If x and y are the smallest and the greatest four digit numbers formed by using 1,3,5 and 9, then find y-x.

Options

A. 5940
B. 8172
C. 3600
D. 6336

Answer

To solve the problem, we need to find the smallest and greatest four-digit numbers that can be formed using the digits 1, 3, 5, and 9. Let's denote the smallest number as \( x \) and the greatest number as \( y \). We will then calculate \( y - x \).

### Step-by-Step Solution:

1. **Identify the digits available**: The digits we can use are 1, 3, 5, and 9.

2. **Find the greatest four-digit number (y)**:
- To form the greatest number, we arrange the digits in descending order.
- The order of the digits from largest to smallest is: 9, 5, 3, 1.
- Therefore, the greatest four-digit number \( y \) is 9531.

3. **Find the smallest four-digit number (x)**:
- To form the smallest number, we arrange the digits in ascending order.
- The order of the digits from smallest to largest is: 1, 3, 5, 9.
- Therefore, the smallest four-digit number \( x \) is 1359.

4. **Calculate \( y - x \)**:
- We now have \( y = 9531 \) and \( x = 1359 \).
- To find \( y - x \):
\[ y - x = 9531 - 1359 \]

5. **Perform the subtraction**:
- Start from the rightmost digit:
- 1 (from 9531) - 9 (from 1359) cannot be done, so we borrow 1 from the next column.
- This makes it 11 - 9 = 2.
- Next column: 2 (after borrowing) - 5 = 7 (since we borrowed 1).
- Next column: 5 - 3 = 2.
- Last column: 9 - 1 = 8.
- So, \( y - x = 8172 \).

### Final Answer:
Thus, \( y - x = 8172 \).

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