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# Sum of digits of a two-digit number is 9. When we interchange the digit it is found that the resulting new number is greater than the original number by 27. What is the two-digit number?  Verified
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Hint: To solve the question, we form equations from the given information and solve those equations to unknow the value of digits of the two-digit number.

Let a, b be the tens and units digit of the two-digit number.
The given sum of the two digits of the two-digit number = 9
$\Rightarrow$ a + b = 9 ………. (1)
The mathematical representation of the two-digit number with a, b at tens and units place respectively = 10a + b
Given that the digits at unit and tens place of two-digit number are reversed, the new two-digit number formed
= 10b + a
The new number formed exceeds the original number by the value 27.
$\Rightarrow$ (10b + a) - (10a + b) = 27
9b - 9a = 27
9(b - a) = 27
b - a = 3 ………. (2)
By adding equations (1) and (2) we get
b - a + a + b = 9 + 3
2b = 12
$\Rightarrow b=\dfrac{12}{2}=6$
By substituting b value in equation (1) we get
a + 6 = 9
a = 9 – 6 = 3
Thus, the original number is equal to 10a + b
$=\left( 10\times 3 \right)+6$
= 30 + 6
= 36
$\therefore$ The two-digit number is equal to 36.

Note: The possibility of mistake can be not using the mathematical representation of a two-digit number to ease the procedure of solving. The alternative procedure of solving can be using a hit-trial method for solving the equation, since a, b are digits of a number their values lie between 0, 9 and since a being the first digit of a two-digit number, it cannot be 0. Thus, we can try the other possible values and calculate the right answer.