lesson 1 homework practice rational numbers

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Rational Numbers Worksheets | Practice Worksheets on Rational Numbers

Most of you might have come across Chapter Rational Numbers while studying the subject. Usually, the word rational conveys logical interpretation followed by a reason. However, when it comes to Maths it is derived from the word Ratio and has a different meaning entirely. To help students prepare effectively we have compiled Rational Numbers Worksheets all in one place.

You can use the Worksheets on Rational Numbers during your practice sessions and test your level of preparation. The Kind of Questions asked in the Worksheets covers various subtopics of Rational Numbers such as Equivalent Rational Numbers, Positive and Negative Rational Numbers, Representing Rational Numbers on the Number Line, etc.

List of Rational Numbers Worksheets to Practice

For a better user experience, we have compiled all of the Worksheets for Rational Numbers in one place. Look no further and begin your practice straight away to score well in your exams. In order to prepare a particular topic, you just need to simply tap on the quick links available to access the corresponding topic worksheet. Solve Problems on your own at first and cross-check the solutions later in order to understand where you went wrong.

• Worksheet on Rational Numbers
• Worksheet on Equivalent Rational Numbers
• Worksheet on the Lowest Form of a Rational Number
• Worksheet on Standard form of a Rational Number
• Worksheet on Equality of Rational Numbers
• Worksheet on Comparison of Rational Numbers
• Worksheet on Representation of Rational Numbers on a Number Line
• Worksheet on Adding Rational Numbers
• Worksheet on Properties of Addition of Rational Numbers
• Worksheet on Subtracting Rational Numbers
• Worksheet on Addition and Subtraction of Rational Numbers
• Worksheet on Rational Expressions Involving Sum and Difference
• Worksheet on Multiplication of Rational Numbers
• Worksheet on Properties of Multiplication of Rational Numbers
• Worksheet on Division of Rational Numbers
• Worksheet on Properties of Division of Rational Numbers
• Worksheet on Finding Rational Numbers between Two Rational Numbers
• Worksheet on Word Problems on Rational Numbers
• Worksheet on Operations on Rational Expressions
• Objective Questions on Rational Numbers

Feel free to use our Online Maths Worksheets available on our Site Worksheetsbuddy.com and ace up your preparation level. You need not worry as you can make use of the worksheets categorized to solve problems that you are looking for.

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Rational & Irrational Numbers Worksheets

What is the Difference Between Rational & Irrational Numbers? It's all about the numbers in mathematics, isn’t it? It is nothing without a number game. But what exactly is a number? A number is an arithmetic value which can be a figure, word or symbol indicating a quantity. There are infinite types in numbers; natural, whole, integers, real, and complex numbers. One of the types which is the real numbers type is further divided into rational and irrational number. In most problems, you will find rational and irrational very commonly. A set of Rational numbers involve having integers and fraction; on the other hand, irrational numbers are numbers that cannot be expressed as fractions. : In mathematics, a rational number is any number that you can represent it in the fractional form like p/q, where q is greater than zero(0). You can use rational numbers as a fraction. But you will write their denominator & numerator as integers, and denominator will be equal to zero(0). Key points about rational numbers: While solving rational numbers (Q), the following points must be in your mind: Real numbers (R) contain all ration numbers (Q) and integers (Z). We can write integers as natural numbers (N). We can express all rational numbers as a whole number because we can write them in a fractional form. How to verify rational numbers: If you want to identify a given number is rational or not, don't forget to check these considerations: The number must be in fraction form like p/q, where q≠0. You can further simplify ratio p/q and express it in decimal form. The set of rational numbers must have +ve & -ve numbers and zero. Example: Verify 1 1/2 is a rational number. Solution: By simplifying, 1 1/2 becomes 3/2. Numerator 3 is an integer. Denominator 2 is an integer that is 2 ≠ 0. Hence proved 3/2 is a rational number. The difference between the two are: 1. Rational numbers can be expressed in a ratio of two integers, while irrational numbers cannot be written or expressed in a ratio of two integers. 2. Rational numbers can be expressed in a fraction; irrational numbers cannot be expressed in fractions. 3. Most of the rational numbers are perfect squares while no irrational number is a perfect square. 4. Rational numbers are finite or recurring decimals, whereas irrational numbers are not.

Basic Lesson

Demonstrates general rules of Rational & Irrational Numbers. Is -72 a rational or irrational number? The number is terminating and can be represented on number line, This indicates that it is a rational number.

Intermediate Lesson

Explores how to approach complex Rational & Irrational Numbers. Is 0.784543189... a rational or irrational number? As this number is not terminating, it goes on and on and on… This is an irrational number.

Independent Practice 1

Determine whether these numbers are rational or irrational. The answers can be found below.

Independent Practice 2

Features another 20 Rational & Irrational Numbers problems.

Homework Worksheet

Rational & Irrational Numbers problems for students to work on at home. Example problems are provided and explained.

10 Rational & Irrational Numbers problems. A math scoring matrix is included.

Homework and Quiz Answer Key

Answers for the homework and quiz.

Lesson and Practice Answer Key

Answers for both lessons and both practice sheets.

The Most Prolific Mathematical Writer?

