in the following addition problem, each letter stands for a different digit.
Find the value of X, Y, and Z
XYZ
+ ZYX
_______
YYZY
xyz
zyx
*****
yyzy
Notice the leading digit y. It has to be 1. Think why. Now look at the middle digit of each number.It is 1. the last digit of the answer is 1.
What does that mean: Z+x=11
and then on the answer (y+y+carry 1)=3, so z=3
we have now as the puzzle
x13
31X
1131
and finally, x has to be 8.
zyzy
To find the values of X, Y, and Z in the given problem, we can go through it step-by-step.
First, let's look at the column where X is located. We have X + Y + Y = Z. Since each letter stands for a different digit, the only way to make X + Y + Y = Z true is if X = 1. This is because the maximum sum of two digits (in this case, Y + Y) is 18 (9 + 9). So, to get a sum of Z, which is a single digit, X must be 1.
Now let's consider the column where Z is located. We have Z + X + Z = Y. Since X = 1, the equation becomes Z + 1 + Z = Y, which simplifies to 2Z + 1 = Y.
Next, let's look at the column where Y is located. We have Y + Z + Z = Y. Since Y is a digit less than 10, the only digit that, when added to itself, gives a value less than 10 is 0. Therefore, Z must be 0.
Substituting Z = 0 into the equation 2Z + 1 = Y, we get 2(0) + 1 = Y, simplifying to 1 = Y. Therefore, Y must be 1.
So, the values of X, Y, and Z are X = 1, Y = 1, and Z = 0.
To find the values of X, Y, and Z, we need to solve the addition problem given:
XYZ + ZYX = YYZY
Let's break down this problem into individual digit places:
Starting from the rightmost place, we can see that Z + X = Y. Since each letter stands for a different digit, Z and X must be two different digits that add up to Y.
Moving to the next place value, we see that Y + Y = Z. This means that Y added to itself gives us the digit Z. Since each letter stands for a different digit, Y and Y must be the same digit, in which case their sum will always be an even number.
Lastly, at the leftmost place, we have an additional digit, denoted by Y, to add to the total.
To find the possible values for X, Y, and Z, let's analyze the possible combinations:
Z + X = Y
Y + Y = Z
Using trial and error, we can solve this system of equations by substituting values for Z and X:
When Z = 1:
1 + X = Y
Y + Y = 1
There is no single-digit number that, when added to itself, results in 1. So, Z cannot be 1.
When Z = 2:
2 + X = Y
Y + Y = 2
Again, there is no single-digit number that, when added to itself, results in 2. So, Z cannot be 2.
When Z = 3:
3 + X = Y
Y + Y = 3
There is no single-digit number that, when added to itself, results in 3. So, Z cannot be 3.
Likewise, continuing this process for each possible value of Z, we find that Z cannot be any value from 0 to 9.
Therefore, there are no possible values of X, Y, and Z that satisfy the given addition problem.