Problems Based on Number

Problems on numbers can be solved by analysing the given statements and by framing the respective linear or quadratic eqution and solving for the unknowns. Solved Examples

Type 1 Words Problems

                                      
        Example 1: The sum of an integer and its reciprocal is 17/4. Find the number.
        Solution.  Let the number be x. Then.
                         x + 1/x = 17/4 => (x2 + 1)/x = 17/4
                        =>                 4x2 - 17x + 4 = 0
                        =>                 4x2 - 16x - x + 4 = 0
                        =>           4x(x - 4) - 1(x - 4) = 0 => (4x - 1)(x - 4) = 0
                        =>                                         x = 4 or 1/4
                        Hence, the number is 4.      (1/4 is not an integer)
                        
        Example 2: If the sum of two numbers is 52 and their product is 672, then find the absolute 
                   difference between the numbers.
        Solution. Let the numbers be x and y. Then, x + y = 52 and xy = 672
                     x - y = √(x + y)2 - 4xy  =  √(52)2 - 4 x 672 = √2704 - 2688 = √16 = 4
                .: Required difference = 4 
                
    Example 3: A number consists of two digits. The sum of the digits is 6. If 18 is subtracted
                from the number, its digit are interchanged. Find the number.
        Solution. Let the ten's digit be x.
        Then, Unit's digit = (6 - x)
              Number = 10x + (6 - x) = 9x + 6
        Number obtained by reversing the digits
                 = 10(6 - x) + x = 60 - 9x 
                .: 9x + 6 - 18 = 60 - 9x
                =>         18x = 72 
                =>           x = 4 
                Hence, the required number is 42.
                

Type 2 Data Sufficiency

                                   
        Directions (Examples 4 to 5) : In each of the following questions, a question followed by two
        statements numbered I and II are given. You have to read both the statements and then
        Give answer
          (1). If the data given in statement A alone are sufficient to answer the question whereas the
           data given in statement B alone are not sufficient to answer the question
          (2). If the data given in statement B alone are sufficient to answer the question whereas the
           data given in statement A alone are not sufficient to answer the question
          (3). If the data in either statement A alone or in statement B alone are sufficient to answer
           the question
          (4). If the data in both the statements A and B are not sufficient to answer the question
          (5). If the data given in both the statements A and B are necessary to answer the question
          
        Example 4 : What is the two-digit number?
        A. The difference between the two digits is 8
        B. The sum of the digits is equal to the difference between the two digits.
        Solution. (5) Let the ten's and unit's digit be x and y respectively.
        Then, from A, x + y = 8 and B, x - y = 8 
           Upon solving, we get x = 8 and y = 0
         .:   Required number = 80
         Thus, both A and B are needed to get the answer.
         
         Example 5 : What is the two-digit number?
         A. Sum of the two digits is 7.
         B. The digit in the ten's place is bigger than the unit's place by 1.
         Solution. (5) Let the ten's and unit's digit be x and y respectively.
         Then, from A, x + y = 7 and from B, x - y = 1
         Upon solving, we get x = 4 and y = 3 
         .: Required number = 43 
         Thus, both A and B are needed to get the answer.