recommended to calculate at least 4 decimal places.
Applet to compute the division of two numbers
Discussion

Repeating Decimals

A repeating decimal number, or recurring decimal number, is a number in decimal form in which some digits after the decimal point repeat infinitely. The digits repeat their values at regular intervals and the infinitely repeated part is not zero. When writing the decimal, a line is drawn over the repeating portion to express such a number.

Repeating Decimal Numbers
Number Notation Fraction
0.666666... 0.6 2/3
1.533333... 1.53 23/15
5.373737... 5.37 532/99
Since they can be expressed as fractions, repeating decimal numbers are rational numbers.
Discussion

Terminating Decimals

A terminating decimal number is a number in decimal form with a finite number of digits. Terminating numbers can be written as fractions, which means that they are rational numbers.

Terminating Decimal Numbers
Number Fraction
0.5 1/2
1.53 153/100
52.372 13 093/250
Pop Quiz

Identifying Types of Decimals

Determine whether the given number is a repeating decimal, a terminating decimal, or neither.

Discussion

Converting a Fraction Into a Repeating Decimal Number

Fractions can be written as decimal numbers with the help of the long division method. However, when the denominator of a fraction has a factor other than 2 or 5, the quotient will be a repeating decimal number. Consider, for example, the following fraction. 7/15 It is possible to write this fraction as a decimal in two steps.
1
Use Long Division
expand_more
Divide the numerator by the denominator using long division.
Dividing 7 by 15
After some point, the remainder does not become zero and a particular digit or set of digits in the quotient keeps repeating. This means that the quotient is a repeating decimal number.
Dividing 7 by 15
2
Use Bar Notation to Write the Repeating Decimal
expand_more

The fraction was found to be a repeating decimal number. 7/15 = 0.4666... The digit that repeats is 6, so a bar is drawn over that digit. 7/15 = 0.46

Example

Favorite Movie Genres

Vincenzo conducts a survey to find out what kind of movies his classmates like. The table shows the information he gathered from the survey.

Genre Number of Students
Action 6
Animation 12
Comedy 4
Science Fiction 5
a What fraction of the students prefer animation?
b What fraction of the students prefer comedy?

Hint
a How many students are there? Write a fraction using the number of students that prefer animation and the total number of the students. Then, convert the fraction into a decimal number.
b Write a fraction using the number of students that prefer comedy and the total number of the students. Convert the fraction into a decimal number.

Solution
a Start by finding the total number of students in the class.
Genre Number of Students
Action 6
Animation 12
Comedy 4
Science Fiction 5
Total 27
There are 27 students in the class. Of those students, 12 prefer animation. This can be represented by a fraction by writing the part, the 12 students who like animation, over the whole, the 27 students in the class. 12/27 Since the given options are decimals, this fraction must be converted into a decimal number. To do so, use long division. Don't stop until the remainder is 0 or a pattern appears.
Dividing 12 by 27
It appears that the remainder will always repeat — in other words, the digit 4 in the quotient will always repeat. This means that the division resulted in a repeating decimal number. 12/27 = 0.444... The same value can also be written by drawing a bar over the repeating digit. 12/27 = 0.4 The answer is A.
b From the table, it can be seen that 4 students like the comedy genre. This means that the fraction of those students is 427. Again, use long division to rewrite it as a decimal. Don't stop until the remainder is 0 or a pattern appears.
Dividing 4 by 27
In this case, more than one digit repeats in the quotient. 4/27 & = 0. 148148... Put a bar above those three digits. This repeating decimal corresponds to option C. 4/27 & = 0.148
Discussion

Converting a Repeating Decimal Number Into a Fraction

Terminating decimal numbers can be converted to fractions by dividing the number by 10^n, where n is the number of decimal places. Terminating Decimal & & Fraction 0.56 & & 56/100 However, it is impossible to count the number of decimal places for repeating decimals because they have an infinite number of digits after the decimal point. Repeating Decimal & & Fraction 0.083 & & ? Even though they have an infinite number of digits, these numbers can be expressed as fractions. Follow these five steps to do so.
1
Assign a Variable for the Repeating Decimal
expand_more

Let x be a variable to represent the given repeating decimal number. An equation can be written by setting x equal to the number. Consider 0.083 as an example. x = 0.083

2
Count the Number of Repeating Digits and Let It Be n
expand_more

For the given decimal, the repeating digit is 3. x = 0.083 There is only one digit with a bar above it. This means n= 1.

