6. Terminating and Repeating Decimals
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Chapter 3
6.
Terminating and Repeating Decimals
| Student Learning Objectives: |
|---|
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Why
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.
Lesson Settings & Tools
| | 12 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
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.
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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 NumbersThe applet below will help convert the fractions to decimal numbers. It is recommended to calculate at least 4 decimal places.
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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 |
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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 | |
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Pop Quiz
Identifying Types of Decimals
Determine whether the given number is a repeating decimal, a terminating decimal, or neither.
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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 
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
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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?
{"type":"choice","form":{"alts":["A. 0.4","B. 0.44","C. 4.44","D. 4.4"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
b What fraction of the students prefer comedy?
{"type":"choice","form":{"alts":["A. 0.148","B. 0.148","C. 0.148","D. 0.148"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
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.
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.
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
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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 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 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.
Since the greatest common factor of 900 and 75 is 75, simplify the fraction by dividing both the numerator and denominator by 75.
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
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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.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{17}{27}"]}}
b Which two movie genres did Vincenzo choose?
{"type":"choice","form":{"alts":["action and comedy","animation and comedy","animation and science fiction","science fiction and action"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}
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.
- Assign a variable to represent the repeating decimal.
- Count the number of repeating digits and call it n.
- Multiply the repeating decimal by 10^n.
- Subtract the equation in Step 1 from the equation in Step 3.
- 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.
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.
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.
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Pop Quiz
Practice Converting Between Fractions and Repeating Decimals
Write the given fraction as a repeating decimal and vice versa.
Extra
How to Submit a Repeating DecimalIf the answer is a repeating decimal, do the following to submit the answer.
- Enter the non-repeating part of the number, including the decimal point, into the first box.
- 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
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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?
{"type":"choice","form":{"alts":["A. No, 3 17 and 3 1071 are greater than \u03c0.","B. Yes, because 3 1071 < \u03c0 < 3 17.","C. No, 3 17 and 3 1071 are less than \u03c0.","D. Yes, because 3 17 < \u03c0 < 3 1071."],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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
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.
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
223/71
Next, divide the numerator 223 by the denominator 71. Calculate to 6 decimal places.
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 MethodThe 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!
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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.
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Terminating and Repeating Decimals
Exercise 2.1
Tearrik's mother makes a pizza with a diameter of 14 13 inches. Tearrik wants to put it in a square box with a side length of 14.39 inches.
External credits: @macrovector
Is the box big enough?
{"type":"choice","form":{"alts":["Yes","No"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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We want to determine whether the given square box is big enough for the pizza. We will start by rewriting the given mixed number as a fraction, then we will convert it to a decimal and, finally, compare both numbers. First, let's take another look at the given picture.
As we can see from the diagram, the box is 14.39 inches wide. Keep in mind that it is a square, which means each side has the same length. From the exercise, we also know that the diameter of the pizza is 14 13 inches. To compare these numbers, we will first rewrite 14 13 as a fraction. Let's do it!
Now that we have written the number as an improper fraction, we can write it in decimal form. We will divide the numerator 43 by the denominator 3. Let's do it!
Notice that the products and remainders repeat, which means that the remainder will never be 0. Because of this, we can say that 14 13 written in decimal form is the repeating decimal 14.3, or 14.33.... 14 13 = 14.33... The diameter of the pizza is smaller than the width of the box because the digit in the hundredths place in 14.3 3... is less than the digit in the hundredths place in 14.3 9. ccc Diameter of Pizza & & Side Length of Square 14.3 3... & < & 14.3 9 Because of this, we can say that the pizza will fit in the box.
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Solution
According to a recent survey, 72.72 % of moviegoers in the United States prefer animated movies. The percent here is a repeating decimal. Write it as a fraction in simplest form.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{8}{11}"]}}
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We are told that according to a survey, 72.72 % of moviegoers prefer animated movie. This percent looks like a repeating decimal, so let's start by writing it as one. We can do this by dividing the number by 100 and removing the percent sign. This will move the decimal point two places to the left. 72.72 % = 0.7272 Since 7 and 2 are repeating digits, it is the same thing as 0.72. Now we can convert this number into a fraction by following five steps.
- Assign a variable to represent the repeating decimal.
- Count the number of repeating digits and call it n.
- Multiply the repeating decimal by 10^n.
- Subtract the equation in Step 1 from the equation in Step 3.
- Solve for the variable and express it as a fraction in simplest form.
Let's do it!
Steps 1 and 2
We will use the variable x to represent the given repeating decimal number. We can write an equation by setting this variable equal to 0.72. x = 0.72 This number has two repeating digits, so the value of n is 2 in this case.
Step 3
Now we multiply both sides of the equation by 10^2, or 100, because n= 2.
Step 4
Now we will subtract the original equation from the equation we found in Step 3. cr & 100x = 72.72 - & x = 0.72 & 99x = 72 0.46cm We found that 99x = 72.
Step 5
Finally, we can now solve the last equation for x by dividing both sides of the equation by 99 and simplifying.
We know that both 99 and 72 are divisible by 9. Let's simplify the fraction!
