In a two digit number the digit in the unit place is 6 less than twice the digit in the tenth place. When tha digits are reversed th number increases by 9. Find the number
u = 2t - 6
10u + t = 10t + u + 9
substituting
... 20t - 60 + t = 10t + 2t - 6 + 9
solve for t, then substitute back to find u
Amazing!
let the tenth digit be x
then the unit digit is 2x-6
the number is 10x + (2x-6)
the number reversed is 10(2x-6) + x
10(2x-6) + x - (10x + (2x-6)) = 9
20x - 60 + x - 10x - 2x + 6 = 9
9x = 63
x=7
then 2x-6 = 8
then number is 78
check:
87-78 = 9 , looks good
To find the number, we'll solve this problem step-by-step:
Let's assume the digit in the unit place is "x," and the digit in the tenth place is "y."
According to the first condition given, "the digit in the unit place is 6 less than twice the digit in the tenth place," we can form the equation:
x = 2y - 6 ------------ (Equation 1)
Now, let's consider the second condition, "when the digits are reversed, the number increases by 9." In this case, the new number can be formed as 10x + y. The original number can be formed as 10y + x. Hence, the equation becomes:
10x + y = 10y + x + 9
9x - 9y = 9 ------------ (Equation 2)
Now, we'll solve the system of equations (Equation 1 and Equation 2) to find the values of x and y.
Step 1: Solve Equation 1 for x in terms of y:
x = 2y - 6
x = 2y - 6 ------------ (Equation 1)
Step 2: Substitute the value of x from Equation 1 into Equation 2:
9x - 9y = 9
9(2y - 6) - 9y = 9
18y - 54 - 9y = 9
9y = 9 + 54
9y = 63
y = 7
Now, substitute the value of y = 7 into Equation 1 to find x:
x = 2y - 6
x = 2(7) - 6
x = 14 - 6
x = 8
Therefore, the digit in the unit place (x) is 8, and the digit in the tenth place (y) is 7.
Finally, combining the digits, the number is 78.
So, the number is 78.