Who was the most prolific mathematical writer of all time? Hint: He made large bounds forward in the study of modern analytic geometry? Answer: Leonhard Euler. We owe Euler for the notation f (x) for a function (1734), e for the base of natural logs (1727), I for the square root of -1 (1777), p for pi, for summation (1755), the notation for finite differences y and 2y and many others.

Curriculum  /  Math  /  7th Grade  /  Unit 2: Operations with Rational Numbers  /  Lesson 1

Operations with Rational Numbers

Lesson 1 of 18

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Lesson Notes

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Represent rational numbers on the number line. Define opposites, absolute value, and rational numbers.

Common Core Standards

Core standards.

The core standards covered in this lesson

The Number System

7.NS.A.1 — Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

Foundational Standards

The foundational standards covered in this lesson

6.NS.C.5 — Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.C.6 — Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

6.NS.C.7 — Understand ordering and absolute value of rational numbers.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Understand that (by general convention) numbers on the number line increase from left to right (or bottom to top), and decrease from right to left (or top to bottom). 
• Represent positive and negative integers, fractions, and decimals on the number line.
• Define opposite numbers as numbers that are the same distance from 0 but on opposite sides on the number line.
• Define the absolute value of a number to be the distance from 0, which, as a measure of distance, is either positive or zero.
• Define rational numbers as numbers that can be written in the form  $$\frac{a}{b}$$ , where  $$b \neq 0$$ , and that can be located on the number line.  

Suggestions for teachers to help them teach this lesson

• The first two lessons are approaching 7.NS.1 and review concepts and skills from sixth-grade standards in the number system domain. These standards are foundational to this seventh-grade unit and will support students in later lessons. Depending on the needs of your students, these lessons can be combined or left as separate lessons. The anchor problems can be used as part of whole-class instruction, diagnostically, or as part of independent practice.
• A physical number line is a very helpful tool for students to have throughout the unit (MP.5). These can be printed out on cardstock or laminated so students can use them over time. A game piece or token to use as a place marker can make the idea of “moving on the number line” more tangible for students. If the number lines are laminated, then white board markers could also be used with them. This blog includes an example with some links to create some for your classroom.

Lesson Materials

• Optional : Laminated number line (1 per student)
• Optional : Dry erase marker (1 per student)
• Optional : Game piece or token (1 per student)

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

25-30 minutes

Darnell thinks that –4 is less than –6 because 4 is smaller than 6, and –4 is closer to 0 than –6 is. Draw a number line to show the numbers 0, –4, and –6. Then explain why Darnell is incorrect.

Guiding Questions

Student response.

Upgrade to Fishtank Plus to view Sample Response.

In each situation, draw a number line and represent the two values as points on the number line. 

a.   deposit of $75 and withdrawal of $50

b.     $$5\frac{3}{4}$$  feet below sea level and 3 feet below sea level

c.   temperature of 25˚C and 12 degrees below 0 on the Celsius scale

Jessica says she’s thinking of two numbers. They are 24 units apart on the number line, and they are opposites. What are the two numbers?

You and a friend are playing a game using a number line. You both place a game token on the number line at 2. You then roll a die and move your token that number of places on the number line. Your friend rolls the die and does the same. 

The two tokens end up at different locations on the number line, however the absolute value of each location is the same, 3. Where are the two tokens on the number line, and what did you and your friend roll on the die to get there?

A set of suggested resources or problem types that teachers can turn into a problem set

15-20 minutes

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

5-10 minutes

Point A is shown on the number line below.

a.   What number does point A represent?

b.   What is the absolute value of the number represented by point A ?

c.   What is the opposite of the number represented by point A ? Indicate this on the number line.

d.   What is the distance between point A and $$-1$$ on the number line?

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic B > Lesson 11 — Absolute Value
• Illustrative Mathematics Integers on the Number Line 2
• Illustrative Mathematics Above and Below Sea Level
• Illustrative Mathematics Jumping Flea — This could be adapted to include fractional or decimal values.
• Open Middle Absolute Value
• EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic A > Lesson 4 — Opposites
• EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic A > Lesson 5 — Opposites
• EngageNY Mathematics Grade 6 Mathematics > Module 3 > Topic A > Lesson 6 — Representing Rational Numbers on the Number Line

Topic A: Adding and Subtracting Rational Numbers

Compare and order rational numbers. Write and interpret inequalities to describe the order of rational numbers.

Describe situations in which opposite quantities combine to make zero.

Model the addition of integers using a number line.

7.NS.A.1.B 7.NS.A.1.D

Determine efficient ways to add rational numbers with and without the number line.

Efficiently add and reason about sums of rational numbers.

Understand subtraction as addition of the opposite value (or additive inverse).

7.NS.A.1.C 7.NS.A.1.D

Find and represent the distance between two rational numbers as the absolute value of their difference.

Subtract rational numbers with and without the number line.

Add and subtract rational numbers efficiently using properties of operations.

Add and subtract rational numbers using a variety of strategies.

7.NS.A.1 7.NS.A.1.A 7.NS.A.1.B 7.NS.A.1.C 7.NS.A.1.D

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Topic B: Multiplying and Dividing Rational Numbers

Determine the rules for multiplying signed numbers.

7.NS.A.2.A 7.NS.A.2.C

Multiply signed rational numbers and interpret products in real-world contexts.

Determine the rules for dividing signed numbers.

