{"id":9912,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9912"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"inverse-and-reciprocal-of-rationals","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/inverse-and-reciprocal-of-rationals\/","title":{"rendered":"Inverse And Reciprocal Of Rationals"},"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>Inverse & Reciprocal of Rational<\/strong><\/h3>\n

Reference: – Introduction to Inverse & Reciprocal, Multiplicative Inverse Definition, Additive Inverse Definition, Difference Between Inverse and Reciprocal, Reciprocal of Rational Numbers, Finding Reciprocal of Fractions, Negative Reciprocals, Reciprocal of Zero (Undefined), Reciprocal of Integers, Reciprocal of Negative Rationals, Reciprocal of Decimal Numbers, Properties of Reciprocals, Applications of Reciprocals, Solved Examples, Odd-One-Out Problems, Common Mistakes, Practice Grid<\/em><\/p>\n

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

    \n
  • Introduction to Additive & Multiplicative Inverses<\/em><\/li>\n
  • Reciprocal as Multiplicative Inverse of Rational Numbers<\/em><\/li>\n
  • Finding Reciprocal of Fractions, Integers, Decimals, and Negative Numbers<\/em><\/li>\n
  • Properties and Applications of Reciprocals<\/em><\/li>\n<\/ul>\n

    Introduction to Inverse & Reciprocal<\/strong><\/p>\n

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

    In the context of rational numbers, “inverse” can refer to two different concepts:<\/p>\n

    \n\n\n\n\n\n\n
    \n

    Type<\/p>\n<\/td>\n

    		\n

    Name<\/p>\n<\/td>\n

    		\n

    Operation<\/p>\n<\/td>\n

    		\n

    Definition<\/p>\n<\/td>\n

    		\n

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

    \n

    Additive Inverse<\/strong><\/p>\n<\/td>\n

    		\n

    Opposite<\/p>\n<\/td>\n

    		\n

    Addition<\/p>\n<\/td>\n

    		\n

    The number that when added to the original gives\u00a00<\/strong><\/p>\n<\/td>\n

    		\n

    Additive inverse of 5 is -5 (5 + (-5) = 0)<\/p>\n<\/td>\n<\/tr>\n

    \n

    Multiplicative Inverse<\/strong><\/p>\n<\/td>\n

    		\n

    Reciprocal<\/p>\n<\/td>\n

    		\n

    Multiplication<\/p>\n<\/td>\n

    		\n

    The number that when multiplied by the original gives\u00a01<\/strong><\/p>\n<\/td>\n

    		\n

    Multiplicative inverse of 5 is 1\/5 (5 \u00d7 1\/5 = 1)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

    When we study\u00a0reciprocal of rational, we specifically study the\u00a0multiplicative inverse.<\/p>\n

    When we classify inverses\/reciprocals, we essentially ask:<\/p>\n

    “What number combines with the given number to yield the identity element (0 for addition, 1 for multiplication)?”<\/p>\n

    Once we identify the inverse, we can solve equations, simplify expressions, and understand number relationships.<\/p>\n

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

      \n
    • Essential for solving linear equations (dividing by a number = multiplying by reciprocal)<\/li>\n
    • Used in ratio and proportion problems<\/li>\n
    • Critical for understanding division of fractions<\/li>\n
    • Appears in physics (resistance, parallel circuits, optics)<\/li>\n
    • Used in financial mathematics (interest rates, exchange rates)<\/li>\n
    • Foundation for calculus (derivatives of reciprocal functions)<\/li>\n<\/ul>\n

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

      Group:<\/strong>\u00a0{2, 3, 4, 5}
      \nReciprocals:<\/strong>\u00a0{1\/2, 1\/3, 1\/4, 1\/5}
      \nCommon Property:<\/strong>\u00a0Each is the multiplicative inverse of the original.<\/p>\n

      So, if “0” was given as a number, its reciprocal is undefined (does not belong).<\/p>\n

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

      1. Concept of Additive Inverse<\/strong><\/p>\n

      The additive inverse of a rational number\u00a0a<\/strong>\u00a0is\u00a0-a<\/strong>.<\/p>\n

