For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is
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Official Correct Answer: B. First, let's restate the problem to make sure I understand it correctly. We have a 4-digit number, which means it's between 1000 and 9999. The number has four digits: thousands, hundreds, tens, and units places. Let's denote them as A, B, C, and D respectively. So, the number is ABCD.
Now, the problem gives us three conditions:
1. The sum of the digits in the thousands, hundreds, and tens places is 15. So, A + B + C = 15.
2. The sum of the digits in the hundreds, tens, and units places is 16. So, B + C + D = 16.
3. The digit in the tens place is 6 more than the digit in the units place. So, C = D + 6.
Our goal is to find the difference between the largest and smallest possible values of this number ABCD.
We have these equations to organize the information:
1. A + B + C = 15
2. B + C + D = 16
3. C = D + 6
We find the possible values of A, B, C, D that satisfy these equations and then determine the maximum and minimum numbers ABCD can be.
Let me start by expressing all variables in terms of one variable to reduce the number of variables. Since equation 3 relates C and D, I can express D in terms of C: D = C - 6.
Now, let's substitute D = C - 6 into equation 2:
B + C + (C - 6) = 16
Simplify:
B + 2C - 6 = 16
B + 2C = 22
So, B = 22 - 2C
Now, substitute B = 22 - 2C into equation 1:
A + (22 - 2C) + C = 15
Simplify:
A + 22 - C = 15
A = 15 - 22 + C
A = C - 7
So now, we have expressions for A and B in terms of C:
A = C - 7
B = 22 - 2C
Now, let's remember that A, B, C, D are digits, meaning they must be integers between 0 and 9, inclusive. However, since ABCD is a 4-digit number, A cannot be 0. So, A must be between 1 and 9.
Let's write down the constraints for each variable:
1. A = C - 7 ≥ 1 ⇒ C - 7 ≥ 1 ⇒ C ≥ 8
2. A = C - 7 ≤ 9 ⇒ C ≤ 16, but since C is a digit, C ≤ 9
3. B = 22 - 2C ≥ 0 ⇒ 22 - 2C ≥ 0 ⇒ 2C ≤ 22 ⇒ C ≤ 11, but since C is a digit, C ≤ 9
4. B = 22 - 2C ≤ 9 ⇒ 22 - 2C ≤ 9 ⇒ 2C ≥ 13 ⇒ C ≥ 6.5, but since C is an integer, C ≥ 7
5. D = C - 6 ≥ 0 ⇒ C - 6 ≥ 0 ⇒ C ≥ 6
6. D = C - 6 ≤ 9 ⇒ C ≤ 15, but since C is a digit, C ≤ 9
Now, let's compile all these constraints:
From A: C must be ≥8 and ≤9
From B: C must be ≥7 and ≤9
From D: C must be ≥6 and ≤9
Combining all these, the most restrictive constraints are C ≥8 and C ≤9. So, C can be either 8 or 9.
Let's consider each case:
Case 1: C = 8
Compute A, B, D:
A = 8 - 7 = 1
B = 22 - 2*8 = 22 - 16 = 6
D = 8 - 6 = 2
So, the number is A B C D = 1 6 8 2 ⇒ 1682
Case 2: C = 9
Compute A, B, D:
A = 9 - 7 = 2
B = 22 - 2*9 = 22 - 18 = 4
D = 9 - 6 = 3
So, the number is A B C D = 2 4 9 3 ⇒ 2493
Is that all? Let me check if there are other possible values of C. From the constraints, C can only be 8 or 9 because C must be ≥8 and ≤9. So, only two possible numbers: 1682 and 2493.
But wait, the question is asking for the difference between the largest and smallest possible values. So, 2493 is larger than 1682. Computing the difference:
2493 - 1682 = 811
Hmm, that's one of the options. Option B is 811. So, that seems to be the answer.
But Did I miss any possible value of C?
From the constraints, C must be between 8 and 9 inclusive. So, only 8 and 9. So, only two numbers: 1682 and 2493.
Wait, but let me check if these numbers satisfy all the given conditions.
For 1682:
- A + B + C = 1 + 6 + 8 = 15 ✔️
- B + C + D = 6 + 8 + 2 = 16 ✔️
- C = D + 6 ⇒ 8 = 2 + 6 ⇒ 8 = 8 ✔️
For 2493:
- A + B + C = 2 + 4 + 9 = 15 ✔️
- B + C + D = 4 + 9 + 3 = 16 ✔️
- C = D + 6 ⇒ 9 = 3 + 6 ⇒ 9 = 9 ✔️
Both numbers satisfy all conditions. So, the maximum number is 2493, and the minimum is 1682. Their difference is 811, which is