7.NS.A.2.B 7.NS.A.2.C

Divide signed rational numbers and interpret quotients in real-world contexts.

Convert rational numbers to decimals using long division and equivalent fractions.

Multiply and divide with rational numbers using properties of operations.

7.NS.A.2 7.NS.A.2.C

Topic C: Using all Four Operations with Rational Numbers

Solve problems with rational numbers and all four operations.

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Into Math Grade 8 Module 10 Lesson 1 Answer Key Understand Rational and Irrational Numbers

We included  HMH Into Math Grade 8 Answer Key  PDF   Module 10 Lesson 1 Understand Rational and Irrational Numbers to make students experts in learning maths.

HMH Into Math Grade 8 Module 10 Lesson 1 Answer Key Understand Rational and Irrational Numbers

I Can determine whether a number is rational and write a given rational number as a fraction.

Spark Your Learning

Explanation: Given in a softball, a player’s batting average is the number of hits divided by the number of at-bats. Wrote each player’s batting average as a decimal as Jamilla = 0.4, Callie = 0.125, Mayumi = 0.44444, Elena = 0.373737, Kaycee = 0.1944, I notice Mayumi is having greatest batting average.

Turn and Talk Do you think all fractions have decimal representations that either end or have digits that repeat? Try some additional fractions and discuss your conjecture. Answer: Yes,

Explanation: Yes, I think all fractions have decimal representations that either end or have digits that repeat, If the prime factorization of the denominator contains all 2’s, all 5’s, or a combination of 2’s and 5’s only, then the decimal expansion of the fraction will be terminating, also if the denominator is a power of ten to begin with, we don’t have to factor, the decimal expansion will be terminating, For example, 3/10=0.3, 37/100=0.37, 528/1000=0.528, 345/10,000=0.0345 and so on, let’s look at other examples: 1/2=0.5 because the denominator is 2, 3/4=0.75 because 4 = 2×2, 2/5=0.4 because the denominator is 5. If the denominator is a prime number other than a 2 or a 5 or if the prime factorization of the denominator contains anyprime numbers other than 2’s or 5’s, the decimal expansion of the fraction will be repeatin, let’s look at some examples: 2/3=0.666…(the prime number 3 makes the decimal repeat), 5/6=0.8333…(6=2×3) 4/7=0.571428571428571428571428…(prime number 7).

Build Understanding

A rational number is any number that can be written as a ratio in the form \(\frac{a}{b}\), where a and b are integers and b is not 0. Every rational number can be written as a terminating decimal or a repeating decimal. Examples of rational numbers are: \(\frac{3}{8}\) = 0.375; 7 = \(\frac{7}{1}\), 0.2 = \(\frac{1}{5}\); 0.11111…= \(\frac{1}{9}\).

Connect to Vocabulary An irrational number is a number that cannot be written in the form \(\frac{a}{b}\), where a and b are integers and b is not 0.

Explanation: Considering the decimal 1.345345634567…. Yes, the decimal appear to have a repeating pattern as the numbers after the decimal keep going on and on. Yes, I think 1.345345634567… is a rational number because every positive number is a rational number.

B. You have learned that pi (π) is the ratio of the circumference of any circle to its diameter. The decimal value of pi is shown below. π = 3.145926535897932…….. Pi is an irrational number, but it can be written as a ratio. How can this be? __________________________ There are two ways to write a repeating decimal. You can use an ellipsis (three dots) to show that the repeating pattern continues: 0.111… ,0.235235235…, and 0,244414444… Or, you can write an overbar over the part of the decimal that repeats: \(0 . \overline{1}\), \(0 . \overline{235}\), and \(0.2 \overline{4}\) Answer: Yes, Pi is an irrational number, but it can be written as a ratio,

Explanation: As a ratio can be expressed as a:b, regardless of [ir]rational nature of a or b, or both. Hence, ratios involving pi or any other irrational quantities can be mani-pulated like any other ratios. Now, when you divide a rational number by another rational number, you get a rational number.

Turn and Talk Convert \(\frac{1}{36}\) to a decimal. Write the decimal using an ellipsis and using an overbar. Explain the process you used. Answer: 0.0277…. and \(0 . 02\overline{7}\),

Explanation: Given to convert \(\frac{1}{36}\) to a decimal using an ellipsis as we have \(\frac{1}{36}\) = 0.027777777, as ellipsis is three dots to show repeating pattern continues, as we have 7 is repeating so it is 0.02777…. and wrote overbar for 7 as repeating \(0 . 02\overline{7}\).

Explanation: Identifying the place value of the last digit in the terminating decimal as thousandth, Using this to determine the denominator of the fraction as 1000 and the digits to the right of the decimal point 825 are the numerator of the fraction therefore 0.825 = \(\frac{825}{1000}\).

Explanation: To write the fraction \(\frac{825}{1000}\) in lowest terms, we identify the greatest common factor (GCF) of the numerator and denominator as the prime factorization of 825, 825 = 3 × 5 × 5 × 11, the prime factorization of 1000, 1000 = 2 × 2 × 2 × 5 × 5 × 5 finding the GCF, multiplying all the prime factors common to both numbers: therefore, GCF = 5 × 5 =25. Now dividing the numerator and denominator by the GCF as \(\frac{825}{1000}\) ÷ \(\frac{25}{25}\) = \(\frac{33}{40}\).

