{"id":9137,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9137"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","slug":"finding-a-rational-between-two-number","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/finding-a-rational-between-two-number\/","title":{"rendered":"Finding A Rational Between Two Number"},"content":{"rendered":"

Unit: <\/strong>Number System<\/strong><\/h2>\n

Chapter: <\/strong>Finding a Rational Between Two Numbers<\/strong><\/h3>\n

Reference: – Introduction to Rational Numbers, Finding Rational Numbers Between Two Rational Numbers, Finding Rational Numbers Between Two Irrational Numbers, Finding Rational Numbers Between a Rational and an Irrational Number, Average Method, Denominator Equalization Method, Formula-Based Method, Fraction Insertion Technique, Decimal Approach, Number Line Method, Finding Multiple Rational Numbers, Density Property of Rational Numbers, Solved Examples, Common Mistakes, Practice Grid<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • Introduction to Finding Rational Numbers Between Two Numbers<\/em><\/li>\n
  • Methods: Average Method, Denominator Equalization, Formula Method<\/em><\/li>\n
  • Finding Multiple Rational Numbers Between Two Given Numbers<\/em><\/li>\n
  • Rational Numbers Between Rational, Irrational, and Mixed Pairs<\/em><\/li>\n<\/ul>\n

    Introduction to Finding a Rational Number Between Two Numbers<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    Finding a rational number between two given numbers means identifying or constructing a number that is:<\/p>\n

      \n
    1. Rational (can be expressed as p\/q, q ≠ 0)<\/li>\n
    2. Strictly between the two given numbers (greater than the smaller and less than the larger)<\/li>\n<\/ol>\n

      This is a fundamental skill based on the Density Property of rational numbers.<\/p>\n

      When we find a rational number between two numbers, we essentially ask:<\/p>\n

      "Can I find a fraction that lies between these two values?"<\/p>\n

      The answer is always YES for any two distinct real numbers — and there are infinitely many such rational numbers.<\/p>\n

      Importance<\/u><\/strong><\/p>\n

        \n
      • Demonstrates the density property of rational numbers (rationals are dense on the number line)<\/li>\n
      • Builds foundational skills for limits, sequences, and calculus<\/li>\n
      • Essential for approximation and interpolation in science\/engineering<\/li>\n
      • Frequently asked in competitive exams (finding n rational numbers between two numbers)<\/li>\n
      • Helps understand that between any two numbers, no matter how close, there exists another number<\/li>\n<\/ul>\n

        Example<\/strong><\/p>\n

        Given:<\/strong> 1 and 2
        \nRational number between them:<\/strong> 1.5 = 3\/2<\/p>\n

        \nCommon Property:<\/strong> 1 < 3\/2 < 2 and 3\/2 is rational.<\/p>\n

        So, if someone asked for a rational number between 1\/2 and 3\/4, we could say 5\/8.<\/p>\n

        Subtopics<\/u><\/strong><\/p>\n

        1. Concept of Density of Rational Numbers<\/strong><\/p>\n

        The set of rational numbers is dense<\/strong> in the set of real numbers.<\/p>\n

        Density Property:<\/strong>
        \nBetween any two distinct real numbers (no matter how close), there exists infinitely many<\/strong> rational numbers.<\/p>\n

        Key Points:<\/strong><\/p>\n

          \n
        • Even between 0.123456 and 0.123457, there are rational numbers.<\/li>\n
        • This is true for rational-rational, rational-irrational, and irrational-irrational pairs.<\/li>\n
        • The average (mean) of two distinct numbers always lies strictly between them.<\/li>\n<\/ul>\n
            \n
          1. Finding the Group Basis (Method Selection)<\/strong><\/li>\n<\/ol>\n

            The method you choose depends on the type of numbers given:<\/p>\n\n\n\n\n\n\n\n\n\n
            \n

            Given Numbers<\/p>\n<\/td>\n

            		\n

            Best Method<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

            \n

            Two rational numbers with same denominator<\/p>\n<\/td>\n

            		\n

            Directly pick numerator between<\/p>\n<\/td>\n<\/tr>\n

            \n

            Two rational numbers with different denominators<\/p>\n<\/td>\n

            		\n

            Denominator equalization or average<\/p>\n<\/td>\n<\/tr>\n

            \n

            One rational, one irrational<\/p>\n<\/td>\n

            		\n

            Use decimal approximation<\/p>\n<\/td>\n<\/tr>\n

            \n

            Two irrational numbers<\/p>\n<\/td>\n

            		\n

            Square them or use decimal approximations<\/p>\n<\/td>\n<\/tr>\n

            \n

            Need multiple rational numbers<\/p>\n<\/td>\n

            		\n

            Insert equally spaced fractions<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