      \n\n\n\n\n\n\n\n
      \n

      Original<\/p>\n<\/td>\n

      		\n

      Additive Inverse<\/p>\n<\/td>\n

      		\n

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

      \n

      7<\/p>\n<\/td>\n

      		\n

      -7<\/p>\n<\/td>\n

      		\n

      7 + (-7) = 0<\/p>\n<\/td>\n<\/tr>\n

      \n

      -3\/4<\/p>\n<\/td>\n

      		\n

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

      		\n

      -3\/4 + 3\/4 = 0<\/p>\n<\/td>\n<\/tr>\n

      \n

      0<\/p>\n<\/td>\n

      		\n

      0<\/p>\n<\/td>\n

      		\n

      0 + 0 = 0<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

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

        \n
      • Every rational number has a unique additive inverse.<\/li>\n
      • Additive inverse is also called the\u00a0opposite.<\/li>\n
      • On the number line, additive inverses are symmetric about 0.<\/li>\n<\/ul>\n

        2. Concept of Multiplicative Inverse (Reciprocal)<\/strong><\/p>\n

        The multiplicative inverse (reciprocal) of a non-zero rational number\u00a0a<\/strong>\u00a0is\u00a01\/a<\/strong>.<\/p>\n

        The reciprocal of a fraction\u00a0p\/q<\/strong>\u00a0(p \u2260 0, q \u2260 0) is\u00a0q\/p<\/strong>.<\/p>\n

        \n\n\n\n\n\n\n\n\n
        \n

        Original<\/p>\n<\/td>\n

        		\n

        Reciprocal<\/p>\n<\/td>\n

        		\n

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

        \n

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

        		\n

        1\/5<\/p>\n<\/td>\n

        		\n

        5 \u00d7 1\/5 = 1<\/p>\n<\/td>\n<\/tr>\n

        \n

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

        		\n

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

        		\n

        2\/3 \u00d7 3\/2 = 1<\/p>\n<\/td>\n<\/tr>\n

        \n

        -4\/7<\/p>\n<\/td>\n

        		\n

        -7\/4<\/p>\n<\/td>\n

        		\n

        (-4\/7) \u00d7 (-7\/4) = 1<\/p>\n<\/td>\n<\/tr>\n

        \n

        0.25 = 1\/4<\/p>\n<\/td>\n

        		\n

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

        		\n

        0.25 \u00d7 4 = 1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

        Special Note:<\/strong><\/p>\n

          \n
        • Zero has no reciprocal\u00a0(1\/0 is undefined).<\/li>\n
        • The reciprocal of 1 is 1.<\/li>\n
        • The reciprocal of -1 is -1.<\/li>\n<\/ul>\n

          \u00a0<\/p>\n

          Reciprocal of Rational Numbers<\/strong><\/p>\n

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

          The\u00a0reciprocal\u00a0of a rational number is obtained by\u00a0interchanging the numerator and denominator\u00a0(for fractional form) or dividing 1 by the number (for decimal form).<\/p>\n

          Formal Definition:<\/strong>
          \nFor any non-zero rational number a, the reciprocal is the number b such that a \u00d7 b = 1.<\/p>\n

          Importance of Reciprocal<\/strong><\/p>\n

            \n
          • Converts division into multiplication: a \u00f7 b = a \u00d7 (1\/b)<\/li>\n
          • Used to simplify complex fractions<\/li>\n
          • Helps in solving proportions<\/li>\n
          • Essential for rate, speed, and work problems<\/li>\n<\/ul>\n

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

            1. Reciprocal of a Proper Fraction<\/strong><\/p>\n

            For a proper fraction (numerator < denominator), the reciprocal is an\u00a0improper fraction<\/strong>\u00a0(greater than 1).<\/p>\n