Explanation: Given the average number of goals scored per game for a soccer team is shown. Wrote this rational number as a fraction or mixed number, Let x be the given decimal. Writing an equation with the decimal. x = 1.62333…. Multiply by a power of 10 so that the first repeating digit appears just to the right of the decimal point. Multiply both sides of the equation by 100. 100x = 162.333….. Multiply by a power of 10 so that the first repeating digit appears just to the left of the decimal point. Multiply both sides of the equation by 1000. we get 1623.33, Subtract the expression equal to from both sides. 1000x = 1623.33 -100x = -162.33 900x = 1,460.997 x = \(\frac{1460}{900}\) or 1\(\frac{561}{900}\).

Check Understanding

Question 1. Convert \(\frac{1}{18}\) to a decimal. A. Write the decimal using an ellipsis. Answer: 0.0555…,

Explanation: Given to convert \(\frac{1}{18}\) to a decimal using an ellipsis as we have \(\frac{1}{18}\) = 0.05555555 as ellipsis is three dots to show repeating pattern continues, as we have 5 is repeating so it is 0.0555….

B. Write the decimal using an overbar. Answer: \(0 . 0\overline{5}\),

Explanation: Given to convert \(\frac{1}{18}\) as we have \(\frac{1}{18}\) = 0.05555 and write overbar for 5 as repeating \(0 . 0\overline{5}\).

Question 2. Use your calculator to find the decimal equivalent of \(\sqrt{17}\). Do you think this is a rational number? Why or why not? Answer: No not rational, \(\sqrt{17}\) is an irrational number,

Explanation: √ 17 is irrational essentially as a consequence of 17 being prime :- that is having no positive factors apart from  1 and itself. Here’s a sketch of a proof: Suppose  √ 17 = p q  for some integers  p ,  q , with  q ≠ 0 . Without loss of generality,  p , q > 0  and  p  and  q have no common factor greater than  1 . If they did have a common factor, then you could divide both by that common factor to get a smaller  p 1  and  q 1  with  √ 17 = p 1 q 1 , Then  p 2   = 17 q 2    and since  p 2    is a multiple of  17  and 17  is prime,  p  must be a multiple of  17 . Let  k = p/ 17, Then  17 q 2   = p 2   = ( 17 k ) 2   = 17 ⋅ 17 k 2 , Divide both ends by  17  to find: q 2   = 17 k 2  hence  q  is a multiple of  17 . So both  p  and  q  are divisible by  17 , contradicting our assumption that  p  and  q have no common factor greater than 1 . So there is no such pair of integers  p  and  q .

For Problems 3-5. write the rational number as a simplified fraction or as a mixed number in simplest form.

Question 3. 1.905 Answer: \(\frac{381}{200}\) or 1\(\frac{181}{200}\),

Explanation: To convert the decimal 1.905 to a fraction, just follow these steps: Step 1 : Write down the number as a fraction of one: 1.905 =  1.905 1, Step 2 : Multiply both top and bottom by 10 for every number after the decimal point: As we have 3 numbers after the decimal point, we multiply both numerator and denominator by 1000. So, 1.905 1  =  (1.905 × 1000) (1 × 1000)  =  1905 1000 . Step 3 : Simplify (or reduce) the above fraction by dividing both numerator and denominator by the GCD( Greatest Common Divisor) between them. In this case, GCD(1905,1000) = 5. So, (1905÷5) (1000÷5)  =  381 200 when reduced to the simplest form. As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus \(\frac{381}{200}\) or 1\(\frac{181}{200}\),when expressed as a mixed number.

Question 4. 0.828282… Answer: \(\frac{82}{99}\),

Question 5. \(0.4 \overline{3}\) Answer: \(\frac{13}{30}\),

On Your Own

Explanation: Given the diagram shows the number of instruments in the string section of an orchestra. There are 7 cellos, 4 double basses, and 11 each of first violins, second violins, and violas. So using ratio notation and decimal notation to describe the relationship between the number of double basses and the total number of instruments in the string section. As a ratio is written with a colon between the two quantities that are being compared as we have double basses 4 and total number are 7 + 4 + 11 + 11 + 11 = 44, So Ratio notation: 4 : 44 and as decimal notation is the representation of a fraction using the base 10 along with a decimal point we have \(\frac{4}{44}\) = \(\frac{1}{11}\) = 0.09090909…. so it is \(0.\overline{09}\).

B. Is the relationship in Part A rational or irrational? Give two different justifications for your answer. Answer: Yes, rational,

Explanation: Relationship is rational Justifications: 1. As we know rational numbers are one very common type of number that we usually study after integers in math. These numbers are in the form of p/q, where p and q can be any integer and q ≠ 0 and The word “Rational” is originated from the word “ratio”. So, rational numbers are very well related to the ratio concept of ratio. As the relationship in part A is ratio notation: 4: 44 or \(\frac{4}{44}\) = \(\frac{1}{11}\) it is rational, 2. As we know non-terminating decimals with repeating patterns of decimals are also rational numbers and we have relationship in part A as \(0.\overline{09}\) a non terminating decimals therefore the relationship is rational.

