Q18 of 186 Page 4

The sum of a two - digit number and the number formed by interchanging the digits is 132. If 12 is added to the number, the new number becomes 5 times the sum of the digits. Find the number.

Let unit’s digit = y

and the ten’s digit = x


So, the original number = 10x + y


After interchanging the digits, New number = x + 10y


The sum of the number = 10x + y


The sum of the digit = x + y


According to the question,


(10x + y) + (x + 10y) = 132


⇒ 11x + 11y = 132


⇒ 11(x + y) = 132


⇒ x + y = 12 …(i)


and 10x + y + 12 = 5(x + y)


⇒ 10x + y + 12 = 5x + 5y


⇒ 10x – 5x + y – 5y = – 12


⇒ 5x – 4y = – 12 …(ii)


From Eq. (i), we get


x = 12 – y …(iii)


On substituting the value of x = 12 – y in Eq. (ii), we get


5(12 – y) – 4y = – 12


⇒ 60 – 5y – 4y = – 12


⇒ – 9y = – 12 – 60


⇒ – 9y = – 72


⇒ y = 8


On putting the value of y = 8 in Eq. (iii), we get


x = 12 – 8 = 4


So, the Original number = 10x + y


= 10×4 + 8


= 48


Hence, the two digit number is 48.


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