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

            \n\n\n\n\n\n\n\n\n
            \n

            Proper Fraction<\/p>\n<\/td>\n

            		\n

            Reciprocal<\/p>\n<\/td>\n

            		\n

            Value of Reciprocal<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

            \n

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

            		\n

            2\/1 = 2<\/p>\n<\/td>\n

            		\n

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

            \n

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

            		\n

            3\/2 = 1.5<\/p>\n<\/td>\n

            		\n

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

            \n

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

            		\n

            5\/3 \u2248 1.667<\/p>\n<\/td>\n

            		\n

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

            \n

            7\/8<\/p>\n<\/td>\n

            		\n

            8\/7 \u2248 1.143<\/p>\n<\/td>\n

            		\n

            > 1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

            Quick Tip:<\/strong>
            \nReciprocal of a proper fraction is always greater than 1.<\/p>\n

            2. Reciprocal of an Improper Fraction<\/strong><\/p>\n

            For an improper fraction (numerator > denominator), the reciprocal is a\u00a0proper fraction\u00a0(less than 1).<\/p>\n

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

            \n\n\n\n\n\n\n\n\n
            \n

            Improper Fraction<\/p>\n<\/td>\n

            		\n

            Reciprocal<\/p>\n<\/td>\n

            		\n

            Value of Reciprocal<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

            \n

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

            		\n

            3\/5 = 0.6<\/p>\n<\/td>\n

            		\n

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

            \n

            7\/4<\/p>\n<\/td>\n

            		\n

            4\/7 \u2248 0.571<\/p>\n<\/td>\n

            		\n

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

            \n

            11\/5<\/p>\n<\/td>\n

            		\n

            5\/11 \u2248 0.455<\/p>\n<\/td>\n

            		\n

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

            \n

            9\/2<\/p>\n<\/td>\n

            		\n

            2\/9 \u2248 0.222<\/p>\n<\/td>\n

            		\n

            < 1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

            Quick Tip:<\/strong>
            \nReciprocal of an improper fraction is always less than 1.<\/p>\n

            3. Reciprocal of an Integer<\/strong><\/p>\n

            An integer n (n \u2260 0) can be written as n\/1, so its reciprocal is 1\/n.<\/p>\n

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

            \n\n\n\n\n\n\n\n\n\n
            \n

            Integer<\/p>\n<\/td>\n

            		\n

            Reciprocal<\/p>\n<\/td>\n

            		\n

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

            \n

            10<\/p>\n<\/td>\n

            		\n

            1\/10 = 0.1<\/p>\n<\/td>\n

            		\u00a0<\/td>\n<\/tr>\n
            \n

            -8<\/p>\n<\/td>\n

            		\n

            -1\/8 = -0.125<\/p>\n<\/td>\n

            		\u00a0<\/td>\n<\/tr>\n
            \n

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

            		\n

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

            		\n

            (self-reciprocal)<\/p>\n<\/td>\n<\/tr>\n

            \n

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

            		\n

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

            		\n

            (self-reciprocal)<\/p>\n<\/td>\n<\/tr>\n

            \n

            0<\/p>\n<\/td>\n

            		\n

            Undefined<\/p>\n<\/td>\n

            		\n

            (no reciprocal)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

            Special Note:<\/strong><\/p>\n

              \n
            • 1 and -1 are the only rational numbers that are their own reciprocals.<\/li>\n
            • Because 1 \u00d7 1 = 1 and (-1) \u00d7 (-1) = 1.<\/li>\n<\/ul>\n

              4. Reciprocal of a Negative Rational Number<\/strong><\/p>\n

              The reciprocal of a negative rational number is also negative.<\/p>\n

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

              \n\n\n\n\n\n\n\n
              \n

              Original<\/p>\n<\/td>\n

              		\n

              Reciprocal<\/p>\n<\/td>\n

              		\n

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

              \n

              -2\/3<\/p>\n<\/td>\n

              		\n

              -3\/2<\/p>\n<\/td>\n

              		\n

              (-2\/3) \u00d7 (-3\/2) = 1<\/p>\n<\/td>\n<\/tr>\n

              \n

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

              		\n

              -1\/5<\/p>\n<\/td>\n

              		\n

              (-5) \u00d7 (-1\/5) = 1<\/p>\n<\/td>\n<\/tr>\n

              \n

              -7\/4<\/p>\n<\/td>\n

              		\n

              -4\/7<\/p>\n<\/td>\n

              		\n

              (-7\/4) \u00d7 (-4\/7) = 1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

              Key Point:<\/strong>
              \nThe sign of the reciprocal is the same as the sign of the original (because product must be positive 1).<\/p>\n