Question 8. Critique Reasoning A student claimed that the number -4 is not rational because it is not a ratio of two integers. Do you agree or disagree? Explain. Answer: Disagree,

Explanation: Given a student claimed that the number -4 is not rational because it is not a ratio of two integers. I disagree because as it the ratio of two integers of -4 and 1.

Question 9. Use Repeated Reasoning Pi (π) is an irrational number. What can you say about the numbers \(\frac{\pi}{2}\), \(\frac{\pi}{3}\), \(\frac{\pi}{4}\)and so on? Are these rational or irrational? Answer: The numbers \(\frac{\pi}{2}\), \(\frac{\pi}{3}\), \(\frac{\pi}{4}\) and so on are irrational only,

Explanation: The number pi(π) is a mathematical constant representing the ratio of a circle’s circumstance to its diameter. It’s an irrational number, meaning that it can’t be represented by a common fraction. We can’t write it down as a non-infinite decimal and is a non-terminating, non-repeating decimal number. Hence the numbers such as \(\frac{\pi}{2}\), \(\frac{\pi}{3}\), \(\frac{\pi}{4}\) and so on are also irrational only as they are multipled by pi(π) the resulting numbers are also non-terminating, non-repeating decimal number.

Question 11. Convert \(\frac{1}{15}\) to a decimal. Write the decimal using an ellipsis, then write the decimal using an overbar. Answer: Ellipsis : 0.0666…, Overbar : \(0 . 0\overline{6}\),

Explanation: Given to convert \(\frac{1}{15}\) to a decimal using an ellipsis as we have \(\frac{1}{15}\) = 0.066666 as ellipsis is three dots to show repeating pattern continues, as we have 6 is repeating so it is 0.0666…. now to write overbar for 6 as repeating \(0 . 0\overline{6}\).

For Problems 13-18, write the number as a fraction or mixed number in simplest form.

Question 13. 4.024 Answer: \(\frac{503}{125}\) or 4\(\frac{3}{125}\),

Question 14. -1.111….. Answer: –\(\frac{10}{9}\) or -1\(\frac{1}{9}\),

Question 15. \(-0 . \overline{39}\) Answer: –\(\frac{33}{99}\) or –\(\frac{13}{33}\),

Question 16. \(0.61 \overline{4}\) Answer: \(\frac{553}{900}\),

Question 17. 2.484848… Answer: \(\frac{82}{33}\) or 2\(\frac{16}{33}\),

Question 18. 0.7222… Answer: \(\frac{65}{90}\),

Explanation: Let x be the given decimal Writing an equation with the decimal. x = 0.72222…. Multiply by a power of 10 so that the first repeating digit appears just to the right of the decimal point. 10x = 7.2222…., Then multiply both sides of the equation by 100. 100x = 72.222….., Subtract the expression equal to from both sides. 100x = 72.2222 -10x = -7.2222 90x = 65 x = \(\frac{65}{90}\).

I’m in a Learning Mindset!

How am I using formative feedback to solve problems about rational numbers? Answer: Focused on complex and challenging problem-solving tasks, Working out practice problems more,

Explanation: While solving problems about rational numbers, Focused on complex and challenging problem-solving tasks, Working out practice problems more.

Lesson 10.1 More Practice/Homework

Explanation: Given Kara and Nathan participated in a 60-minute maze race. As a ratio is written with a colon between the two quantities that are being compared as the relationship between Kara’s time and the total time of the race in ratio notation: 32: 60 and as decimal notation is the representation of a fraction using the base 10 along with a decimal point we have \(\frac{32}{60}\) = \(\frac{8}{15}\) = 0.5333333… so it is \(0.5\overline{3}\).

B. Is the relationship in Part A rational or irrational? Justify your answer. Answer: Yes, rational,

Explanation: Relationship is rational Justification: As we know rational numbers are one very common type of number that we usually study after integers in math. These numbers are in the form of p/q, where p and q can be any integer and q ≠ 0 and The word “Rational” is originated from the word “ratio”. So, rational numbers are very well related to the ratio concept of ratio. As the relationship in part A is ratio notation: 32: 60 or \(\frac{32}{60}\) = \(\frac{8}{15}\) it is rational and as we know non-terminating decimals with repeating patterns of decimals are also rational numbers and we have relationship in part A as \(0.5\overline{3}\) a non terminating decimals therefore the relationship is rational.

Question 2. Math on the Spot Write the decimal \(0 . \overline{63}\) as a fraction in simplest form. Answer: \(\frac{7}{11}\),

Explanation: Given the average number of hourly visitors to an art exhibit is \(45.1 \overline{3}\). The average number of hourly visitors as a mixed number in simplest form is to pass a periodic decimal number to a mixed number, we must first pass it to a fraction, as follows: The number is 45.13333, which is equal to 45.13 ~ where we know that 3 repeats infinitely many times. First we must subtract the number without a comma and the number without a comma without ncluding the periodic number, like this: 4513 – 451 = 4062, Then, divide by a number of nines, for each periodic number together with zeros for each non-periodic number (only the part after the comma). Ie .13 ~, Because there is only one periodic (3) and non-periodic (1) number, it is divided by 90. 4062/90, we take out half 2031/45 then third 677/15, can no longer be reduced. Now pass to a mixed number, we divide, the integer part would be the quotient, and the other part would be the fractional part, which would be the numerator the remainder and the denominator would be the number that divides like this we get 45\(\frac{2}{15}\).

