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Question

# Question 9 A two – digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.

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Solution

## Let the two – digit number = 10x + y Case I: multiplying the sum of the digits by 8 and then subtracting 5 = two – digit number ⇒ 8 × (x + y) – 5 = 10x + y ⇒ 8x + 8y – 5 = 10x + y ⇒ 2x – 7y = - 5...(i) Case II: Multiplying the difference of the digits by 16 and then adding 3 = two – digit number ⇒16 × (x –y) + 3 = 10x + y ⇒16x – 16y + 3 = 10x + y ⇒6x – 17y = - 3...(ii) Now, multiplying Eq. (i) by 3 and then subtracting from Eq. (ii), we get 6x – 17y = - 3 6x – 21y = - 15 4y = 12 ⇒ y = 3 Now put the value of y in Eq. (i) we get 2x – 7 × 3 = - 5 ⇒2x = 21 – 5 = 16 . So, x = 8 Hence, the required two – digit number = 10x + y = 10 × 8 + 3 = 80 + 3 = 83

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