              5. Reciprocal of a Decimal Number<\/strong><\/p>\n

              Convert the decimal to a fraction, then flip.<\/p>\n

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

              \n\n\n\n\n\n\n\n\n\n
              \n

              Decimal<\/p>\n<\/td>\n

              		\n

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

              		\n

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

              \n

              0.5<\/p>\n<\/td>\n

              		\n

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

              		\n

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

              \n

              0.25<\/p>\n<\/td>\n

              		\n

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

              		\n

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

              \n

              0.125<\/p>\n<\/td>\n

              		\n

              1\/8<\/p>\n<\/td>\n

              		\n

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

              \n

              0.75<\/p>\n<\/td>\n

              		\n

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

              		\n

              4\/3 \u2248 1.333<\/p>\n<\/td>\n<\/tr>\n

              \n

              2.5<\/p>\n<\/td>\n

              		\n

              5\/2<\/p>\n<\/td>\n

              		\n

              2\/5 = 0.4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

              Quick Tip:<\/strong>
              \nReciprocal of a decimal between 0 and 1 is > 1.
              \nReciprocal of a decimal > 1 is between 0 and 1.<\/p>\n

              \u00a0<\/p>\n

              Properties of Reciprocals<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n\n\n\n\n
              \n

              Property<\/p>\n<\/td>\n

              		\n

              Statement<\/p>\n<\/td>\n

              		\n

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

              \n

              Non-zero requirement<\/strong><\/p>\n<\/td>\n

              		\n

              0 has no reciprocal<\/p>\n<\/td>\n

              		\n

              1\/0 is undefined<\/p>\n<\/td>\n<\/tr>\n

              \n

              Self-reciprocal<\/strong><\/p>\n<\/td>\n

              		\n

              1 and -1 are their own reciprocals<\/p>\n<\/td>\n

              		\n

              1 \u00d7 1 = 1, (-1) \u00d7 (-1) = 1<\/p>\n<\/td>\n<\/tr>\n

              \n

              Product property<\/strong><\/p>\n<\/td>\n

              		\n

              a \u00d7 (1\/a) = 1<\/p>\n<\/td>\n

              		\n

              5 \u00d7 1\/5 = 1<\/p>\n<\/td>\n<\/tr>\n

              \n

              Reciprocal of reciprocal<\/strong><\/p>\n<\/td>\n

              		\n

              1\/(1\/a) = a (a \u2260 0)<\/p>\n<\/td>\n

              		\n

              1\/(1\/7) = 7<\/p>\n<\/td>\n<\/tr>\n

              \n

              Reciprocal of product<\/strong><\/p>\n<\/td>\n

              		\n

              1\/(a \u00d7 b) = (1\/a) \u00d7 (1\/b)<\/p>\n<\/td>\n

              		\n

              1\/(2\u00d73) = 1\/6 = (1\/2)\u00d7(1\/3)<\/p>\n<\/td>\n<\/tr>\n

              \n

              Reciprocal of quotient<\/strong><\/p>\n<\/td>\n

              		\n

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

              		\n

              1\/(5\/7) = 7\/5<\/p>\n<\/td>\n<\/tr>\n

              \n

              Reciprocal of negative<\/strong><\/p>\n<\/td>\n

              		\n

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

              		\n

              1\/(-4) = -1\/4<\/p>\n<\/td>\n<\/tr>\n

              \n

              Sign preservation<\/strong><\/p>\n<\/td>\n

              		\n

              Sign remains same<\/p>\n<\/td>\n

              		\n

              -3 \u2192 -1\/3<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

              Difference Between Additive Inverse and Multiplicative Inverse (Reciprocal)<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n\n\n\n\n\n
              \n

