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A 2 digit number is reversed and added to the original number to form a new number. If the original number is multiplied by 2 and 18 is added to it, then it becomes equal to the newly formed number. The sum of digits of the original number is 16. What is the ratio of ten's digit to the unit's digit of the original number?

A 2 digit number is reversed and added to the original number to form a new number. If the original number is multiplied by 2 and 18 is added to it, then it becomes equal to the newly formed number. The sum of digits of the original number is 16. What is the ratio of ten's digit to the unit's digit of the original number? Correct Answer 7 : 9

Given:

New number is formed by adding original number and its reverse.

Sum of digits of original number = 16

Concept Used:

A 2 digit number pq can be expressed as 

pq = 10p + q

Calculations:

Let the original 2 digit number be xy.

xy = 10x + y

yx = x + 10y

New number = (10x + y) + (x + 10y)

⇒ New number = 11x + 11y      ----(1)

Also, New number = (2 × xy + 18)

⇒ New number = 2 × (10x + y) + 18      ----(2)

Equating (1) and (2), we get

(11x + 11y) = (20x + 2y) + 18

⇒ 9y = 9x + 18

⇒ -x + y = 2     ----(1)

Also, x + y =16       ----(2)

Adding (1) and (2), we get

y = 18/2 = 9, x = 7

⇒ Original number = 79

⇒ x : y = 7 :9 

∴ The ratio of ten's digit to the unit's digit of the original number is 7 : 9.

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