3
Multiply the Repeating Decimal by 10^n
expand_more
Since there is one repeating digit, multiply each side of the equation by 10^1, or 10.
x = 0.083
MultEqn

LHS * 10=RHS* 10

x * 10 = 0.083 * 10
Multiply

Multiply

10x = 0.83
4
Subtract the Equation in Step 1 From the Equation in Step 3
expand_more

Now subtract the original equation from the equation obtained in Step 3. cl & 10x = 0.83 - & x = 0.083 Writing the decimals without using bar notation can make subtraction easier to understand. cl & 10x = 0.83333 ... - & x = 0.08333 ... & 9x = 0.75000 ... The zeros can be ignored. It is found that 9x is equal to 0.75.

5
Solve for the Variable and Express It as a Fraction in Its Simplest Form
expand_more
Finally, solve the equation for x.
9x = 0.75
DivEqn

.LHS /9.=.RHS /9.

9x/9 = 0.75/9
CancelCommonFac

Cancel out common factors

9x/9 = 0.75/9
SimpQuot

Simplify quotient

x = 0.75/9
Start by eliminating the decimal in the numerator to simplify the fraction. This can be done by multiplying the numerator and denominator by 10^2 because the numerator has 2 decimal places.
x = 0.75/9
ExpandFrac

a/b=a * 100/b * 100

x = 0.75 * 100/9* 100
Multiply

Multiply

x = 75/900
Since the greatest common factor of 900 and 75 is 75, simplify the fraction by dividing both the numerator and denominator by 75.
x = 75/900
ReduceFrac

a/b=.a /75./.b /75.

x = 75/ 75/900 / 75
CalcQuot

Calculate quotient

x = 1/12
It is found that x is equal to 112. Remember that x is also equal to 0.083. x = 0.83 and x = 1/12 Therefore, by the Transitive Property of Equality, the fraction is equal to the given repeating decimal number. 0.083=1/12
Example

Which Two Genres Did Vincenzo Choose?

Vincenzo is organizing the information he gathered from his survey.

Genre Number of Students
Action 6
Animation 12
Comedy 4
Science Fiction 5

He selected two genres at random. He used a calculator to divide the number of students who prefer the two genres by the total number of students.

His calculator showed the result as 0.629629629....

a Write the result as a fraction.
b Which two movie genres did Vincenzo choose?

Hint
a Let x represent the repeating decimal and write an equation. How many repeating digits are there in the decimal number? Multiply the equation by a power of 10.
b What is the numerator of the fraction found in Part A? Determine which of the two types gives a sum equal to this numerator.

Solution
a The number shown on the calculator is a repeating decimal number because some digits after the decimal point repeat themselves.

0. 629 629 629... ⇔ 0.629 There are five steps to write this repeating decimal number as a fraction.

  1. Assign a variable to represent the repeating decimal.
  2. Count the number of repeating digits and call it n.
  3. Multiply the repeating decimal by 10^n.
  4. Subtract the equation in Step 1 from the equation in Step 3.
  5. Solve for the variable and express it as a fraction in its simplest form.

Step 1

Let x be a variable that represents the given repeating decimal number. Write an equation by setting x equal to 0.629. x = 0.629

Step 2

The repeated sequence of digits in the decimal is 629. There are three repeating digits, so n= 3.

Step 3

Since n=3, multiply each side of the equation by 10^3, or 1000.
x = 0.629
MultEqn

LHS * 1000=RHS* 1000

x * 1000 = 0.629 * 1000
Multiply

Multiply

1000x = 629.629

Step 4

Now subtract the original equation from the equation obtained in Step 3. cr & 1000x = 629.629 [0.6em] - & x = 0.629 Consider the expanded forms of the numbers when performing this subtraction. cr & 1000x = 629.629629 ... - & x = 0.629629 ... & 999x = 629.000000 ... Since the zeros can be ignored, 999x is equal to 629.