            Methods to Find a Rational Number Between Two Numbers<\/strong><\/p>\n

            Method 1: Average Method<\/strong><\/p>\n

            Definition<\/strong><\/p>\n

            The average (arithmetic mean) of two distinct numbers always lies between them.
            \nIf the two numbers are rational, their average is also rational.<\/p>\n

            Formula:<\/strong> For numbers a and b (a < b)
            \nRational number = (a + b) \/ 2<\/p>\n

            Example 1 – Two rational numbers:<\/strong>
            \nFind a rational number between 3\/4 and 5\/6.<\/p>\n

            Average = (3\/4 + 5\/6)\/2 = (9\/12 + 10\/12)\/2 = (19\/12)\/2 = 19\/24<\/p>\n

            Check: 3\/4 = 0.75, 19\/24 ≈ 0.7917, 5\/6 ≈ 0.8333<\/p>\n

            Example 2 – Two integers:<\/strong>
            \nBetween 5 and 6: (5+6)\/2 = 5.5 = 11\/2<\/p>\n

            Example 3 – Negative numbers:<\/strong>
            \nBetween -3 and -2: (-3 + -2)\/2 = -2.5 = -5\/2<\/p>\n

            Quick Tip:<\/strong>
            \nThe average method is the simplest and most reliable for finding exactly one rational number between two given numbers.<\/p>\n

            Method 2: Denominator Equalization Method<\/strong><\/p>\n

            Definition<\/u><\/strong><\/p>\n

            To find rational numbers between two rational numbers with different denominators, convert them to equivalent fractions with the same denominator, then find fractions with numerators between.<\/p>\n

            Steps:<\/strong><\/p>\n

              \n
            1. Write both fractions with a common denominator (preferably the LCM)<\/li>\n
            2. Identify integers between the two numerators<\/li>\n
            3. Write new fractions with same denominator<\/li>\n
            4.  <\/li>\n<\/ol>\n

              Example:<\/strong> Find a rational number between 3\/4 and 5\/6<\/p>\n\n\n\n\n\n\n\n
              \n

              Step<\/p>\n<\/td>\n

              		\n

              Calculation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              Common denominator<\/p>\n<\/td>\n

              		\n

              LCM(4,6) = 12<\/p>\n<\/td>\n<\/tr>\n

              \n

              Convert<\/p>\n<\/td>\n

              		\n

              3\/4 = 9\/12, 5\/6 = 10\/12<\/p>\n<\/td>\n<\/tr>\n

              \n

              Between numerators<\/p>\n<\/td>\n

              		\n

              9 and 10 → no integer between!<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              Problem: No integer between 9 and 10. What to do?<\/em><\/p>\n

              Solution – Expand further:<\/strong>
              \nWrite with denominator 24:
              \n3\/4 = 18\/24, 5\/6 = 20\/24
              \nNow numerators: 18 and 20 → integer 19 exists.
              \nRational number = 19\/24<\/p>\n

              To find multiple rational numbers:<\/strong> Choose an even larger denominator.<\/p>\n

              Quick Rule:<\/strong>
              \nIf you need n<\/strong> rational numbers between a\/b and c\/d, choose denominator = (n+1) × LCM(b,d) or larger.<\/p>\n

              Method 3: Formula Method (For Multiple Rational Numbers)<\/strong><\/p>\n

              Definition<\/u><\/strong><\/p>\n

              To find n rational numbers between two rational numbers a and b (where a < b):<\/p>\n

              Let d = (b – a) \/ (n + 1)
              \nThen the n rational numbers are:
              \na + d, a + 2d, a + 3d, …, a + n×d<\/p>\n

              These numbers are equally spaced between a and b.<\/p>\n

              Example:<\/strong> Find 3 rational numbers between 2 and 3.<\/p>\n\n\n\n\n\n\n\n\n
              \n

