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.