温馨提示：本站仅提供公开网络链接索引服务，不存储、不篡改任何第三方内容，所有内容版权归原作者所有
AI智能索引来源：http://www.edu.com/math-glossary/Rational-Numbers-Between-Two-Rational-Numbers-Definition-Examples
点击访问原文链接

Rational Numbers Between Two Rational Numbers: Definition and Examples | EDU.COM

进入原网站 导航首页
Rational Numbers Between Two Rational Numbers: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Rational Numbers Between Two Rational NumbersRational Numbers Between Two Rational Numbers: Definition and ExamplesTable of ContentsRational Numbers Between Two Rational Numbers Definition of Rational Numbers Between Two Rational Numbers

Rational numbers are numbers that can be written in the form pq\frac{p}{q}qp​, where ppp and qqq are integers, and qqq ≠ 0. The set of rational numbers includes integers, fractions, terminating decimals, and non-terminating repeating decimals. There are infinitely many rational numbers between any two distinct rational numbers. Unlike natural numbers where we have a finite number of values between two numbers (e.g., between 8 and 14, there are only five natural numbers: 9, 10, 11, 12, and 13), with rational numbers, we can always find infinitely many values between any two points on the number line.

There are several methods to find rational numbers between two given rational numbers. When the two rational numbers have the same denominator, we can simply find values with the same denominator but numerators in between. When they have different denominators, we first convert them to equivalent fractions with the same denominator using the LCM method. Alternatively, we can use the arithmetic mean method, where we find the average of the two rational numbers to get a rational number that lies between them. This value is always a rational number between the two given numbers.

Examples of Rational Numbers Between Two Rational Numbers Example 1: Finding Rational Numbers with Same Denominator Problem:

Find the rational numbers between 35\frac{3}{5}53​ and 95\frac{9}{5}59​.

Step-by-step solution:

Step 1, Notice that both rational numbers have the same denominator 5.

Step 2, Look at the numerators. We have 3 and 9.

Step 3, Find all natural numbers between 3 and 9. These are 4, 5, 6, 7, and 8.

Step 4, Create fractions using these numbers as numerators while keeping the same denominator (5).

Step 5, Write down all the rational numbers: 45,55,65,75\frac{4}{5}, \frac{5}{5}, \frac{6}{5}, \frac{7}{5}54​,55​,56​,57​, and 85\frac{8}{5}58​.

Step 6, For more rational numbers between 35\frac{3}{5}53​ and 95\frac{9}{5}59​, convert to equivalent fractions with larger denominators. For example, multiply both numerator and denominator by 10: 3×105×10=3050\frac{3 \times 10}{5 \times 10} = \frac{30}{50}5×103×10​=5030​ and 9×105×10=9050\frac{9 \times 10}{5 \times 10} = \frac{90}{50}5×109×10​=5090​

Step 7, Now find more rational numbers between 3050\frac{30}{50}5030​ and 9050\frac{90}{50}5090​, such as: 3150,3250,3750\frac{31}{50}, \frac{32}{50}, \frac{37}{50}5031​,5032​,5037​, etc.

Example 2: Finding Rational Numbers with Different Denominators Problem:

Find three rational numbers between 27\frac{2}{7}72​ and 13\frac{1}{3}31​.

Step-by-step solution:

Step 1, Notice that the rational numbers have different denominators (3 and 7).

Step 2, Find the LCM of the denominators: LCM(3, 7) = 21

Step 3, Convert both fractions to equivalent fractions with denominator 21:

2×37×3=621\frac{2 \times 3}{7 \times 3} = \frac{6}{21}7×32×3​=216​

1×73×7=721\frac{1 \times 7}{3 \times 7} = \frac{7}{21}3×71×7​=217​

Step 4, Since the difference between numerators is small (7 - 6=1), let's find equivalent fractions with larger denominator. Multiply the numerator and denominator by 10:

6×1021×10=60210\frac{6 \times 10}{21 \times 10} = \frac{60}{210}21×106×10​=21060​

7×1021×10=70210\frac{7 \times 10}{21 \times 10} = \frac{70}{210}21×107×10​=21070​

Step 5, Now you can find three rational numbers between 60210\frac{60}{210}21060​ and 70210\frac{70}{210}21070​, such as 61210,65210\frac{61}{210}, \frac{65}{210}21061​,21065​ and 69210\frac{69}{210}21069​.

Example 3: Using Arithmetic Mean Method Problem:

Find three rational numbers between 27\frac{2}{7}72​ and 13\frac{1}{3}31​ using the arithmetic mean method.