              Step<\/p>\n<\/td>\n

              		\n

              Calculation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              Given<\/p>\n<\/td>\n

              		\n

              a = 2, b = 3, n = 3<\/p>\n<\/td>\n<\/tr>\n

              \n

              d = (b-a)\/(n+1)<\/p>\n<\/td>\n

              		\n

              d = (3-2)\/(3+1) = 1\/4 = 0.25<\/p>\n<\/td>\n<\/tr>\n

              \n

              Numbers<\/p>\n<\/td>\n

              		\n

              2.25, 2.50, 2.75<\/p>\n<\/td>\n<\/tr>\n

              \n

              As fractions<\/p>\n<\/td>\n

              		\n

              9\/4, 5\/2, 11\/4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              Check: 2 < 9\/4=2.25 < 5\/2=2.5 < 11\/4=2.75 < 3<\/p>\n

               <\/p>\n

              Example:<\/strong> Find 4 rational numbers between 1 and 2.<\/p>\n

              d = (2-1)\/(4+1) = 1\/5 = 0.2
              \nNumbers: 1.2, 1.4, 1.6, 1.8 = 6\/5, 7\/5, 8\/5, 9\/5<\/p>\n

               <\/p>\n

              Method 4: Fraction Insertion Technique<\/strong><\/p>\n

              Definition<\/u><\/strong><\/p>\n

              A special trick: For fractions a\/b and c\/d (in lowest terms),
              \na\/b < (a+c)\/(b+d) < c\/d<\/p>\n

              This is called the mediant or Farey sum.<\/p>\n

              Example:<\/strong> Between 3\/4 and 5\/6<\/p>\n

              Mediant = (3+5)\/(4+6) = 8\/10 = 4\/5 = 0.8<\/p>\n

              Check: 3\/4=0.75 < 0.8 < 5\/6≈0.8333<\/p>\n

               <\/p>\n

              To find more rational numbers:<\/strong><\/p>\n

                \n
              • Between 3\/4 and 4\/5 → (3+4)\/(4+5)=7\/9≈0.7778<\/li>\n
              • Between 4\/5 and 5\/6 → (4+5)\/(5+6)=9\/11≈0.8182<\/li>\n<\/ul>\n

                Special Note:<\/strong>
                \nThe mediant is always between the two fractions, but not necessarily the average. It works well for finding additional fractions.<\/p>\n

                Method 5: Decimal Approach<\/strong><\/p>\n

                Definition<\/u><\/strong><\/p>\n

                Convert both numbers to decimals, then choose a terminating or repeating decimal between them. Convert back to fraction.<\/p>\n

                Example 1 – Rational numbers:<\/strong>
                \nBetween 1\/3 (0.3333…) and 1\/2 (0.5)
                \nChoose 0.4 = 2\/5, or 0.45 = 9\/20, or 0.375 = 3\/8<\/p>\n

                Example 2 – Rational and Irrational (√2):<\/strong>
                \nBetween 1.4 and √2 ≈ 1.4142135…
                \nChoose 1.41 = 141\/100<\/p>\n

                Example 3 – Two Irrationals (π and e):<\/strong>
                \nπ ≈ 3.14159, e ≈ 2.71828 — wait, π > e. But careful!
                \nActually π ≈ 3.1416, e ≈ 2.7183. So e < π.
                \nBetween 2.7183 and 3.1416 → choose 3.0 = 3<\/p>\n

                Quick Tip:<\/strong>
                \nThe decimal method works for any pair of real numbers (rational or irrational), but you must round the irrationals carefully.<\/p>\n

                Special Cases<\/strong><\/p>\n

                Case 1: Finding Rational Numbers Between Two Rational Numbers with Same Denominator<\/strong><\/p>\n

                Example:<\/strong> Between 5\/8 and 7\/8<\/p>\n

                Numerators: 5 and 7 → integer 6 exists
                \nRational number = 6\/8 = ¾<\/p>\n

                To find multiple:<\/strong> Increase denominator
                \nWrite as 10\/16, 14\/16 → between: 11\/16, 12\/16=3\/4, 13\/16<\/p>\n

                Case 2: Negative Rational Numbers<\/strong><\/p>\n

                Example:<\/strong> Between -2\/3 and -1\/3<\/p>\n

                With denominator 3, numerators: -2 and -1 → integer ? No integer -1.5? Wait, integers are -2 and -1. There is no integer between -2 and -1. So expand denominator.<\/p>\n

