MathematicsThe sum of a two-digit number and the number formed by interchanging its digits is 110. If 10 is subtracted from the first number, the new number is 4 more than 5 times the sum of its digits in the first number. Find the first number

The sum of a two-digit number and the number formed by interchanging its digits is 110. If 10 is subtracted from the first number, the new number is 4 more than 5 times the sum of its digits in the first number. Find the first number


  1. A
    60
  2. B
    61
  3. C
    62
  4. D
    64 

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    Solution:

    let the unit place digit be x and tens place digit be y
    then the two-digit number will be 10y + x
    and the number formed by interchanging the unit place and tens place digits will be 10x + y
    according to the first condition given in the qs i.e, the sum of two numbers is 110 that  is
    10y + x + 10x + y = 110
    Or, 11x + 11y = 110
    divide the above equation by 11 we get
    x + y = 10
    Or, x = 10 - y ....(i)
    now according to the second equation,
    if 10 is subtracted from the first number i.e, the new number is 10y + x - 10
    given that the new number is 4 more than 5 time the sum of its digits in the first number i.e
    the sum of its digits in the first number is x + y, now 5 times of its, 5(x + y), and now 4 more that is, 4 + 5(x + y)
    therefore, new number = 4 + 5(x + y)
    10y + x - 10 = 4 +5(x + y)
    10y - 5y + x = 4 +10 +5x
    5y = 14 + 4x.....(ii)
    substitute the value of x from (i) to (ii)
    we  get , 5y = 14 + 4(10 - y)
    5y = 14 + 40 - 4y
    y = 6
    and from (i)
    x = 4
    then the first number 10y + x = 10x6 + 4 = 64
    The first number is 64.
    Hence, the correct option is 4.
     
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