Step-by-step solution:

Step 1, Find the arithmetic mean (average) of the two given rational numbers: 12(27+13)\frac{1}{2}(\frac{2}{7} + \frac{1}{3})21​(72​+31​)

Step 2, Find a common denominator for 27\frac{2}{7}72​ and 13\frac{1}{3}31​: 12(6+721)\frac{1}{2}(\frac{6 + 7}{21})21​(216+7​)

Step 3, Add the numerators: 12(1321)\frac{1}{2}(\frac{13}{21})21​(2113​)

Step 4, Simplify the expression: 1342\frac{13}{42}4213​

Step 5, To find the second rational number, find the arithmetic mean of 13\frac{1}{3}31​ and 1342\frac{13}{42}4213​:

12(13+1342)\frac{1}{2}(\frac{1}{3} + \frac{13}{42})21​(31​+4213​)

=12(14+1342)= \frac{1}{2}(\frac{14 + 13}{42})=21​(4214+13​)

=2784= \frac{27}{84}=8427​

Step 6, To find the third rational number, find the arithmetic mean of 27\frac{2}{7}72​ and 1342\frac{13}{42}4213​:

12(27+1342)\frac{1}{2}(\frac{2}{7} + \frac{13}{42})21​(72​+4213​)

=12(12+1342)= \frac{1}{2}(\frac{12 + 13}{42})=21​(4212+13​)

=2584= \frac{25}{84}=8425​

Step 7, So the three rational numbers between 27\frac{2}{7}72​ and 13\frac{1}{3}31​ are 1342\frac{13}{42}4213​, 2784\frac{27}{84}8427​, and 2584\frac{25}{84}8425​.

Comments(5)FFoodieEllieNovember 6, 2025I've used this glossary page to teach rational numbers to my students. The examples and methods really made it easy for them to grasp!

EEngineerChrisNovember 6, 2025I've used this glossary page to teach rational numbers. The methods and examples are super helpful, making it easy for students to grasp!

VVolleyballLoverRyanNovember 5, 2025I've used this glossary page to teach rational numbers between two rational numbers. It's a great resource with clear examples that really help students understand!

BBadmintonEnthusiastWyattNovember 5, 2025I've used this glossary page to teach rational numbers. The examples and methods are super helpful, making it easy for students to grasp!

MCMs. CarterSeptember 17, 2025I’ve used this page to help my kids understand rational numbers better. The examples and step-by-step explanations made it so much easier for them to grasp the concept. Highly recommend!

Explore More TermsAlternate Exterior AnglesArea of A SectorDirect VariationDivision by ZeroDifference Between Square and RectangleStraight AngleView All Math TermsRecommended Interactive LessonsUnderstand the Commutative Property of Multiplication3Math3.OA.B.5Divide by 13Math3.OA.C.7Multiply by 53Math3.OA.C.7Multiply by 73Math3.OA.C.7Solve the subtraction puzzle with missing digits3Math3.NBT.A.2Write Multiplication Equations for Arrays3Math3.OA.A.1View All Interactive LessonsRecommended VideosCombine and Take Apart 3D Shapes1Math1.G.A.2Parallel and Perpendicular Lines4Math4.G.A.1Estimate products of two two-digit numbers4Math4.NBT.B.5Find Angle Measures by Adding and Subtracting4Math4.MD.C.7Place Value Pattern Of Whole Numbers5Math5.NBT.A.1Use Models and The Standard Algorithm to Divide Decimals by Whole Numbers5Math5.NBT.B.7View All VideosRecommended WorksheetsClassify and Count ObjectsKMathK.MD.B.3Find 10 more or 10 less mentally1Math1.NBT.C.5Subtract 10 And 100 Mentally2Math2.NBT.B.8Understand and Estimate Liquid Volume3Math3.MD.A.2Generate and Compare Patterns5Math5.OA.B.3Analyze The Relationship of The Dependent and Independent Variables Using Graphs and Tables6Math6.EE.C.9View All WorksheetsRecommended Coloring PagesCat with butterfly wings surrounded by sparkling starsPre-K – KAll SubjectsGlobe on a stand with a simple classroom deskPre-K – KAll SubjectsTwo sea turtles swimming above a rocky seabed with a couple of corals1 – 2All SubjectsYogurt parfait being prepared with a hand holding a spoon3 – 4All SubjectsA scene with several fruit trees, a small wooden sign that reads 'Orchard', and a path winding through3 – 4All SubjectsAnnulus decorated as a wreath with ribbons and ornaments5 – 6All SubjectsView All Coloring PagesRecommended BlogsVR Education: Building Equitable Learning for All K-6 StudentsNovember 11, 2025Nearpod Cheats: Smart Ways to Maximize Learning SuccessOctober 15, 2025Hiragana Keyboard Setup Guide for K-6 Japanese LearningOctober 14, 2025Exploring the Antonym of Tutor: Understanding Independent LearningOctober 12, 2025Tutoring Vacation: 5 Smart Ways to Keep Kids LearningOctober 12, 2025Changes to the ACT: What K-6 Families Need to KnowOctober 11, 2025View All Blog PostsQUICK LINKSAbout UsPrivacy PolicyTerms of ServiceTOOLSHomework HelperGuide DesignerPodcast MakerPlan BuilderRESOURCESMath GlossaryEnglish GlossaryEnglish Language ArtsMathematicsScienceBook InsightsFun with WordsBlog© 2025 EDU.COM. All rights reserved.

智能索引记录