Step 5

Lastly, solve the obtained equation for x.
999x = 629
DivEqn

.LHS /999.=.RHS /999.

999x/999 = 629/999
CancelCommonFac

Cancel out common factors

999x/999 = 629/999
SimpQuot

Simplify quotient

x = 629/999
Check whether this fraction can be simplified by finding the greatest common factor (GCF) of the numerator and denominator. 629 = 17 * 37 999 = 3^3 * 37 Then, GCF(629,999) is equal to 37. Finally, simplify the fraction by dividing both numerator and denominator by 37.
x = 629/999
ReduceFrac

a/b=.a /37./.b /37.

x = 629/37/999 /37
CalcQuot

Calculate quotient

x = 17/27
As a result, x is equal to 1727, which is equal to the given repeating number. 0.629=17/27
b In Part A, it was found that 0.629 is equivalent to 1727. Notice that the denominator is the same as the total number of students, so 1727 should be the fraction that Vincenzo entered into the calculator. The numerator of this fraction is the sum of the students who prefer two specific genres.

Now take a look at the table and determine which of the two genres gives a sum equal to 17.

Genre Number of Students
Action 6
Animation 12
Comedy 4
Science Fiction 5

The number of students who prefer animation and science fiction is 17. Therefore, Vincenzo selected the animation and science fiction genres.

Pop Quiz

Practice Converting Between Fractions and Repeating Decimals

Write the given fraction as a repeating decimal and vice versa.
Randomly generated fraction or decimal number

Extra

 How to Submit a Repeating Decimal

If the answer is a repeating decimal, do the following to submit the answer.

  1. Enter the non-repeating part of the number, including the decimal point, into the first box.
  2. Enter the repeating part into the second box.

Here are some examples. Repeating Decimal & & How to Submit 0.409 & ⇒ & 0.4 09 [0.6em] 0.9 & ⇒ & 0. 9


Example

Was Archimedes Correct?

Vincenzo is watching a movie after school. In one scene, the leading character was explaining some of Archimedes's contributions to mathematics. One of his contributions was a method he used to approximate the value of π. π = 3.141592 ... Although this number cannot be expressed exactly as a ratio of two integers, Archimedes claimed that π was between 3 17 and 3 1071. Was Archimedes correct?

Hint

Write the mixed numbers as decimals.

Solution

Archimedes believed that π was between two rational numbers, 3 17 and 3 1071. π = 3.141592... To check whether Archimedes was correct, both of the numbers will be written as a decimal. Since the first six decimal places of π are given, the numbers will be calculated up to 6 decimal places. In this way, the numbers can be compared.

Rewriting 3 17

First, rewrite 3 17 as an improper fraction.
3 17
▼
Write mixed number as a fraction
MixedToFrac

a bc=a* c+b/c

3 * 7 + 1/7
Multiply

Multiply

21 + 1/7
AddTerms

Add terms

22/7
Next, use long division to divide the numerator by the denominator of the fraction. Divide up to six decimal places because the first six decimal places of π is given.
Dividing 22 by 7
As seen, the mixed number is equal to 3.142857.... 3 17 & = 3.142857...

Rewriting 3 1071

The same steps as before can be followed to write 3 1071 as a decimal. Start by rewriting the mixed number as an improper fraction.
3 1071
▼
Write mixed number as a fraction
MixedToFrac

a bc=a* c+b/c

3 * 71 + 10/71
Multiply

Multiply

213 + 10/71
AddTerms

Add terms

223/71
Next, divide the numerator 223 by the denominator 71. Calculate to 6 decimal places.
Dividing 223 by 71
As shown, 3 1071 is equal to 3.140845 .... 3 1071 = 3.140845...

Comparing the Numbers

Now that the decimal forms of both numbers have been found, the numbers can be compared. 3 17 &= 3.142857... 3 1071 &= 3.140845... π &= 3.141592... Notice that the first three digits of the numbers are all the same. 3 17 &= 3.142857... 3 1071 &= 3.140845... π &= 3.141592... This means that the digits in the thousandth place must be compared. 3 17 &= 3.14 2857... 3 1071 &= 3.14 0845... π &= 3.14 1592... The number with the greatest digit in the thousandth place is the greatest, and the number with the least digit is the least. 3.140845... < 3.141592... < 3.142857... ⇓ 3 1071 < π < 3 17 Therefore, π is between 3 1071 and 3 17. Archimedes was correct. The answer is B.

