9. Sum of two numbers is 97. If the larger number is divided by the smaller, the quotient is 7 and the remainder is 1. Find the numbers.


Solution. Let the larger number be x and smaller number be y.
As per the first given condition,
x + y = 97 ........eq. no. (1)
As per the second condition,
Dividend = Divisor × Quotient + Remainder
x = y × 7 + 1
∴  x = 7y + 1
∴  x – 7y = 1 .......(2)
Subtracting (2) from (1),
x + y = 97
x – 7y = 1
(-)  (+)   (-)
      8y = 98
∴ y = 96/8
∴ y = 12
Substituting y = 12 in equation (1)
∴ x + y = 97
∴ x + 12 = 97
∴ x = 97 – 12
∴ x = 85

The two numbers are 85 and 12.