In a three digit number, the hundreds digit is twice the units digit. If the digits are reversed, the new number is 396 less than the original number. Find the number.
a b c = 100 a + 10 b + c
a = 2 c
reversed c b a = 100 c + 10 b + a
100 a + 10 b + c - 100 c - 10 b - a = 396
99 a - 99 c = 396
but a = 2 c so
198 c - 99 c = 396
99 c = 396
c = 4
then a = 8
so
8 b 4
what if b is 1 ?
is 814 - 418 = 396 ????
yes
what if b is 3?
is 834 - 438 = 396 ???
yes
In fact I suspect the middle digit can be about any :)
a b c = 100 a + 10 b + c
a = 2 c
reversed c b a = 100 c + 10 b + a
100 a + 10 b + c - 100 c - 10 b - a = 396
99 a - 99 c = 396
but a = 2 c so
198 c - 99 c = 396
99 c = 396
c = 4
then a = 8
b might be 1 or 3 however the difference is same
Let's solve this step-by-step:
Step 1: Let's assume the units digit is x.
Step 2: Since the hundreds digit is twice the units digit, the hundreds digit is 2x.
Step 3: The original number can be written as 100(2x) + 10(y) + x, where y is the tens digit.
Step 4: The new number, with the digits reversed, is 100x + 10(y) + 2x.
Step 5: According to the problem, the new number is 396 less than the original number, so we can write the equation as:
100(2x) + 10(y) + x - (100x + 10(y) + 2x) = 396.
Simplifying the equation:
200x + 10y + x - 100x - 10y - 2x = 396.
99x - 99x = 396.
0 = 396.
Step 6: The equation is contradictory, which means that there is no three-digit number that satisfies the given conditions.
Therefore, there is no solution.
To find the three-digit number, let's assume that the units digit is x. According to the problem, the hundreds digit is twice the units digit, which means the hundreds digit is 2x.
So, the original number can be represented as 2x * 100 + x * 10 + x, which simplifies to 200x + 10x + x, or 211x.
Now, let's reverse the digits and create the new number. The units digit will be 2x and the hundreds digit will be x. Therefore, the new number can be represented as 2x * 100 + 0 * 10 + x, which simplifies to 200x + x, or 201x.
According to the problem, the new number is 396 less than the original number. So, we can write the equation:
211x - 201x = 396
Now, let's solve for x:
10x = 396
Dividing both sides by 10:
x = 39.6
Since x represents a digit, it must be a whole number. Given that the units digit cannot be a decimal, we can conclude that there is no three-digit number that satisfies the given conditions.