Questions: The letters W, X, Y, and Z represent different digits. If the subtraction operation is correct, what is the value of W+X+Y+Z ? (A) 21 (B) 18 (C) 14 (D) 10

Solution Steps

To solve this problem, we need to determine the digits represented by W, X, Y, and Z such that the subtraction operation is correct. We then sum these digits to find the value of W + X + Y + Z.

1. Set up the subtraction equation based on the given problem.
2. Solve for the digits W, X, Y, and Z.
3. Sum the digits to find the final answer.

We are given a subtraction problem involving four-digit numbers where each letter represents a different digit. The subtraction is set up as follows:

\[ \begin{array}{r} 4 \mathrm{~W} \\ \mathrm{X} Y \\ -\mathrm{Y} 53 \mathrm{Z} \\ \hline 20024 \end{array} \]

We need to determine the values of \( \mathrm{W}, \mathrm{X}, \mathrm{Y}, \) and \( \mathrm{Z} \) and then find the sum \( \mathrm{W} + \mathrm{X} + \mathrm{Y} + \mathrm{Z} \).

The subtraction can be written as:

\[ 4WXY - Y53Z = 20024 \]

Since \( 4WXY \) and \( Y53Z \) are four-digit numbers, we can infer the following:

• \( 4WXY \) must be a number in the range 4000 to 4999.
• \( Y53Z \) must be a number in the range 1000 to 1999.

Given the result of the subtraction is 20024, we can infer that \( Y \) must be 1 because subtracting a number in the 1000s from a number in the 4000s should result in a number in the 2000s.

Substituting \( Y = 1 \) into the equation, we get:

\[ 4W1X - 153Z = 20024 \]

Rewriting the equation, we have:

\[ 4W1X = 20024 + 153Z \]

Since \( 4W1X \) is a four-digit number, \( 153Z \) must be a three-digit number. Therefore, \( Z \) must be a digit such that \( 153Z \) is a three-digit number. The possible values for \( Z \) are 0 through 6.

We test each possible value of \( Z \) to find a valid four-digit number for \( 4W1X \):

1. \( Z = 0 \): \[ 4W1X = 20024 + 153 \cdot 0 = 20024 \] This is not a valid four-digit number.

2. \( Z = 1 \): \[ 4W1X = 20024 + 153 \cdot 1 = 20177 \] This is not a valid four-digit number.

3. \( Z = 2 \): \[ 4W1X = 20024 + 153 \cdot 2 = 20330 \] This is not a valid four-digit number.

4. \( Z = 3 \): \[ 4W1X = 20024 + 153 \cdot 3 = 20483 \] This is not a valid four-digit number.

5. \( Z = 4 \): \[ 4W1X = 20024 + 153 \cdot 4 = 20636 \] This is not a valid four-digit number.

6. \( Z = 5 \): \[ 4W1X = 20024 + 153 \cdot 5 = 20789 \] This is not a valid four-digit number.

7. \( Z = 6 \): \[ 4W1X = 20024 + 153 \cdot 6 = 20942 \] This is not a valid four-digit number.

Since none of the values for \( Z \) resulted in a valid four-digit number, we need to re-evaluate our approach. We might have made an error in our assumptions or calculations.

Final Answer