lesson 1 problem solving practice terminating and repeating decimals

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Catch-Up and Review

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

• Long Division
• Equivalent Fractions
• Dividing Decimals

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.

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?

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.

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 .

Identifying Types of Decimals

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

Converting a Fraction Into a Repeating Decimal Number

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.

Converting a Repeating Decimal Number Into a Fraction

LHS ⋅ 1 0 = RHS ⋅ 1 0

LHS / 9 = RHS / 9

Cancel out common factors

Simplify quotient

b a ​ = b ⋅ 1 0 0 a ⋅ 1 0 0 ​

b a ​ = b / 7 5 a / 7 5 ​

Calculate quotient

Which Two Genres Did Vincenzo Choose?

Vincenzo is organizing the information he gathered from his survey.

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 . 6 2 9 6 2 9 6 2 9 … .

• Assign a variable to represent the repeating decimal.
• Count the number of repeating digits and call it n .
• Multiply the repeating decimal by 1 0 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.

The repeated sequence of digits in the decimal is 6 2 9 . There are three repeating digits, so n = 3 .

LHS ⋅ 1 0 0 0 = RHS ⋅ 1 0 0 0

LHS / 9 9 9 = RHS / 9 9 9

b a ​ = b / 3 7 a / 3 7 ​

Now take a look at the table and determine which of the two genres gives a sum equal to 1 7 .

The number of students who prefer animation and science fiction is 1 7 . Therefore, Vincenzo selected the animation and science fiction genres.

Practice Converting Between Fractions and Repeating Decimals

If 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.

Was Archimedes Correct?

Write the mixed numbers as decimals .

Rewriting 3 7 1 ​

a c b ​ = c a ⋅ c + b ​

Rewriting 3 7 1 1 0 ​

Comparing the numbers, why is π so special.

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.

Terminating and Repeating Decimals

The decimal representation of a rational number is converting a rational number into a decimal number that has the same mathematical value as the rational number. A rational number can be represented as a decimal number with the help of the long division method. We divide the given rational number in the long division form and the quotient which we get is the decimal representation of the rational number. A rational number can have two types of decimal representations (expansions):

Terminating Non-terminating but repeating

While dividing a number by the long division method, if we get zero as the remainder, the decimal expansion of such a number is called terminating. And while dividing a number, if the decimal expansion continues and the remainder does not become zero, it is called non-terminating or repeating. The decimal form of a fraction usually represented by a bar over the repeating numbers.

1. If the number is a mixed number, convert it into an improper fraction. 2. Just divide the numerator by the denominator. If you end up with a remainder of 0 , then you have a terminating decimal. Otherwise, the remainders will begin to repeat after some point, and you have a repeating decimal. 3. Determine which answers are repeating decimals and put a bar over the repeating numbers in the decimal.

Practice Terminating and Repeating Decimals

Practice Problem 1

Practice Problem 2

Practice Problem 3

Terminating Decimal – a decimal which can be expressed in a finite number of figures or for which all figures to the right of some place are zero.

Repeating Decimal – a decimal in which after a certain point a particular digit or sequence of digits repeats itself indefinitely.

Pre-requisite Skills Division With Remainders Convert Between Mixed Numbers and Improper Fractions Write Fractions as Decimals Decimals, Fractions, and Mixed Numbers Write Fractions as Decimal Numbers Write Improper Fractions as Mixed Number Write Mixed Number as Improper Fraction

Related Skills Compare and Order Rational Numbers Add and Subtract Like Fractions Add and Subtract Unlike Fractions Add and Subtract Mixed Numbers Multiply Rational Numbers Divide Rational Numbers Convert Units

Worksheet on Repeating Decimals

Practice the questions given in the worksheet on repeating decimals or recurring decimals. The questions are based in expressing the decimal form.

1. Identify pure recurring decimals and mixed recurring decimals.

2. Identify whether the following are terminating or non-terminating:

3. Convert the pure recurring decimals into vulgar fractions:

4. Express each of the infinite repeating decimal or mixed recurring decimals in the form of a/b also a/b should be positive integers.

5. Find the value of the infinite repeating decimals:

Answers for the worksheet on repeating decimals or recurring decimals are given below to check the exact answers of the above decimal form.

1. (a) Pure recurring decimals

(b) Mixed recurring decimals

(c) Pure recurring decimals

(d) Mixed recurring decimals

(e) Pure recurring decimals

2. (a) Terminating

(b) Terminating

(c) Terminating

(d) Non-terminating

(e) Terminating

(f) Terminating

(g) Non-terminating

(d) 1792/333

(h) 473/999

4. (a) 8441/9900

(b) 713/1650

(c) 2711/900

(d)5561/4950

(f) 611/495

(g) 193/495

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Terminating and Repeating Decimals

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Terminating and Repeating Decimals

Any rational number (that is, a fraction in lowest terms) can be written as either a terminating decimal or a repeating decimal . Just divide the numerator by the denominator . If you end up with a remainder of 0 , then you have a terminating decimal. Otherwise, the remainders will begin to repeat after some point, and you have a repeating decimal.

