{"id":9913,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9913"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","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":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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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\u00a0(can be expressed as p\/q, q \u2260 0)<\/li>\n
    2. Strictly between\u00a0the two given numbers (greater than the smaller and less than the larger)<\/li>\n<\/ol>\n

      This is a fundamental skill based on the\u00a0Density Property\u00a0of 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\u00a0YES\u00a0for any two distinct real numbers \u2014 and there are infinitely many such rational numbers.<\/p>\n

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

        \n
      • Demonstrates the\u00a0density property\u00a0of 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>\u00a01 and 2
        \nRational number between them:<\/strong>\u00a01.5 = 3\/2<\/p>\n

        \nCommon Property:<\/strong>\u00a01 < 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\u00a0dense<\/strong>\u00a0in the set of real numbers.<\/p>\n

        Density Property:<\/strong>
        \nBetween any two distinct real numbers (no matter how close), there exists\u00a0infinitely many<\/strong>\u00a0rational 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
            \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

            Source: Kapdec.com<\/p>\n<\/div>\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>\u00a0For numbers a and b (a < b)
            \nRational number = (a + b) \/ 2<\/p>\n

            Example 1 \u2013 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 \u2248 0.7917, 5\/6 \u2248 0.8333<\/p>\n

            Example 2 \u2013 Two integers:<\/strong>
            \nBetween 5 and 6: (5+6)\/2 = 5.5 = 11\/2<\/p>\n

            Example 3 \u2013 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\u00a0one\u00a0rational 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\u00a0different denominators, convert them to\u00a0equivalent fractions with the same denominator, then find fractions with numerators between.<\/p>\n

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

              \n
            1. Write both fractions with a\u00a0common denominator\u00a0(preferably the LCM)<\/li>\n
            2. Identify integers between the two numerators<\/li>\n
            3. Write new fractions with same denominator<\/li>\n
            4. \u00a0<\/li>\n<\/ol>\n

              Example:<\/strong>\u00a0Find a rational number between 3\/4 and 5\/6<\/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

              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 \u2192 no integer between!<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              Source: Kapdec.com<\/p>\n<\/div>\n

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

              Solution \u2013 Expand further:<\/strong>
              \nWrite with denominator 24:
              \n3\/4 = 18\/24, 5\/6 = 20\/24
              \nNow numerators: 18 and 20 \u2192 integer 19 exists.
              \nRational number = 19\/24<\/p>\n

              To find multiple rational numbers:<\/strong>\u00a0Choose an even larger denominator.<\/p>\n

              Quick Rule:<\/strong>
              \nIf you need\u00a0n<\/strong>\u00a0rational numbers between a\/b and c\/d, choose denominator = (n+1) \u00d7 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\u00a0n rational numbers\u00a0between two rational numbers\u00a0a\u00a0and\u00a0b\u00a0(where a < b):<\/p>\n

              Let d = (b – a) \/ (n + 1)
              \nThen the n rational numbers are:
              \na + d, a + 2d, a + 3d, \u2026, a + n\u00d7d<\/p>\n

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

              Example:<\/strong>\u00a0Find 3 rational numbers between 2 and 3.<\/p>\n

              \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

              Source: Kapdec.com<\/p>\n<\/div>\n

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

              \u00a0<\/p>\n

              Example:<\/strong>\u00a0Find 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

              \u00a0<\/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\u00a0mediant\u00a0or Farey sum.<\/p>\n

              Example:<\/strong>\u00a0Between 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\u22480.8333<\/p>\n

              \u00a0<\/p>\n

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

                \n
              • Between 3\/4 and 4\/5 \u2192 (3+4)\/(4+5)=7\/9\u22480.7778<\/li>\n
              • Between 4\/5 and 5\/6 \u2192 (4+5)\/(5+6)=9\/11\u22480.8182<\/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 \u2013 Rational numbers:<\/strong>
                \nBetween 1\/3 (0.3333\u2026) and 1\/2 (0.5)
                \nChoose 0.4 = 2\/5, or 0.45 = 9\/20, or 0.375 = 3\/8<\/p>\n

                Example 2 \u2013 Rational and Irrational (\u221a2):<\/strong>
                \nBetween 1.4 and \u221a2 \u2248 1.4142135\u2026
                \nChoose 1.41 = 141\/100<\/p>\n

                Example 3 \u2013 Two Irrationals (\u03c0 and e):<\/strong>
                \n\u03c0 \u2248 3.14159, e \u2248 2.71828 \u2014 wait, \u03c0 > e. But careful!
                \nActually \u03c0 \u2248 3.1416, e \u2248 2.7183. So e < \u03c0.
                \nBetween 2.7183 and 3.1416 \u2192 choose 3.0 = 3<\/p>\n

                Quick Tip:<\/strong>
                \nThe decimal method works for\u00a0any\u00a0pair 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>\u00a0Between 5\/8 and 7\/8<\/p>\n

