Let's solve this problem step by step, using the relationships given:

Let the four-digit number be $ABCD$, where:

• $A$ is the thousands digit,
• $B$ is the hundreds digit,
• $C$ is the tens digit,
• $D$ is the ones digit.

The problem gives us the following relationships:

1. The thousands digit $A$ is 4 greater than the hundreds digit $B$: $A = B + 4$

2. The tens digit $C$ is twice the thousands digit $A$: $C = 2A$

3. The ones digit $D$ is one half of the thousands digit $A$: $D = \frac{A}{2}$

Step 1: Solve for $A$ and $B$

Since $A = B + 4$, $B$ must be less than $A$. Also, $A$ must be between 4 and 9 (because $B$ must be a single digit between 0 and 9).

Step 2: Test possible values for $A$

We need to test the values of $A$ from 4 to 9 and see if $C$ and $D$ are valid digits.

• For $A = 4$, $B = 0$: $C = 2 \times 4 = 8 \quad \text{and} \quad D = \frac{4}{2} = 2$ So, the number would be 4082.

• For $A = 5$, $B = 1$: C = 2 \times 5 = 10 \quad \text{(not valid, as C must be a single digit)}

• For $A = 6$, $B = 2$: C = 2 \times 6 = 12 \quad \text{(not valid, as C must be a single digit)}

• For $A = 7$, $B = 3$: C = 2 \times 7 = 14 \quad \text{(not valid, as C must be a single digit)}

• For $A = 8$, $B = 4$: C = 2 \times 8 = 16 \quad \text{(not valid, as C must be a single digit)}

• For $A = 9$, $B = 5$: C = 2 \times 9 = 18 \quad \text{(not valid, as C must be a single digit)}

Step 3: Determine the correct number

The only valid solution occurs when $A = 4$, $B = 0$, $C = 8$, and $D = 2$.

Thus, the number is 4082.

Would you like further details or have any questions?

Here are some related questions for further practice:

1. What would be the number if the thousands digit were 5 less than the hundreds digit?
2. What if the ones digit were twice the hundreds digit?
3. How would the number change if the tens digit were half the hundreds digit?
4. What if the number was five digits long, with the new digit equal to the sum of the tens and hundreds digits?
5. How would the number change if the ones digit were 3 more than the tens digit?

Tip: When working with digit problems, it is important to check all possible values to ensure you find the only valid solution.