Question

The sum of the digits of a 2 digit number is 7. The number obtained by interchanging the digits exceeds the original number by 27. Find the number?

Answer 1

Hint: Assuming both the digits as separate variables, we make two equations one of which is addition of two digits and the other has a relation which states interchanging of digits exceeding the number by 27. We use the concept of a general way of writing a number knowing its place as one or tenths.

Complete step-by-step answer:
It is given that there is a two digit number. Let us consider the two digits number be x and y, where x is the number on the tenth place and y is the number at one place. The two digit number is written as 10x+y.
Sum of two digits of a two digit number is 7.
$x + y = 7$ … (1)
By interchanging the digits, y is the number on the tenth place and x is the number at one place. The interchanged two digit number is written as 10y+x.
Number obtained by interchanging the digits exceeds the original number by 27.
\[\left( {10y + x} \right) - \left( {10x + y} \right) = 27\] ... (2)
Solving Equation (2) results into
\[10y + x - 10x - y = 27\]
\[9y - 9x = 27\] ... (3)
Dividing both sides of the equation b 9
\[y - x = 3\]
\[y = x + 3\] … (4)
Using Equation (4) in Equation (1)
$x + x + 3 = 7$ ... (5)
$2x + 3 = 7$
$2x = 7 - 3$
Shift all constant values to one side of the equation.
$2x = 4$
Divide both sides by 2
$x = 2$ … (6)
Using Equation (6), calculate the value of y from Equation (1)
$2 + y = 7$
$y = 5$ … (7)

Using the value of x and y the two digit number which we have represented as 10x+y is 25.
The answer of this question can be checked by interchanging the digits and then using the interchanged number in Equation (3) and checking for the value.
Interchanged number is 52. So y is 5 and x is 2.
\[9y - 9x = 27\]
\[9 \times 5 - 9 \times 2 = 27\]
\[45 - 18 = 27\]
\[27 = 27\]
Hence, the calculated answer is correct.

Note: Students are likely to get confused when we assume the digits as variables and might write the number as xy which is wrong because then value of x is not showing at tenths place.