The difference between the two digits of a two digit number is 2. If the digits swap places,the number is 18 less than the original number. What is the original number?
let the tens digit be x, then the unit digit is x-2
x ≥ 2
the value of the number is 10x + x-2
the value of the number reversed is 10(x-2) + x
10x + x-2 - ( 10(x-2) + x) = 18
11x - 2 - (11x - 20) = 18
11x - 2 - 11x + 20 = 18
18 = 18 , which is a true statement.
So it will work for any number of x
and the original number could have been:
20, 31, 42, 53, 64, 75, 86, or 97
check:
20 - 02 = 18
31 - 13 = 18
42 - 24 = 18
53 - 35 = 18
...
97 - 79 = 18
Well, well, looks like we have a little puzzle here! Let's put on our thinking caps and solve this riddle, shall we?
Let's start by assuming the two digits of the original number are x and y. If the difference between the two digits is 2, then we have two possibilities: x - y = 2 or y - x = 2. But we'll go with x - y = 2 for now, just to keep things simple.
So, if we swap the places of the two digits, we get a new number with the digits y and x. According to the problem, this new number is 18 less than the original number.
Now, let's look at the equation we just made: (10y + x) = (10x + y) - 18. Simplifying it further, we get 10y + x = 10x + y - 18.
Let's simplify it even more: 10y - y = 10x - x - 18. That becomes 9y = 9x - 18.
Finally, we can simplify it one last time: y = x - 2.
Now, combining this with our equation x - y = 2, we can substitute y with x - 2: x - (x - 2) = 2.
Simplifying it further, we get x - x + 2 = 2. Well, isn't that handy! The x's cancel each other out.
Therefore, 2 = 2. That's always true! It means any value of x would work. So, the original number could be any number where the difference between the digits is 2.
For example, if x is 4, then y would be 2, and the original number would be 42. If x is 3, then y would be 1, and the original number would be 31. You get the idea!
So, my dear friend, the original number could be 42, 31, or any other two-digit number where the difference between the digits is 2. Happy number hunting!
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Let's assume the original number has a tens digit of x and a units digit of y.
According to the given information, we can form two equations:
1. The difference between the two digits is 2:
x - y = 2
2. If the digits swap places, the number is 18 less than the original number:
10y + x = 10x + y - 18
Now we can solve these equations simultaneously to find the values of x and y.
1. Rearranging the first equation:
x = y + 2
2. Substituting the value of x in the second equation:
10y + y + 2 = 10(y + 2) + y - 18
11y + 2 = 10y + 20 + y - 18
11y + 2 = 11y + 2
0 = 0
The equation simplifies to 0 = 0, which means that both equations are satisfied for any value of y. This implies that there are infinitely many solutions for the original number.
However, let's assume y = 4 for further calculations.
Substituting y = 4 in equation 1, we get:
x = 4 + 2
x = 6
So, if y = 4, then x = 6. Therefore, the original number is 64.
In conclusion, the original number is 64.
To solve this problem, let's consider the original two-digit number as "10a + b", where "a" represents the tens digit, and "b" represents the units digit.
Given that the difference between the two digits is 2, we can write the following equation:
a - b = 2 (equation 1)
If the digits swap places, the new number is represented as "10b + a". We are told that this new number is 18 less than the original number. We can represent this as:
10b + a = 10a + b - 18 (equation 2)
To solve this system of equations, we can use the method of substitution or elimination. Let's use substitution.
From equation 1, we can express "a" in terms of "b":
a = b + 2
Now let's substitute "a" into equation 2:
10b + (b + 2) = 10(b + 2) + b - 18
Simplifying this equation, we get:
11b + 2 = 10b + 20 + b - 18
11b + 2 = 11b + 2
Since the variables cancel out, we are left with a constant equation:
2 = 2
This means that the value represented by "b" can be any number. Hence, there are infinitely many solutions to this problem.
Therefore, there is not a unique answer to the question of "What is the original number?"