We found that x is equal to 811. We can conclude that the given repeating decimal is equivalent to 811. 0.72=8/11 Since 0.72 is also equal to 72.72 %, we can also express this percent as 811. This fraction can be interpreted in the context of the question as 8 out of every 11 people surveyed prefer animated movies.
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Solution
The following table shows some decimal numbers and their fractional equivalents.
| Decimal | Fraction |
|---|---|
| 0.2 | 2/9 |
| 0.3 | 3/9 |
| 0.4 | 4/9 |
| 0.5 | 5/9 |
Use the table to find the fraction equivalent of 0.6.
{"type":"choice","form":{"alts":["A. 60\/99","B. 6\/9","C. 3\/5","D. 55\/90"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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We are asked to determine which of the given fractions represents 0.6. Let's first analyze the given table that shows decimal numbers and their fraction equivalents.
| Decimal | Fraction |
|---|---|
| 0.2 | 2/9 |
| 0.3 | 3/9 |
| 0.4 | 4/9 |
| 0.5 | 5/9 |
We can see that each decimal from the table is equal to a fraction with 9 in the denominator and the repeating digit in the numerator. Let's use this pattern to rewrite 0.6.
| Decimal | Fraction |
|---|---|
| 0.2 | 2/9 |
| 0.3 | 3/9 |
| 0.4 | 4/9 |
| 0.5 | 5/9 |
| 0.6 | 6/9 |
According to the pattern, 0.6 should be equal to 69. The answer is B.
We can check our answer by writing 69 as a decimal. To do so, we will divide the numerator by the denominator.
We can see that the pattern starts to repeat itself after a few steps. This means that the decimal is a repeating decimal. 6/9 = 0. 6 We found that 0.6 is equivalent to 69, so the correct option is B.
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Solution
{"type":"multichoice","form":{"alts":["1\/4","1\/6","1\/8","1\/16","1\/12","1\/5","1\/15","1\/18"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":[0,2,3,5]}
{"type":"multichoice","form":{"alts":["I","II","III ","IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[2,3]}
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Let's list the given fractions. 1/4, 1/5, 1/6, 1/8, 1/12, 1/15, 1/16, and 1/18 We want to write these fractions as decimals to see which ones are terminating. Recall that if the denominator of a fraction is a power of 10, then the fraction is a terminating number. Let's check if we can find a number that, when multiplied by the denominator, is equal to some power of 10. 10 & = 5 * 2 100 & = 4 * 25 1000 & = 8 * 125 10 000 & = 16 * 625 We were able to find factors for four of the denominators that made their product equal a power of 10. This means that the fractions 1 4, 1 5, 1 8, and 1 16 are terminating decimals.
| Fraction | Expand | Decimal Equivalent |
|---|---|---|
| 1/5 | 1 * 2/5 * 2 = 2/10 | 0.2 |
| 1/4 | 1 * 25/4 * 25 = 25/100 | 0.25 |
| 1/8 | 1 * 125/8 * 125 = 125/1000 | 0.125 |
| 1/16 | 1 * 625/16 * 625 = 625/10 000 | 0.0625 |
The denominators of the other fractions — 6, 12, 15, and 16 — are not factors of any power of 10. This suggests that they are not terminating decimals. We can use long division confirm our suspicions. Let's find 118.
As we can see, it is a repeating decimal. We can find the decimal form of the other fractions in the same way.
| Fraction | Decimal |
|---|---|
| 1/6 | 0.16 |
| 1/12 | 0.83 |
| 1/15 | 0.06 |
| 1/18 | 0.05 |
As a result, the fractions with terminating decimals are as follows. 1/4, 1/5, 1/8, and 1/16
Let's make a table to organize what we found in the previous part.
| Denominators 4, 5, 8, and 16 | Denominators 6, 12, 15, and 18 | ||
|---|---|---|---|
| Fraction | Terminating Decimal | Fraction | Repeating Decimal |
| 1/4 | 0.25 | 1/6 | 0.16 |
| 1/5 | 0.2 | 1/12 | 0.83 |
| 1/8 | 0.125 | 1/15 | 0.06 |
| 1/16 | 0.0625 | 1/18 | 0.05 |
It makes sense to say that a fraction's denominator determines whether it has a terminating or repeating decimal. If the denominator of a fraction in simplest form is a factor of some power of 10, then we can be sure that the result will be a terminating decimal. We can see that the numbers 4, 5, 8, and 16 are factors of some power of 10. 10 & = 5 * 2 100 & = 4 * 25 1000 & = 8 * 125 10 000 & = 16 * 625 Therefore, Statement III is true. On the other hand, the denominators 6, 12, 15, and 18 are not factors of any power of 10. Notice that they are all multiples of 3, and no power of 10 is divisible by 3. 6 & = 3 * 2 12 & = 3 * 4 15 & = 3 * 5 18 & = 3 * 6 As we can see in the table, simplified fractions with a prime factor other than 2 or 5 in the denominator produce repeating decimals. This means that Statement IV is also true. The first two statements are not correct. The following examples disprove the first two expressions. 4/12 = 0.3 and 3/15 = 0.2
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Terminating and Repeating Decimals
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