Question

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

Hint: Let the two digit number be 10x + y. And the number with reversed digit be 10y+x.

Assuming the two digit number as 10x+y. Now, the first part of the question is saying that number (10x + y) is 4 more than 6 times the sum (x + y). So according to given condition, we have
$\Rightarrow {\text{10x + y = 6(x + y) + 4}}$
Simplifying the equation , we get
$\Rightarrow {\text{10x - 6x + y - 6y = 4}} \\ \Rightarrow {\text{4x - 5y = 4 }} \cdots \left( 1 \right) \\$
Further, the question says, if we subtract 18 from the number 10x+y , the digits of the number are reversed. On writing the equation for the same, we get
$\Rightarrow {\text{10x + y - 18 = 10y + x}} \\ \Rightarrow {\text{10x - x + y - 10y - 18 = 0}} \\ \Rightarrow {\text{9x - 9y = 18}} \\ \Rightarrow {\text{x - y = 2 }} \cdots \left( 2 \right) \\$
We have two equations in two variables. On solving them , we can easily get x and y. Multiplying equation (2) with 4 and subtracting it from equation(1),
$\Rightarrow {\text{4x - 5y - 4x + 4y = 4 - 8}} \\ \Rightarrow {\text{ - y = - 4}} \\ \Rightarrow {\text{y = 4}} \\$
Substituting y = 4 in equation (2) , we get
$\Rightarrow {\text{x - 4 = 2}} \\ \Rightarrow {\text{x = 6}} \\$
Hence , the required number is ${\text{10x + y = 60 + 4 = 64}}$. The answer is 64.

Note: In these types of questions , we need to remember the basic rule of digits. One’s place will have 1 as multiple, whereas 10’s place has 10 , 100’s place has 100 and so on. And reversing the number will lead to change of digit’s places.Hence, multiples will be multiplied accord