                Use denominator 6: -2\/3 = -4\/6, -1\/3 = -2\/6
                \nNumerators: -4, -3, -2 → integer -3 exists → -3\/6 = -1\/2<\/p>\n

                Check: -2\/3 ≈ -0.667 < -0.5 < -0.333 \u2713<\/p>\n

                Case 3: Irrationals That Are Square Roots<\/strong><\/p>\n

                Example:<\/strong> Find a rational number between √2 and √3<\/p>\n

                √2 ≈ 1.4142, √3 ≈ 1.7320
                \nChoose 1.5 = 3\/2, or 1.6 = 8\/5, or 1.7 = 17\/10<\/p>\n

                Check: √2 ≈1.414 < 1.5 < 1.732 ≈ √3<\/p>\n

                Alternative method (squaring):<\/strong>
                \nSince 2 < (rational)² < 3. Find a perfect square between 2 and 3? None. But 2.25 is between, so √2.25 = 1.5 works. So pick rational = 3\/2.<\/p>\n

                Case 4: Very Close Numbers<\/strong><\/p>\n

                Example:<\/strong> Between 0.1234 and 0.1235<\/p>\n

                Average = (0.1234 + 0.1235)\/2 = 0.12345 = 12345\/100000 = 2469\/20000<\/p>\n

                Even closer:<\/strong> Between 1\/1000 and 1\/1001<\/p>\n

                Average = (1\/1000 + 1\/1001)\/2 = (1001+1000)\/(1000×1001×2) = 2001\/2002000<\/p>\n

                Finding Multiple Rational Numbers (General Formula)<\/strong><\/p>\n

                To find n rational numbers between two rational numbers p and q (p < q):<\/strong><\/p>\n

                Step 1:<\/strong> Compute d = (q – p) \/ (n + 1)<\/p>\n

                Step 2:<\/strong> Required numbers: p + d, p + 2d, …, p + n×d<\/p>\n

                Example:<\/strong> Find 5 rational numbers between 1\/2 and 3\/4<\/p>\n\n\n\n\n\n\n\n\n
                \n

                Step<\/p>\n<\/td>\n

                		\n

                Calculation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                \n

                Convert to decimals (optional)<\/p>\n<\/td>\n

                		\n

                0.5 and 0.75<\/p>\n<\/td>\n<\/tr>\n

                \n

                d = (0.75-0.5)\/(5+1)<\/p>\n<\/td>\n

                		\n

                d = 0.25\/6 = 0.041666… = 1\/24<\/p>\n<\/td>\n<\/tr>\n

                \n

                Numbers (decimals)<\/p>\n<\/td>\n

                		\n

                0.54167, 0.58333, 0.625, 0.66667, 0.70833<\/p>\n<\/td>\n<\/tr>\n

                \n

                As fractions<\/p>\n<\/td>\n

                		\n

                13\/24, 14\/24=7\/12, 15\/24=5\/8, 16\/24=2\/3, 17\/24<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                 <\/p>\n

                Check:<\/strong> 1\/2 = 12\/24 < 13\/24 < 14\/24 < 15\/24 < 16\/24 < 17\/24 < 18\/24 = 3\/4<\/p>\n

                Solved Examples<\/strong><\/p>\n

                Example 1:<\/strong> Find one rational number between 2\/5 and 3\/5.<\/p>\n

                Solution (Average):<\/strong> (2\/5 + 3\/5)\/2 = (5\/5)\/2 = 1\/2 = 0.5
                \nSolution (Denominator method):<\/strong> Numerator between 2 and 3? None. Expand to denominator 10: 4\/10 and 6\/10 → 5\/10 = ½<\/p>\n

                Answer:<\/strong> 1\/2<\/p>\n

                Example 2:<\/strong> Find three rational numbers between -1 and 0.<\/p>\n

                Solution (Formula method):<\/strong> a = -1, b = 0, n = 3
                \nd = (0 – (-1))\/(3+1) = 1\/4 = 0.25
                \nNumbers: -0.75, -0.5, -0.25 = -3\/4, -1\/2, -1\/4<\/p>\n

