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A two digit number is 3 more than 4 times the sum of its digits. If 18 is added to this number, the sum is equal to the number obtained by interchanging the digits. Find the number

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Let the digit in unit’s place be ‘x’ and the digit in ten’s place be ‘y’.

Digit in tens place Digit in units place Number Sum of the digits
Original number y x 10y + x y + x
Number obtained by interchanging the digits  x y 10x + y x + y

According to the first condition,

A two digit number is 3 more than 4 times the sum of its digits. 

10y + x = 4(x + y) + 3 

∴ 10y + x = 4x + 4y + 3 

∴ x – 4x + 10y – 4y = 3 

∴ – 3x + 6y = 3 

Dividing both sides by -3,

 x – 2y = -1 …(i) 

According to the second condition,

 if 18 is added to the number, the sum is equal to the number obtained by interchanging the digits.

10y + x + 18= 10x + y 

∴ x – 10x + 10y – y = -18 

∴ – 9x + 9y = -18 

Dividing both sides by – 9,

 x – y = 2 ……(ii) 

Subtracting equation (ii) from (i),

∴ y = 3

 Substituting y = 3 in equation (ii),

 x – y = 2 

∴ x – 3 = 2 

∴ x = 2 + 3 = 5 

∴ Original number = 10y + x

 = 10(3) + 5 

= 30 + 5 

= 35 

The required number is 35.

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