              Feature<\/p>\n<\/td>\n

              		\n

              Additive Inverse (Opposite)<\/p>\n<\/td>\n

              		\n

              Multiplicative Inverse (Reciprocal)<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              Operation<\/strong><\/p>\n<\/td>\n

              		\n

              Addition<\/p>\n<\/td>\n

              		\n

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

              \n

              Identity element<\/strong><\/p>\n<\/td>\n

              		\n

              0<\/p>\n<\/td>\n

              		\n

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

              \n

              Formula<\/strong><\/p>\n<\/td>\n

              		\n

              -a<\/p>\n<\/td>\n

              		\n

              1\/a (a \u2260 0)<\/p>\n<\/td>\n<\/tr>\n

              \n

              For a = 5<\/strong><\/p>\n<\/td>\n

              		\n

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

              		\n

              1\/5<\/p>\n<\/td>\n<\/tr>\n

              \n

              For a = -3<\/strong><\/p>\n<\/td>\n

              		\n

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

              		\n

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

              \n

              For a = 0<\/strong><\/p>\n<\/td>\n

              		\n

              0 (exists)<\/p>\n<\/td>\n

              		\n

              Undefined (does not exist)<\/p>\n<\/td>\n<\/tr>\n

              \n

              For a = 1<\/strong><\/p>\n<\/td>\n

              		\n

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

              		\n

              1 (same)<\/p>\n<\/td>\n<\/tr>\n

              \n

              For a = -1<\/strong><\/p>\n<\/td>\n

              		\n

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

              		\n

              -1 (same)<\/p>\n<\/td>\n<\/tr>\n

              \n

              Product of a and its inverse<\/strong><\/p>\n<\/td>\n

              		\n

              a + (-a) = 0<\/p>\n<\/td>\n

              		\n

              a \u00d7 (1\/a) = 1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

              Example to distinguish:<\/strong><\/p>\n

              Given number:\u00a02\/3<\/strong><\/p>\n

                \n
              • Additive inverse =\u00a0-2\/3<\/strong>\u00a0\u2192 2\/3 + (-2\/3) = 0<\/li>\n
              • Multiplicative inverse (reciprocal) =\u00a03\/2<\/strong>\u00a0\u2192 2\/3 \u00d7 3\/2 = 1<\/li>\n<\/ul>\n

                Applications of Reciprocals<\/strong><\/p>\n

                1. Division of Fractions<\/strong><\/p>\n

                a\/b \u00f7 c\/d = a\/b \u00d7 d\/c = (a \u00d7 d)\/(b \u00d7 c)<\/p>\n

                Example:<\/strong>\u00a03\/4 \u00f7 2\/5 = 3\/4 \u00d7 5\/2 = 15\/8<\/p>\n

                2. Solving Equations<\/strong><\/p>\n

                If 5x = 15, multiply both sides by reciprocal of 5 (which is 1\/5):
                \nx = 15 \u00d7 1\/5 = 3<\/p>\n

                3. Rate and Work Problems<\/strong><\/p>\n

                If a pipe fills a tank in 3 hours, its filling rate is 1\/3 tank per hour.<\/p>\n

                4. Electrical Resistance (Parallel Circuits)<\/strong><\/p>\n

                1\/R Total = 1\/R\u2081 + 1\/R\u2082 + 1\/R\u2083<\/p>\n

                5. Speed and Time<\/strong><\/p>\n

                If speed = distance\/time, then time = distance \u00d7 (1\/speed)<\/p>\n

                6. Fractions and Ratios<\/strong><\/p>\n

                To find the ratio inverted property: a\/b = c\/d \u21d2 b\/a = d\/c<\/p>\n

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

                Example 1:<\/strong>\u00a0Find the reciprocal of 3\/7.<\/p>\n

                Solution:<\/strong>\u00a0Reciprocal = 7\/3<\/p>\n

                Answer:<\/strong>\u00a07\/3<\/p>\n

                Example 2:<\/strong>\u00a0Find the reciprocal of -5.<\/p>\n

                Solution:<\/strong>\u00a0-5 = -5\/1 \u2192 Reciprocal = -1\/5<\/p>\n

                Answer:<\/strong>\u00a0-1\/5<\/p>\n

                Example 3:<\/strong>\u00a0Find the product of a number and its reciprocal for a = 4\/9.<\/p>\n

