Question

Amanda is thinking of a 5 digit number, the digit in the hundreds place is 7. The digit in the thousands place is the greatest possible digit. The digit in the ones place is double the digit in the ten thousands place. The digits in he thousands place and ones place add up to 17. The digits in the tens and ten thousands place adds up to the digit in the hundreds place. What is Amanda thinking of?

Answer 1

Let's denote the 5-digit number as \( ABCDE \), where:

• \( A \) is the ten thousands place,
• \( B \) is the thousands place,
• \( C \) is the hundreds place (which is given as \( 7 \)),
• \( D \) is the tens place,
• \( E \) is the ones place.

From the problem, we have the following information:

1. \( C = 7 \).
2. \( B \) (the thousands place) is the greatest possible digit, so we will consider \( B = 9 \) initially.
3. The ones place \( E \) is double the ten thousands place \( A \): \( E = 2A \).
4. The digits \( B \) and \( E \) add up to 17: \( B + E = 17 \).
5. The digits in the tens and ten thousands place add up to the digit in the hundreds place: \( D + A = C = 7 \).

Now let's analyze these conditions step by step.

Step 1: Start with \( B = 9 \).

When \( B = 9 \): \[ 9 + E = 17 \implies E = 17 - 9 = 8. \]

Step 2: Substitute \( E \) to find \( A \).

Now, from the condition \( E = 2A \): \[ 8 = 2A \implies A = \frac{8}{2} = 4. \]

Step 3: Find \( D \).

Using the equation \( D + A = 7 \): \[ D + 4 = 7 \implies D = 3. \]

Summary of Digits:

Now we have the digits:

• \( A = 4 \),
• \( B = 9 \),
• \( C = 7 \),
• \( D = 3 \),
• \( E = 8 \).

Thus, the number Amanda is thinking of is: \[ 49738. \]

Verification:

1. The digit in the hundreds place is \( 7 \). (Correct)
2. The digit in the thousands place is \( 9 \), the greatest possible digit. (Correct)
3. The digit in the ones place \( 8 \) is double the digit in the ten thousands place \( 4 \). (Correct)
4. The digits in the thousands place and ones place \( 9 + 8 = 17 \). (Correct)
5. The digits in the tens and ten thousands place \( 3 + 4 = 7 \). (Correct)

All conditions are satisfied, hence the number Amanda is thinking of is:

\[ \boxed{49738}. \]