Convert the fraction 5 8 to a decimal.

The division is as follows:

            0.625 8 5.000                 48 _                       20                       16 _                             40                             40 _                                   0

So, 5 8 = 0.625 . This is a terminating decimal.

Convert the fraction 7 12 to a decimal.

                0.5833 12 7.0000                     6   0 _                     1   00                             96 _                                   40                                   36 _                                         40                                         36 _                                               4

7 12 = 0.58 3 ¯

This is a repeating decimal.

The bar over the number, in this case 3 , indicates the number or block of numbers that repeat unendingly.

See also Converting Repeating Decimals to Fractions .

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Converting Fractions to Terminating and Repeating Decimals (A)

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Repeating Decimals – Definition, Types, Examples, Facts, FAQs

What are repeating decimals, types of decimals, how to convert recurring decimals to fractions, solved examples on repeating decimals, practice problems on repeating decimals, frequently asked questions on repeating decimals.

Repeating decimals are decimals in which a digit or a group of digits after the decimal point repeats indefinitely and at regular intervals such that the decimal representation becomes periodic. 

Repeating decimals are also known as “recurring decimals.”

Examples of repeating decimals: 0.3333…, 1.454545…, 0.626262…, etc.

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Definition of Repeating Decimals

Repeating decimals or recurring decimals are the non-terminating decimals in which a digit or a group of digits are repeated at equal intervals after the decimal point.

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Decimals can be classified into two categories: terminating decimals and non-terminating decimals.

1. What are terminating decimals?  

Terminating decimals are decimals in which there are a finite number of digits after the decimal point . The digits after the decimal point terminate after a finite number of digits. 

2. What are non-terminating decimals? 

Non-terminating decimals are the decimals in which there are an infinite number of digits after the decimal point.

The non-terminating decimals are further classified as

a) Repeating decimals: Digits or a group of digits repeat themselves

b) Non-repeating decimals: Digits do not follow any repeating pattern

Representation of Recurring Decimals

We generally write repeating decimals using an ellipsis (a set of dots, i.e., “…”). To express that a digit or a group of digits is repeating, we can also write a dot or a bar on the top of the recurring part. 

Period of a Repeating Decimal Number

The recurring part (the recurring digit or the group of recurring digits) in a non-terminating and repeating decimal is called the period of the decimal.

Example: What is the period of the decimal expansion of $\frac{1}{7}$?

$\frac{1}{7} = 0.\overline{142857}$

Period $= 142857$

Periodicity of a Repeating Decimal

The number of digits that repeat in a recurring decimal is called the periodicity (or the length of the period) of the repeating decimal.

Example: The periodicity of $0.\overline{142857}$ is 6.

Repeating Decimals as Rational Numbers

All repeating decimals are rational numbers . All rational numbers can be written in the decimal form that has the same mathematical value, with the help of the long division method. 

The decimal expansion of a rational number can be of two types only:

• Terminating decimal expansion
• Non-terminating but repeating decimal expansion

Rational Numbers with Repeating Decimal Expansion

We divide the numerator of the rational number by its denominator using the long division method to find the decimal expansion. The quotient obtained represents the decimal representation of that rational number. 

Let’s consider an example.

Example: What is the decimal expansion of the rational number $\frac{4}{9}$ ?

Divide 4 by 9.

$\frac{4}{9} = 0.444..=0.\overline{4}$

Thus, the rational number $\frac{4}{9}$ has a non-terminating and repeating decimal expansion. 

Repeating or recurring decimals have a repetitive pattern of digits after the decimal point. Let’s understand the steps to convert a repeating decimal as a fraction.

Step 1: Identify the repeating digits in the given decimal number. 

Step 2: Equate the decimal number to any variable. Make sure that only repeating digits come after the decimal point. 

Step 3: Multiply both sides of the equation by a power of ten equal to the number of repeating digits. 

Step 4: Subtract the equation obtained in step 3 from the equation obtained in step 2. 

Step 5: Simplify to get the answer. 

Example 1: Write 0.3333… as a fraction.

Let $x = 0.3333$… …(i)

Repeating digit $= 3$

Number of repeating digits $= 1$

$10x = 3.3333$… …(ii)

Subtracting (ii) from (i), we get 

$10x\;-\;x = 3.333…\;-\;0.3333…$

$x = \frac{3}{9}$

$x = \frac{1}{3}$

Example 2: Convert 0.5232323… into a fraction.