                Solution:<\/strong>\u00a0Reciprocal = 9\/4 \u2192 Product = 4\/9 \u00d7 9\/4 = 1<\/p>\n

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

                Example 4:<\/strong>\u00a0Find the additive inverse and multiplicative inverse of -7\/2.<\/p>\n

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

                  \n
                • Additive inverse = 7\/2<\/li>\n
                • Multiplicative inverse (reciprocal) = -2\/7<\/li>\n<\/ul>\n

                  Answer:<\/strong>\u00a0Additive: 7\/2, Multiplicative: -2\/7<\/p>\n

                  Example 5:<\/strong>\u00a0If the reciprocal of (x\/3) is 12\/5, find x.<\/p>\n

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

                  Reciprocal of x\/3 = 3\/x
                  \nGiven: 3\/x = 12\/5
                  \nCross multiply: 3 \u00d7 5 = 12 \u00d7 x \u2192 15 = 12x \u2192 x = 15\/12 = 5\/4<\/p>\n

                  Answer:<\/strong>\u00a0x = 5\/4<\/p>\n

                  Example 6:<\/strong>\u00a0Find the reciprocal of 0.2.<\/p>\n

                  Solution:<\/strong>\u00a00.2 = 2\/10 = 1\/5 \u2192 Reciprocal = 5<\/p>\n

                  Answer:<\/strong>\u00a05<\/p>\n

                  Example 7:<\/strong>\u00a0Which is greater: the reciprocal of 2\/3 or the reciprocal of 3\/4?<\/p>\n

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

                  Reciprocal of 2\/3 = 3\/2 = 1.5
                  \nReciprocal of 3\/4 = 4\/3 \u2248 1.333
                  \n1.5 > 1.333, so 3\/2 > 4\/3<\/p>\n

                  Answer:<\/strong>\u00a0Reciprocal of 2\/3 is greater.<\/p>\n

                  Example 8 \u2013 Odd One Out Style Problem:<\/strong><\/p>\n

                  Examine the five items below. Each row shows a number and its claimed reciprocal. Exactly one row has an INCORRECT reciprocal. Identify it and give three independent reasons (A) definition check, (B) product verification, (C) property-based reasoning).<\/strong><\/p>\n

                  \n\n\n\n\n\n\n\n\n\n
                  \n

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

                  		\n

                  Original Number<\/p>\n<\/td>\n

                  		\n

                  Claimed Reciprocal<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                  \n

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

                  		\n

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

                  		\n

                  4\/3<\/p>\n<\/td>\n<\/tr>\n

                  \n

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

                  		\n

                  -7<\/p>\n<\/td>\n

                  		\n

                  -1\/7<\/p>\n<\/td>\n<\/tr>\n

                  \n

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

                  		\n

                  0<\/p>\n<\/td>\n

                  		\n

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

                  \n

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

                  		\n

                  2.5<\/p>\n<\/td>\n

                  		\n

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

                  \n

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

                  		\n

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

                  		\n

                  -1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

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

                  (A) Definition check (interchanging numerator\/denominator or 1\/a):<\/strong><\/p>\n

                    \n
                  1. 3\/4 \u2192 4\/3 \u2713 Correct<\/li>\n
                  2. -7 \u2192 -1\/7 \u2713 Correct<\/li>\n
                  3. 0 \u2192 0 ? 1\/0 is undefined, so 0 is NOT the reciprocal of 0 \u2717 Incorrect<\/li>\n
                  4. 2.5 = 5\/2 \u2192 2\/5 = 0.4 \u2713 Correct<\/li>\n
                  5. -1 \u2192 -1 \u2713 Correct (since -1 \u00d7 -1 = 1)<\/li>\n<\/ol>\n

                    (B) Product verification (original \u00d7 reciprocal should = 1):<\/strong><\/p>\n

