Question

A two-digit number is 4 more than 6 times the sum of its digit. If 18 is subtracted from the number, the digits are reversed. Find the number.

Answer 1

Hint: To solve this let us consider 2 digit number such that the units place is y and the tens place is x. then the number will be 10x + y. Now we have the first condition two-digit number is 4 more than 6 times the sum of its digit. Hence we get 10x + y = 6(x + y) + 4 and then we have the second condition if 18 is subtracted from the number, the digits are reversed. This means 10x + y – 18 = 10y + x. Now we have two equations in two variables hence we can solve them simultaneously to find x and y.

Complete step-by-step solution:
Now let us assume the given number is such that its units place has y and tens place has x.
Hence the number will be 10x + y.
Let us try to understand this by an example.
Now 23 has 2 in its tens place and 3 in its units place and we have 23 = 2 × 10 + 3.
Hence we say our required number is 10x + y.
Now let us check the first condition which is the two-digit number is 4 more than 6 times the sum of its digit.
Now our two-digit number is 10x + y and it has digits x and y.
Hence the sum of digits will be (x + y) and 6 times the sum of digits will be 6(x + y).
4 more than 6 times the sum of its digit means 6(x + y) + 4.
Hence we have 10x + y = 6(x + y) + 4
Now let us open the bracket in RHS.
10x + y = 6x + 6y + 4.
Taking 6x and 6y to LHS we get.
10x – 6x + y – 6y = 4
4x – 5y = 4 ………………………. (1)
Now consider the second condition which is if 18 is subtracted from the number, the digits are reversed.
Now our number is 10x + y. the number with digits reversed will be 10y + x.
Hence according to the given condition we have,
10x + y – 18 = 10y + x.
Now raking – 18 to RHS and 10y, x to LHS we get.
10x – x + y – 10y = 18
9x – 9y = 18
Dividing the equation by 9 we get
x – y = 2………………………………. (2)
Now multiplying equation (2) by 4 we get
4x – 4y = 8 ……………………. (3)
Now subtracting equation (1) from equation (2) we get
4x – 4y – (4x – 5y) = 8 – 4
4x – 4y – 4x + 5y = 4
Hence we get y = 4
Now substituting y = 4 in equation (2) we get.
x – 4 = 2
Taking 4 to RHS we get
x = 4 + 2 = 6
 Hence we have x = 6.
Hence we have x = 6 and y = 4.
Now the number was 10x + y = 10 × 6 + 4 = 60 + 4 = 64.
Hence the number is 64.

Note: Now we have taken that the number is 10x + y note that here tens place is x and units place is y. also for most of the problems where we have to deal with a digit of number always consider the number to be of the form 10x + y. where x is tens place and y is units place. sometimes students make mistakes by assuming the number as XY where x is the tenth digit and y is the unit digits.