# 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?

Last updated date: 29th Mar 2023

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Answer

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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.

Complete step-by-step answer:

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.

Complete step-by-step answer:

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.

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