                Numerators: 5 and 7 \u2192 integer 6 exists
                \nRational number = 6\/8 = \u00be<\/p>\n

                To find multiple:<\/strong>\u00a0Increase denominator
                \nWrite as 10\/16, 14\/16 \u2192 between: 11\/16, 12\/16=3\/4, 13\/16<\/p>\n

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

                Example:<\/strong>\u00a0Between -2\/3 and -1\/3<\/p>\n

                With denominator 3, numerators: -2 and -1 \u2192 integer ? No integer -1.5? Wait, integers are -2 and -1. There is\u00a0no integer\u00a0between -2 and -1. So expand denominator.<\/p>\n

                Use denominator 6: -2\/3 = -4\/6, -1\/3 = -2\/6
                \nNumerators: -4, -3, -2 \u2192 integer -3 exists \u2192 -3\/6 = -1\/2<\/p>\n

                Check: -2\/3 \u2248 -0.667 < -0.5 < -0.333 \u2713<\/p>\n

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

                Example:<\/strong>\u00a0Find a rational number between \u221a2 and \u221a3<\/p>\n

                \u221a2 \u2248 1.4142, \u221a3 \u2248 1.7320
                \nChoose 1.5 = 3\/2, or 1.6 = 8\/5, or 1.7 = 17\/10<\/p>\n

                Check: \u221a2 \u22481.414 < 1.5 < 1.732 \u2248 \u221a3<\/p>\n

                Alternative method (squaring):<\/strong>
                \nSince 2 < (rational)\u00b2 < 3. Find a perfect square between 2 and 3? None. But 2.25 is between, so \u221a2.25 = 1.5 works. So pick rational = 3\/2.<\/p>\n

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

                Example:<\/strong>\u00a0Between 0.1234 and 0.1235<\/p>\n

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

                Even closer:<\/strong>\u00a0Between 1\/1000 and 1\/1001<\/p>\n

                Average = (1\/1000 + 1\/1001)\/2 = (1001+1000)\/(1000\u00d71001\u00d72) = 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>\u00a0Compute d = (q – p) \/ (n + 1)<\/p>\n

                Step 2:<\/strong>\u00a0Required numbers: p + d, p + 2d, \u2026, p + n\u00d7d<\/p>\n

                Example:<\/strong>\u00a0Find 5 rational numbers between 1\/2 and 3\/4<\/p>\n

                \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\u2026 = 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

                Source: Kapdec.com<\/p>\n<\/div>\n

                \u00a0<\/p>\n

                Check:<\/strong>\u00a01\/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>\u00a0Find one rational number between 2\/5 and 3\/5.<\/p>\n

                Solution (Average):<\/strong>\u00a0(2\/5 + 3\/5)\/2 = (5\/5)\/2 = 1\/2 = 0.5
                \nSolution (Denominator method):<\/strong>\u00a0Numerator between 2 and 3? None. Expand to denominator 10: 4\/10 and 6\/10 \u2192 5\/10 = \u00bd<\/p>\n

                Answer:<\/strong>\u00a01\/2<\/p>\n

                Example 2:<\/strong>\u00a0Find three rational numbers between -1 and 0.<\/p>\n

                Solution (Formula method):<\/strong>\u00a0a = -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>\u00a0-3\/4, -1\/2, -1\/4<\/p>\n

                Example 3:<\/strong>\u00a0Find two rational numbers between \u221a5 and \u221a6.<\/p>\n

                Solution (Decimal method):<\/strong>
                \n\u221a5 \u2248 2.23607, \u221a6 \u2248 2.44949
                \nChoose 2.3 = 23\/10 and 2.4 = 12\/5<\/p>\n

                Check:<\/strong>\u00a02.236 < 2.3 < 2.4 < 2.449<\/p>\n

                Answer:<\/strong>\u00a023\/10 and 12\/5<\/p>\n

                Example 4:<\/strong>\u00a0Find four rational numbers between 2\/3 and 5\/6.<\/p>\n

                Solution:<\/strong>
                \nCommon denominator: LCM(3,6)=6 \u2192 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 \u2192 between numerators 8 and 10 \u2192 9\/12=3\/4 (only one)
                \nNeed 4 numbers \u2192 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>\u00a07\/10, 11\/15, 23\/30, 4\/5<\/p>\n

                Example 6 \u2013 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

                \u00a0<\/p>\n

                \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

                \u221a2 and \u221a3<\/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

                Source: Kapdec.com<\/p>\n<\/div>\n

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

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

                  \n
                1. 1\/4=0.25 < 1\/3\u22480.333 < 0.5 \u2713 True<\/li>\n
                2. \u221a2\u22481.414 < 1.7 < 1.732\u2248\u221a3 \u2713 True<\/li>\n
                3. 3\/5=0.6, 4\/5=0.8, 5\/8=0.625 \u2192 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

                  \u00a0<\/p>\n

                  \u00a0<\/p>\n

                  <\/p>\n