Question 4. Critique Reasoning Katrina said the number 0.101100111000… is a rational number because it consists only of the digits 0 and 1, and these digits repeat. Do you agree or disagree? Explain. Answer: Disagree,

Explanation: Given Katrina said the number 0.101100111000… is a rational number because it consists only of the digits 0 and 1 and these digits repeat. I disagree as 0.101100111000… is a infinite acyclic decimal, i.e, we cannot write it in form of p/q, p, q are  integers q ≠ 0, As it is not a rational number it is irrational number.

Question 5. Convert \(\frac{1}{27}\) to a decimal. Write the decimal using an ellipsis, then write the decimal using an overbar. Answer: Ellipsis : 0.037037…, Overbar : \(0 . 0\overline{037}\),

Explanation: Given to convert \(\frac{1}{27}\) to a decimal using an ellipsis as we have \(\frac{1}{27}\) = 0.0370370 as ellipsis is three dots to show repeating pattern continues, as we have 037 is repeating so it is 0.037037…. now to write overbar for 037 as repeating \(0 . 0\overline{037}\).

Write the rational number as a fraction or mixed number in simplest form.

Question 6. 0.0606060… Answer: \(\frac{2}{33}\),

Explanation: Given to convert 0.0606060… as a fraction Step 1:  To convert 0. 06 repeating into a fraction, begin writing this simple equation: n = 0. 06 (equation 1), Step 2:  Notice that there are 2 digitss in the repeating block  (06) , so multiply both sides by 1 followed by 2 zeros, i.e., by 100. 100 × n = 6. 06 (equation 2), Step 3:  Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out. 100 × n = 6. 06, 1 × n = 0. 06,  99 × n = 6, 6/ 99 could be the answer, but it still we can put in the simplest form, i.e., reduced. To simplify this fraction, divide the numerator and denominator by 3 (the GCF – greatest common factor). n =  6 99  =  6 ÷ 3 99 ÷ 3  =  2 33 . So, 0. 060606….  =  \(\frac{2}{33}\) as the lowest possible fraction.

Question 7. -8.725 Answer: –\(\frac{349}{40}\) or -8\(\frac{29}{40}\),

Question 8. 0.424242… Answer: \(\frac{42}{99}\),

Explanation: Let x be 0.424242……., since same number is repeating after two digtits, we just multiply 10 2 with x i.e 100x=42.424242……., 100x – x = 42.424242…….                    -0.424242……. 99x = 42, therefore x = \(\frac{42}{99}\).

Question 9. 1.52888… Answer: \(\frac{2369}{225}\) or 10\(\frac{119}{225}\),

Question 10. \(-10 . \overline{7}\) Answer: –\(\frac{3233}{300}\) or -10\(\frac{233}{300}\),

Question 11. \(0 . \overline{57}\) Answer: \(\frac{19}{33}\),

Question 12. Midori wrote a ratio of two integers. Which of the following must be true about the number Midori wrote? (A) The decimal form of the number is a terminating decimal. (B) The decimal form of the number is a repeating decimal. (C) The number is rational. (D) The number is irrational. Answer: (A) The decimal form of the number is a terminating decimal, (B) The decimal form of the number is a repeating decimal, (C) The number is rational,

Explanation: Given Midori wrote a ratio of two integers ince any integer can be written as the ratio of two integers, all integers are rational numbers. Remembering that all the counting numbers and all the whole numbers are also integers and so they, too, are rational, And any number that can be represented as a repeating repeating or terminating decimal terminating, A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. Therefore bits (A) The decimal form of the number is a terminating decimal, (B) The decimal form of the number is a repeating decimal and (C) The number is rational.

Question 13. Which of the following is \(0 . \overline{15}\) written as a fraction in simplest form? (A) \(\frac{1}{15}\) (B) \(\frac{1}{9}\) (C) \(\frac{3}{20}\) (D) \(\frac{5}{33}\) Answer: (D) \(\frac{5}{33}\),

Question 14. Paolo keeps track of his favorite baseball player’s batting average, and notices the average is 0.4727272… . If his player has 55 at bats, what is the number of hits? _____________ hits Answer: Number of hits of Paolo favorite baseball player is 26,

Explanation: Given Paolo keeps track of his favorite baseball player’s batting average, and notices the average is 0.4727272…. If his player has 55 at bats, the number of hits are average of favorite baseball player is 0.47272727…, Number of bats = 55, We need to find the number of hits. Let the number of hits be ‘x’, Now we know that batting average can be calculated by dividing number of hits from number of bats framing in equation form we get \(\frac{x}{55}\) = 0.47272727…, Multiplying both side by 55 we get \(\frac{x}{55}\) X 55 = 0.47272727 X 55, x = 26, Hence Number of hits of Paolo favorite baseball player is 26.

In a survey of 120 people who visited a lake, 20% went fishing. Of those who went fishing, 75% also went swimming. Of those who did not go fishing, 50% went swimming. Use this information to solve Problems 15-16.

Explanation: Given in a survey of 120 people who visited a lake 20% went fishing. Of those who went fishing, 75% also went swimming. Of those who did not go fishing, 50% went swimming. So constructed a two-way frequency table to display the data as  shown above.