Extra

 Archimedes's Method
The Greek mathematician Archimedes devised a technique to approximate π. His idea was quite simple — he drew regular polygons inside and outside of a circle. He used a circle with diameter of 1 because its circumference is π.
The circumference of the circle would be somewhere between the perimeters of the polygons. Archimedes calculated the perimeters of the polygons by means of geometric arguments.
His final estimate for π use a shape with 96 sides. His calculations gave a range between 3 1071 and 3 17. 3 1071 < π < 3 17 Of course, he found this result with pen and paper, without the help of a computer or calculator. Amazing!
Closure

Why Is π So Special?

This lesson presented two types of decimal numbers, terminating and repeating decimals. The lesson also introduced methods for converting both fractions. Now comes a tough question! Are there decimal numbers that have an infinite number of places but do not repeat? The answer is yes and they are called irrational numbers. Take, for example, π. π = 3.141592653 ... No repeating patterns are seen in this type of decimal number, so their decimal representations have no end. Here are a few other commonly used irrational numbers.

Some Irrational Numbers
Euler's number, e 2.718281 ...
Golden ratio, ϕ 1.618033 ...
sqrt(2) 1.414213 ...

Irrational numbers cannot be converted into rational numbers. All that can be done is to get better and better approximations. The diagram shows how decimal numbers are classified.


Level 1
Level 2
Level 3
Terminating and Repeating Decimals
Exercise 3.1

Zosia knows that three one-thirds is equal to one. 1/3 + 1/3 + 1/3 = 1 She then remembers that 13 can also be written as a repeating decimal. She adds these numbers as decimals again and gets 0.999999.... cr & 0.333333 ... & 0.333333 ... +& 0.333333 ... & 0.999999 ... Consider the following question to determine whether Zosia's calculations are correct.

Write 0.999999 ... in the form of a fraction.
Are Zosia's calculations correct?

Let's write the given repeating decimal as a fraction. 0.999999 ... We will follow five steps to rewrite it.

  1. Assign a variable to represent the repeating decimal.
  2. Count the number of repeating digits and call it n.
  3. Multiply the repeating decimal by 10^n.
  4. Subtract the equation in Step 1 from the equation in Step 3.
  5. Solve for the variable and express it as a fraction in its simplest form.

Let's do it!

Steps 1 and 2

Let x represent the given repeating decimal number. x = 0.999999 ... Notice that there is only one repeating digit, 9. This means that n= 1.

Step 3

We will multiply both sides of the equation by 10^1, or 10, because n= 1.

Step 4

Now we will subtract the original equation from the equation written in Step 3. cr & 10x = 9.999999 ... - & x = 0.999999 ... & 9x = 9.000000 ... We found that 9x = 9.

Step 5

Finally, we will divide both sides of the equation by 9 to find the value of x.

We found that x is equal to 99, or simply 1. We can conclude that 0.999999 ... is equal to 1. 0.999999 ... = 1

Alternatively, we can use the decimal expansion of 19, which is equal to 0.111111.... 1/9 = 0.111111... When we multiply both sides by 9, we get the following. 9 * 1/9 = 0.999999... On the left hand side, we get 99, or 1.


In Part A, we found that 0.999999 ... is equal to 1. 0.999999 ... = 1 At first glance, it would be easy for us to say that the left-hand side will never equal the right. This is because we cannot easily get a sense of infinitely many 9s. However, the equation is actually true because of the infinite number of 9s on the left. We showed how this is possible with our calculations in Part A. cr & 0.333333 ... & 0.333333 ... +& 0.333333 ... & 0.999999 ... cr & 0.333333 ... & 0.333333 ... +& 0.333333 ... & 1.000000 ... Therefore, all of Zosia's calculations are correct. Three one-thirds, or 33, can be written as 0.999999... or as 1.000000 ....

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Terminating and Repeating Decimals

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