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.