                      \n
                    1. 3\/4 \u00d7 4\/3 = 1 \u2713<\/li>\n
                    2. -7 \u00d7 (-1\/7) = 1 \u2713<\/li>\n
                    3. 0 \u00d7 0 = 0 \u2717 (should be 1, but 0 \u2260 1)<\/li>\n
                    4. 2.5 \u00d7 0.4 = 1 \u2713<\/li>\n
                    5. -1 \u00d7 -1 = 1 \u2713<\/li>\n<\/ol>\n

                      (C) Property-based reasoning:<\/strong><\/p>\n

                        \n
                      • The reciprocal of a non-zero number is defined as 1\/a.<\/li>\n
                      • 0 has no reciprocal because 1\/0 is undefined in rational numbers.<\/li>\n
                      • Saying “reciprocal of 0 is 0” violates the fundamental property that a \u00d7 (reciprocal) = 1.<\/li>\n
                      • Among all items, only item 3 involves zero, which is the only rational number without a reciprocal.<\/li>\n<\/ul>\n

                        Conclusion:<\/strong>\u00a0Item 3 is the odd one out because it incorrectly claims that 0 is the reciprocal of 0 (when in fact 0 has no reciprocal).<\/p>\n

                        Example 9 \u2013 Odd One Out (More Complex):<\/strong><\/p>\n

                        Examine the six rational numbers below. Exactly one does NOT have a reciprocal that belongs to a specific property group. Identify the odd one out and justify.<\/strong><\/p>\n

                        Items:<\/strong>\u00a0{ 1\/2, 2\/3, 3\/4, 4\/5, 5\/6, 1 }<\/p>\n

                        Property to check:<\/strong>\u00a0“Reciprocal is greater than the original number”<\/p>\n

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

                        \n\n\n\n\n\n\n\n\n\n\n
                        \n

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

                        		\n

                        Reciprocal<\/p>\n<\/td>\n

                        		\n

                        Original vs Reciprocal<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

                        \n

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

                        		\n

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

                        		\n

                        0.5 < 2 \u2192 Reciprocal > Original \u2713<\/p>\n<\/td>\n<\/tr>\n

                        \n

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

                        		\n

                        3\/2 = 1.5<\/p>\n<\/td>\n

                        		\n

                        0.667 < 1.5 \u2192 Reciprocal > Original \u2713<\/p>\n<\/td>\n<\/tr>\n

                        \n

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

                        		\n

                        4\/3 \u2248 1.333<\/p>\n<\/td>\n

                        		\n

                        0.75 < 1.333 \u2192 Reciprocal > Original \u2713<\/p>\n<\/td>\n<\/tr>\n

                        \n

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

                        		\n

                        5\/4 = 1.25<\/p>\n<\/td>\n

                        		\n

                        0.8 < 1.25 \u2192 Reciprocal > Original \u2713<\/p>\n<\/td>\n<\/tr>\n

                        \n

                        5\/6<\/p>\n<\/td>\n

                        		\n

                        6\/5 = 1.2<\/p>\n<\/td>\n

                        		\n

                        0.833 < 1.2 \u2192 Reciprocal > Original \u2713<\/p>\n<\/td>\n<\/tr>\n

                        \n

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

                        		\n

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

                        		\n

                        1 = 1 \u2192 Reciprocal = Original \u2717<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

                        Three independent reasons why 1 is the odd one out:<\/strong><\/p>\n

                        (A) Numerical comparison:<\/strong>\u00a0For all proper fractions (1\/2 to 5\/6), reciprocal > original. For 1, reciprocal = original.<\/p>\n

                        (B) Fraction property:<\/strong>\u00a0Proper fractions (numerator < denominator) always have reciprocals > 1, while the original is < 1. For 1, both original and reciprocal equal 1.<\/p>\n

                        (C) Self-reciprocal uniqueness:<\/strong>\u00a01 is one of only two numbers (1 and -1) that are their own reciprocals. All other items (proper fractions) are not self-reciprocal.<\/p>\n

                        Conclusion:<\/strong>\u00a01 is the odd one out.<\/p>\n

                        \u00a0<\/p>\n

                        <\/p>\n