a number consists of two digits whose sum is 9. If three times the number is equal to eight times the number formed by interchanging the digits, find the difference between the number and its adverse.
The answer is wrong
AAAaannndd the bot gets it wrong yet again!
x+y=9
3(10x+y) = 8(10y+x)
The number is 72
check: 3*72 = 8*27 ✅
UwU!
To solve this problem, let's break it down into steps.
Step 1: Representing the Number
Let's assume the tens digit is x and the units digit is y. Since the sum of the digits is 9, we have the equation: x + y = 9.
Step 2: Forming the Number
The number can be represented as 10x + y, where 10x represents the tens digit multiplied by 10, and y represents the units digit.
Step 3: Forming the Reversed Number
The reversed number can be represented as 10y + x, where 10y represents the units digit multiplied by 10, and x represents the tens digit.
Step 4: Writing the Equation
According to the problem statement, three times the original number (3*(10x + y)) is equal to eight times the reversed number (8*(10y + x)). So, we have the equation:
3*(10x + y) = 8*(10y + x).
Step 5: Solving the Equation
Let's simplify the equation:
30x + 3y = 80y + 8x.
We can rearrange this equation to isolate either x or y:
30x - 8x = 80y - 3y,
22x = 77y.
Step 6: Finding the Possible Digits
Since x and y represent the digits, they must be integers between 0 and 9. Let's find the possible values for x and y that satisfy the equation.
Dividing both sides of the equation by 77, we get:
22x/77 = y/1.
Simplifying further, we have:
2x/7 = y.
Since x and y are digits, x can be 1, 2, 3, and 4 (since only these values satisfy the equation when substituted for x and y).
Step 7: Finding the Numbers and their Difference
Using the values for x, we can find the corresponding values for y:
When x = 1, y = 2.
When x = 2, y = 4.
When x = 3, y = 6.
When x = 4, y = 8.
The numbers and their reversed counterparts are:
For x = 1, y = 2:
Number: 10*1 + 2 = 12
Reversed Number: 10*2 + 1 = 21
For x = 2, y = 4:
Number: 10*2 + 4 = 24
Reversed Number: 10*4 + 2 = 42
For x = 3, y = 6:
Number: 10*3 + 6 = 36
Reversed Number: 10*6 + 3 = 63
For x = 4, y = 8:
Number: 10*4 + 8 = 48
Reversed Number: 10*8 + 4 = 84
To find the difference between the number and its reversed counterpart, subtract the smaller number from the larger number:
For the numbers 12 and 21: 21 - 12 = 9.
For the numbers 24 and 42: 42 - 24 = 18.
For the numbers 36 and 63: 63 - 36 = 27.
For the numbers 48 and 84: 84 - 48 = 36.
Therefore, the differences between each number and its reversed counterpart are 9, 18, 27, and 36.
Let the number be xy.
Then, 3xy = 8(y x).
Therefore, 3x = 8y and 3y = 8x.
Subtracting the two equations, we get 5x = 5y.
Therefore, x = y.
The number is therefore xx.
The difference between the number and its adverse is 11.