Question 16. What percent of those surveyed did not go swimming? Answer: 45% did not go swimming,

Explanation: From the two-way frequency table we got information that out of 120 people who visited a lake 54 people did not go swimming, so percent of those surveyed did not go swimming are \(\frac{54}{120}\) X 100 = 45%.

Corresponding Angles: It should be noted that the pair of corresponding angles are equal in measure. In the figure, there are four pairs of corresponding angles that is, ∠a = ∠e, ∠b = ∠f, ∠c = ∠g, and ∠d = ∠h, Alternate Interior Angles: Alternate interior angles are formed on the inside of two parallel lines that are intersected by a transversal. They are equal in measure, In this figure, ∠c = ∠e, ∠d = ∠f, Alternate Exterior Angles: Alternative exterior angles are formed on either side of the transversal and they are equal in measure. In this figure, ∠a = ∠g, ∠b = ∠h, Consecutive Interior Angles: Consecutive interior angles or co-interior angles are formed on the inside of the transversal and they are supplementary. Here, ∠c + ∠f = 180°, and ∠d + ∠e = 180°, Vertically Opposite Angles: Vertically opposite angles are formed when two straight lines intersect each other and they are equal in measure. Here, ∠a = ∠c, ∠b = ∠d, ∠e = ∠g, ∠f = ∠h, If we see the given figure (3y + 5)° and (4y – 30)° are corresponding angles so we have  (3y + 5)° = (4y – 30)° solving y = 5°+ 30° = 35°, Now ∠c = (3y + 5)° = (3 X 35° + 5)° = 110°, So ∠g = (4y – 30)° = (4 X 35° – 30°)°= 140°- 30° = 110° so ∠c = ∠g. As we have alternative exterior angles as ∠c = ∠e and we have ∠a = ∠e as they are corresponding angles, therefore ∠c = ∠g means ∠c = ∠e = ∠a = ∠g are equal it means that ∠1 = 110°.

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7.1 Rational and Irrational Numbers

Learning objectives.

By the end of this section, you will be able to:

• Identify rational numbers and irrational numbers
• Classify different types of real numbers

Be Prepared 7.1

Before you get started, take this readiness quiz.

Write 3.19 3.19 as an improper fraction. If you missed this problem, review Example 5.4 .

Be Prepared 7.2

Write 5 11 5 11 as a decimal. If you missed this problem, review Example 5.30 .

Be Prepared 7.3

Simplify: 144 . 144 . If you missed this problem, review Example 5.69 .

Identify Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six chapters of this book! It's time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers , and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.

In this chapter, we'll make sure your skills are firmly set. We'll take another look at the kinds of numbers we have worked with in all previous chapters. We'll work with properties of numbers that will help you improve your number sense. And we'll practice using them in ways that we'll use when we solve equations and complete other procedures in algebra.

We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?

Rational Numbers

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

A rational number is a number that can be written in the form p q , p q , where p p and q q are integers and q ≠ 0 . q ≠ 0 .

All fractions, both positive and negative, are rational numbers. A few examples are

Each numerator and each denominator is an integer.

We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.

Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let's look at a few to see if we can write each of them as the ratio of two integers. We've already seen that integers are rational numbers. The integer −8 −8 could be written as the decimal −8.0 . −8.0 . So, clearly, some decimals are rational.

Think about the decimal 7.3 . 7.3 . Can we write it as a ratio of two integers? Because 7.3 7.3 means 7 3 10 , 7 3 10 , we can write it as an improper fraction, 73 10 . 73 10 . So 7.3 7.3 is the ratio of the integers 73 73 and 10 . 10 . It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.3 7.3 or −1.2684 ) −1.2684 ) is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction.

Example 7.1

Write each as the ratio of two integers: ⓐ −15 −15 ⓑ 6.81 6.81 ⓒ −3 6 7 . −3 6 7 .

Write each as the ratio of two integers: ⓐ −24 −24 ⓑ 3.57 . 3.57 .

Write each as the ratio of two integers: ⓐ −19 −19 ⓑ 8.41 . 8.41 .

Let's look at the decimal form of the numbers we know are rational. We have seen that every integer is a rational number , since a = a 1 a = a 1 for any integer, a . a . We can also change any integer to a decimal by adding a decimal point and a zero.

Integer −2 , −1 , 0 , 1 , 2 , 3 Decimal −2.0 , −1.0 , 0.0 , 1.0 , 2.0 , 3.0 These decimal numbers stop. Integer −2 , −1 , 0 , 1 , 2 , 3 Decimal −2.0 , −1.0 , 0.0 , 1.0 , 2.0 , 3.0 These decimal numbers stop.

We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.

Ratio of Integers 4 5 , − 7 8 , 13 4 , − 20 3 Decimal Forms 0.8 , −0.875 , 3.25 , −6.666… These decimals either stop or repeat. −6 . 66 — Ratio of Integers 4 5 , − 7 8 , 13 4 , − 20 3 Decimal Forms 0.8 , −0.875 , 3.25 , −6.666… These decimals either stop or repeat. −6 . 66 —

What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.

Irrational Numbers

Are there any decimals that do not stop or repeat? Yes. The number π π (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat.

Similarly, the decimal representations of square roots of whole numbers that are not perfect squares never stop and never repeat. For example,

A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number .

Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

Let's summarize a method we can use to determine whether a number is rational or irrational.

If the decimal form of a number

• stops or repeats, the number is rational.
• does not stop and does not repeat, the number is irrational.

Example 7.2

Identify each of the following as rational or irrational:

• ⓐ 0.58 3 – 0.58 3 –
• ⓑ 0.475 0.475
• ⓒ 3.605551275… 3.605551275…

ⓐ 0.58 3 – 0.58 3 – The bar above the 3 3 indicates that it repeats. Therefore, 0.58 3 – 0.58 3 – is a repeating decimal, and is therefore a rational number.

ⓑ 0.475 0.475 This decimal stops after the 5 5 , so it is a rational number.

ⓒ 3.605551275… 3.605551275… The ellipsis (…) (…) means that this number does not stop. There is no repeating pattern of digits. Since the number doesn't stop and doesn't repeat, it is irrational.

ⓐ 0.29 0.29 ⓑ 0.81 6 – 0.81 6 – ⓒ 2.515115111… 2.515115111…

ⓐ 0.2 3 – 0.2 3 – ⓑ 0.125 0.125 ⓒ 0.418302… 0.418302…

Let's think about square roots now. Square roots of perfect squares are always whole numbers , so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.

Example 7.3

ⓐ The number 36 36 is a perfect square, since 6 2 = 36 . 6 2 = 36 . So 36 = 6 . 36 = 6 . Therefore 36 36 is rational.

ⓑ Remember that 6 2 = 36 6 2 = 36 and 7 2 = 49 , 7 2 = 49 , so 44 44 is not a perfect square.

This means 44 44 is irrational.

Classify Real Numbers

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers . Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers , we get the set of real numbers .

Figure 7.2 illustrates how the number sets are related.

• Real Numbers

Real numbers are numbers that are either rational or irrational.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real”, and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. You won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.

Example 7.4

Determine whether each of the numbers in the following list is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number.

ⓐ The whole numbers are 0 , 1 , 2 , 3 ,… 0 , 1 , 2 , 3 ,… The number 8 8 is the only whole number given.

ⓑ The integers are the whole numbers, their opposites, and 0 . 0 . From the given numbers, −7 −7 and 8 8 are integers. Also, notice that 64 64 is the square of 8 8 so − 64 = −8 . − 64 = −8 . So the integers are −7 , 8 , − 64 . −7 , 8 , − 64 .

ⓒ Since all integers are rational, the numbers −7 , 8 , and − 64 −7 , 8 , and − 64 are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so 14 5 and 5.9 14 5 and 5.9 are rational.

ⓓ The number 5 5 is not a perfect square, so 5 5 is irrational.

ⓔ All of the numbers listed are real.

We'll summarize the results in a table.

Determine whether each number is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number: −3 , − 2 , 0 . 3 – , 9 5 , 4 , 49 . −3 , − 2 , 0 . 3 – , 9 5 , 4 , 49 .

Determine whether each number is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number: − 25 , − 3 8 , −1 , 6 , 121 , 2.041975… − 25 , − 3 8 , −1 , 6 , 121 , 2.041975…

ACCESS ADDITIONAL ONLINE RESOURCES

• Sets of Real Numbers

Section 7.1 Exercises

Practice makes perfect.

In the following exercises, write as the ratio of two integers.

• ⓑ 3.19 3.19
• ⓑ −1.61 −1.61
• ⓑ 9.279 9.279
• ⓑ 4.399 4.399

In the following exercises, determine which of the given numbers are rational and which are irrational.

0.75 0.75 , 0.22 3 – 0.22 3 – , 1.39174… 1.39174…

0.36 0.36 , 0.94729… 0.94729… , 2.52 8 – 2.52 8 –

0 . 45 — 0 . 45 — , 1.919293… 1.919293… , 3.59 3.59

0.1 3 – , 0.42982… 0.1 3 – , 0.42982… , 1.875 1.875

In the following exercises, identify whether each number is rational or irrational.

Classifying Real Numbers

In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.

−8 −8 , 0 , 1.95286.... 0 , 1.95286.... , 12 5 12 5 , 36 36 , 9 9

−9 −9 , −3 4 9 −3 4 9 , − 9 − 9 , 0.4 09 — 0.4 09 — , 11 6 11 6 , 7 7

− 100 − 100 , −7 −7 , − 8 3 − 8 3 , −1 −1 , 0.77 0.77 , 3 1 4 3 1 4

Everyday Math

Field trip All the 5 th 5 th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 147 people. Each bus holds 44 44 people.

ⓐ How many buses will be needed?

ⓑ Why must the answer be a whole number?

ⓒ Why shouldn't you round the answer the usual way?

Child care Serena wants to open a licensed child care center. Her state requires that there be no more than 12 12 children for each teacher. She would like her child care center to serve 40 40 children.

ⓐ How many teachers will be needed?

Writing Exercises

In your own words, explain the difference between a rational number and an irrational number.

Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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• Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Prealgebra 2e
• Publication date: Mar 11, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/prealgebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/prealgebra-2e/pages/7-1-rational-and-irrational-numbers

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Chapter 3: Operations with Rational Numbers

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• Math in Motion
• Standardized Test Practice
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