The repeating digits in the given decimal number are 23. 

Let $x = 0.5232323$….(i)

Notice that we need to shift the non-repeating digit 5 to the left of the decimal point. Multiply (i) by 10.

$10x = 5.232323$….(ii)

There are two repeating digits, so we multiply the above equation by 100. We will get,

$1000x = 523.232323$…(iii)

Subtract (ii) from (iii).

$990x = 518$

$x = \frac{518}{990}$

$x = \frac{259}{495}$

Example 3: Convert 7.324242.. into a fraction.

Let $x = 7.3242424$…

Let’s keep only repeating digits after the decimal point.

$10x = 73.2424$… …(i)

Multiply both the sides by the power of 10 equal to the number of repeating digits. Thus, we multiply by $10^2 = 100$.

$1000x = 7324.2424$… …(ii)

Subtracting (ii) from (i), we get

$1000x \;-\; 10x = 7324.2424…\;-\;73.2424$..

$\Rightarrow 990x = 7251$

$\Rightarrow x = \frac{7251}{990}$

$\Rightarrow x = \frac{2417}{330}$

Properties of Repeating Decimals

• All repeating decimals are rational numbers. They can be expressed in the form of $\frac{p}{q}$, where $q \neq 0,\; p$ and q are integers. 
• A real number is rational if and only if its decimal expansion is repeating or terminating. 
• Repeating decimals have an infinite number of digits.
• Repeating decimals are non-terminating decimals that show a certain repetitive pattern of digits after the decimal point. 
• If a digit or a group of digits after the decimal point in a non-terminating decimal are not repeating, the decimal expansion represents an irrational number.

1. Convert the repeating decimal 1.8888… into a fraction.

Solution:  

Let $x = 1.8888$… — (1)

Multiplying both the sides by 10, we get

$10x = 18.8888$… — (2)

Subtracting (1) from (2), we get

$10x \;-\; x = 18.888$…$-1.888$…

$9x = \frac{1}{7}$

$x = \frac{17}{9}$

2. What fraction is equivalent to the repeating decimal number 0.434343…?

Let $x = 0.434343$… — (1)

Multiplying both the sides by 100, we get

$100x = 43.434343$…. — (2)

$100x \;-\; x = 43.434343…\;-\;0.434343….$

$x = \frac{43}{99}$

3. Convert 0.567567567… into a fraction.

Solution: 

Let $x = 0.567567567$… — (1)

Multiplying both the sides by 1000

$1000x = 567.567567$… — (2)

$1000x\;-\;x = 567.567567567$…$-0.567567567$…

$999x = 567$

$x= \frac{567}{999} = \frac{189}{333} = \frac{63}{111} = \frac{21}{37}$

4. Which fraction is equivalent to 2.34444…?

Let $x = 2.34444$… 

Multiplying both the sides by 10

$10x = 23.4444$… — (1)

Multiplying both sides by 10 

$100x = 234.4444$… — (2)

$100x\;-\;10x = 234.444$…$-23.444$…

$90x = 211$

$x = \frac{211}{90}$

5. What fraction is equivalent to the repeating decimal number 24.579579…?  

Let $x = 24.579579$… — (1)

$1000x = 24579.579579$… — (2)

$1000x\;-\;x = 24579.579579$…$-24.579579$…

$999x = 24,555$

$x = \frac{24555}{999} = \frac{8185}{333}$

Repeating Decimals - Definition, Types, Examples, Facts, FAQs

Attend this quiz & Test your knowledge.

On converting 6.7777… into a fraction, we get ____

Which is the period of 0.123123123…, on converting 0.67777... into a fraction, we get ____, all repeating decimals are ______., what is the periodicity of $\frac{2}{7}$.

Can we convert non-terminating and non-repeating decimals into fractions?

No, we can never convert a non terminating decimal into a fraction. Such decimals are irrational numbers .

What is the difference between terminating and non-terminating decimal expansion?

Terminating decimal expansion is the expansion in which the remainder $= 0$ and non-terminating decimal expansion is the expansion in which the remainder $0$.

How can we convert fractions into decimals?

Fractions can be converted into decimals by dividing the numerator by the denominator .

How do we represent the non-terminating repeating decimal expansion?

We represent the non-terminating repeating decimal expansion using a bar. For example, 0.232323… can be represented as $0.\overline{23}$.

What is a mixed recurring decimal?

A decimal in which there is at least one non-repeating digit before the recurring part is known as a mixed recurring decimal .Example: $1.235555… = 1.2\overline{35}$

What is the symbol for repeating digits in recurring decimals?

Dot notation or bar notation on the top of the recurring digits is used.

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