A two digit number is seven times the sum of it's digits. The number formed by reversing the digits is 6 more than half of the original number. Find the difference of the digits of the given number .

a = first number

b = second number

Your number = 10 a + b

A two digit number is seven times the sum of it's digits meaans:

10 a + b = 7 ( a + b )

The number formed by reversing the digits is 10 b + a

The number formed by reversing the digits is 6 more than half of the original number means:

10 b + a = ( 10 a + b ) / 2 + 6

Now you must solve system of two equations:

10 a + b = 7 ( a + b )

10 b + a = ( 10 a + b ) / 2 + 6
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First equation:

10 a + b = 7 a + 7 b

Subtract 7 a to both sides

3 a + b = 7 b

Subtract b to both sides

3 a = 6 b

Divide both sides by 3

a = 2 b

Replace a with 2 b in equation:

10 b + a = ( 10 a + b ) / 2 + 6

10 b + 2 b = ( 10 • 2 b + b ) / 2 + 6

12 b = ( 20 b + b ) / 2 + 6

12 b = 21 b / 2 + 6

Multiply both sides by 2

24 b = 21 b + 12

Subtract 21 b to both sides

3 b = 12

Divide both sides by 3

b = 4

a = 2 b

a = 2 • 4

a = 8

a - b = 8 - 4 = 4

Check result:

A two digit number is seven times the sum of it's digits.

Your number = 10 a + b = 10 • 8 + 4 = 84

Sum of digits = 8 + 4 = 12

84 = 7 •12

The number formed by reversing the digits is 6 more than half of the original number.

The number formed by reversing the digits is 10 b + a = 10 • 4 + 8 = 48

Half of the original number =

84 / 2 = 42

48 = 42 + 6

a = first digit

b = second digit

To solve this problem, let's break it down step by step.

Step 1: Understand the problem
We are given a two-digit number that satisfies two conditions: (1) it is seven times the sum of its digits, and (2) the number formed by reversing the digits is 6 more than half of the original number. We need to find the difference between the digits of the given number.

Step 2: Let's assume the number
Let's assume that the given two-digit number is represented as 10x + y, where x represents the tens digit and y represents the units digit.

Step 3: Express the conditions as equations
Condition 1: "A two-digit number is seven times the sum of its digits."
This can be expressed as the equation:
10x + y = 7(x + y)

Condition 2: "The number formed by reversing the digits is 6 more than half of the original number."
The number formed by reversing the digits is represented as 10y + x. The original number is 10x + y.
So, we can express condition 2 as the equation:
10y + x = (1/2)(10x + y) + 6

Step 4: Solve the equations
Simplify equation 1 by expanding the multiplication:
10x + y = 7x + 7y

Simplify equation 2 by expanding the multiplication and distributing the fraction:
10y + x = 5x/2 + y/2 + 6

Now we can solve the equations simultaneously. Subtract 7x from both sides of equation 1, and subtract (5x/2 + y/2) from both sides of equation 2 to eliminate x from both equations:
3x - 6y = 0 [Equation A]
-9y + x - (5x/2 + y/2) = 6
-x/2 - 11y/2 = 6
Multiply through by -2 to eliminate the fractions:
x + 11y = -12 [Equation B]

Now we have a system of two equations with two variables. We can solve this system using the method of substitution or elimination.

Step 5: Solve the system of equations
Let's solve the system of equations by substitution. Solve equation A for x:
3x - 6y = 0
3x = 6y
x = 2y

Substitute x = 2y into equation B:
2y + 11y = -12
13y = -12
y = -12/13

Substitute y = -12/13 into equation A to find x:
3x - 6(-12/13) = 0
3x + 72/13 = 0
3x = -72/13
x = -24/13

So, the solution to the system of equations is x = -24/13 and y = -12/13.

Step 6: Find the difference between the digits
The difference between the digits, represented as |x - y|, can be calculated as:
|x - y| = |(-24/13) - (-12/13)| = |-24/13 + 12/13|

Simplifying the expression:
|-24/13 + 12/13| = |-12/13|

The absolute value of -12/13 is simply 12/13.

Therefore, the difference between the digits of the given number is 12/13.