What is the original number if the sum of the digits of a two digit number is seven the number obtained by interchanging the digits is 27 more than the original number?

Answer

Verified

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.