The letters A,B,and C represent three different digits. Determine the number ABC if ABC
+ ACB
----------
BCA
What three different digits are represented by X,Y,Z in this addition problem?
XZY + XYZ = YZX
Just another perspective:
1--First, from the last column, we could have either Y + Z = X or Y + Z = 10 + X
2--From the second column, we could have either Y + Z = Z or Y + Z + 1 = 10 + Z
3--But we can't have (Y + Z) equaling both X and Z, so Y and Z must add up to more than 10.
3--Now we have Y + Z = 10 + X and Y + Z + 1 = 10 + Z or Y + Z = Z + 9.
4--Equating (Y + Z)'s, we get 10 + X = Z + 9 or X = Z - 1
5--Knowing that Y + Z exceeds 10, we can say 2X + 1 = Y
6--Substituting, we get 2X + 1 + Z = 10 + X or X = 9 - Z
7--Equating our expressions for X we get 9 - Z = Z - 1 or 2Z = 10 and Z = 5.
I thing the rest should fall out fairly easy for you.
To determine the number ABC, we need to solve the equation:
ABC + ACB = BCA
Since the letters A, B, and C represent three different digits, we can start by assigning values to each letter and find the corresponding digit.
Let's assume A = 1, B = 2, and C = 3.
Now, we can substitute these values into the equation and see if it holds true:
123 + 132 = 231
This equation is not true. Therefore, the values A = 1, B = 2, and C = 3 do not give us the correct number.
We need to try different combinations until we find the correct values for A, B, and C. Let's try A = 2, B = 1, and C = 3.
213 + 231 = 432
This equation is true. Therefore, the number ABC is equal to 213.
To find the value of ABC, we need to solve the equation:
ABC + ACB = BCA
Let's break it down step by step:
1. The first digit (A) can be any digit from 1 to 9 since it cannot be zero. Let's assume A = 1.
2. Now let's substitute A = 1 into the equation:
1BC + 1CB = BCA
3. Since B and C are different digits, they can each take any value from 0 to 9, excluding 1. Let's consider the possibilities:
a) If B = 0:
Substituting B = 0 into the equation, we get:
10C + 1C = C10
This equation does not produce a valid result since C cannot be 10.
b) If B = 2:
Substituting B = 2 into the equation, we get:
12C + 1C = C12
This equation also does not produce a valid result since C cannot be 12.
c) If B = 3:
Substituting B = 3 into the equation, we get:
13C + 1C = C13
This equation also does not produce a valid result since C cannot be 13.
d) If B = 4:
Substituting B = 4 into the equation, we get:
14C + 1C = C14
This equation does not produce a valid result since C cannot be 14.
e) If B = 5:
Substituting B = 5 into the equation, we get:
15C + 1C = C15
This equation also does not produce a valid result since C cannot be 15.
f) If B = 6:
Substituting B = 6 into the equation, we get:
16C + 1C = C16
This equation does not produce a valid result since C cannot be 16.
g) If B = 7:
Substituting B = 7 into the equation, we get:
17C + 1C = C17
This equation does not produce a valid result since C cannot be 17.
h) If B = 8:
Substituting B = 8 into the equation, we get:
18C + 1C = C18
This equation does not produce a valid result since C cannot be 18.
i) If B = 9:
Substituting B = 9 into the equation, we get:
19C + 1C = C19
This equation does not produce a valid result since C cannot be 19.
Since none of the cases give us a valid solution, there is no three-digit number ABC that satisfies the given equation.