                Answer:<\/strong> -3\/4, -1\/2, -1\/4<\/p>\n

                Example 3:<\/strong> Find two rational numbers between √5 and √6.<\/p>\n

                Solution (Decimal method):<\/strong>
                \n√5 ≈ 2.23607, √6 ≈ 2.44949
                \nChoose 2.3 = 23\/10 and 2.4 = 12\/5<\/p>\n

                Check:<\/strong> 2.236 < 2.3 < 2.4 < 2.449<\/p>\n

                Answer:<\/strong> 23\/10 and 12\/5<\/p>\n

                Example 4:<\/strong> Find four rational numbers between 2\/3 and 5\/6.<\/p>\n

                Solution:<\/strong>
                \nCommon denominator: LCM(3,6)=6 → 2\/3=4\/6, 5\/6=5\/6 (no integer between 4 and 5)
                \nUse denominator 12: 4\/6=8\/12, 5\/6=10\/12 → between numerators 8 and 10 → 9\/12=3\/4 (only one)
                \nNeed 4 numbers → use denominator 30 (since n+1=5, multiply):
                \n2\/3 = 20\/30, 5\/6 = 25\/30<\/p>\n

                \nNumerators between 20 and 25: 21,22,23,24
                \nNumbers: 21\/30=7\/10, 22\/30=11\/15, 23\/30, 24\/30=4\/5<\/p>\n

                Answer:<\/strong> 7\/10, 11\/15, 23\/30, 4\/5<\/p>\n

                Example 6 – Odd One Out Style Problem:<\/strong><\/p>\n

                Examine the five pairs below. In each pair, a rational number is claimed to lie between the two given numbers. Exactly one of these claims is FALSE. Identify the false claim and justify with three independent reasons.<\/p>\n

                 <\/p>\n\n\n\n\n\n\n\n\n\n
                \n

                Item<\/p>\n<\/td>\n

                		\n

                Given Numbers<\/p>\n<\/td>\n

                		\n

                Claimed Rational Number<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                \n

                1<\/p>\n<\/td>\n

                		\n

                1\/4 and 1\/2<\/p>\n<\/td>\n

                		\n

                1\/3<\/p>\n<\/td>\n<\/tr>\n

                \n

                2<\/p>\n<\/td>\n

                		\n

                √2 and √3<\/p>\n<\/td>\n

                		\n

                1.7<\/p>\n<\/td>\n<\/tr>\n

                \n

                3<\/p>\n<\/td>\n

                		\n

                3\/5 and 4\/5<\/p>\n<\/td>\n

                		\n

                5\/8<\/p>\n<\/td>\n<\/tr>\n

                \n

                4<\/p>\n<\/td>\n

                		\n

                -2 and -1<\/p>\n<\/td>\n

                		\n

                -1.5<\/p>\n<\/td>\n<\/tr>\n

                \n

                5<\/p>\n<\/td>\n

                		\n

                0.99 and 1.01<\/p>\n<\/td>\n

                		\n

                1.0<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                Solution:<\/strong><\/p>\n

                (A) Numerical verification (inequality check):<\/strong><\/p>\n

                  \n
                1. 1\/4=0.25 < 1\/3≈0.333 < 0.5 \u2713 True<\/li>\n
                2. √2≈1.414 < 1.7 < 1.732≈√3 \u2713 True<\/li>\n
                3. 3\/5=0.6, 4\/5=0.8, 5\/8=0.625 → 0.6 < 0.625 < 0.8 \u2713 True<\/li>\n
                4. -2 < -1.5 < -1 \u2713 True<\/li>\n
                5. 0.99 < 1.0 < 1.01 \u2713 True<\/li>\n<\/ol>\n

                   <\/p>\n

                   <\/p>\n","protected":false},"excerpt":{"rendered":"

                  Unit: Number System Chapter: Finding a Rational Between Two Numbers Reference: – Introduction to Rational Numbers, Finding Rational Numbers Between Two Rational Numbers, Finding Rational Numbers Between Two Irrational Numbers, Finding Rational Numbers Between a Rational and an Irrational Number, Average Method, Denominator Equalization Method, Formula-Based Method, Fraction Insertion Technique, Decimal Approach, Number Line Method, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[593],"tags":[],"class_list":["post-9137","post","type-post","status-publish","format-standard","hentry","category-grade-8"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9137","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9137"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9137\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}