a number rounds off to 4000.the digits in the hundreds place is twice the digits in the tens place.the sum of the digits is 12.the number uses only two different digits.find the number.
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To solve this problem, let's break it down into steps:
Step 1: Identify the information given in the problem.
- The number rounds off to 4000.
- The digits in the hundreds place are twice the digits in the tens place.
- The sum of the digits is 12.
- The number uses only two different digits.
Step 2: Determine the possible values for the digits.
Since we are given that the sum of the digits is 12, and the digits cannot be negative or zero, we can narrow down the possible values for the digits. Let's list the possible pairs of digits that sum to 12 and are not zero:
{0, 12}, {1, 11}, {2, 10}, {3, 9}, {4, 8}, {5, 7}, {6, 6}
Step 3: Analyze the conditions given in the problem.
- The number rounds off to 4000. Therefore, the number must be greater than or equal to 4000 and less than 5000.
Step 4: Apply the conditions to the possible values of digits.
- For the number to be greater than or equal to 4000, the thousands digit must be 4.
- Using the condition that the hundreds digit is twice the tens digit, we can try the following pairs of digits: {2, 1}, {4, 2}, and {8, 4}.
Step 5: Calculate the possible numbers using the pairs of digits.
1. Consider the pair {2, 1}:
Thousands digit = 4
Hundreds digit = 2 * 1 = 2
Tens digit = 1
Units digit = sum of digits - (thousands digit + hundreds digit + tens digit)
= 12 - (4 + 2 + 1)
= 5
Therefore, the number is 4215, which is not within the range of 4000.
This case is not valid.
2. Consider the pair {4, 2}:
Thousands digit = 4
Hundreds digit = 2 * 2 = 4
Tens digit = 2
Units digit = sum of digits - (thousands digit + hundreds digit + tens digit)
= 12 - (4 + 4 + 2)
= 2
Therefore, the number is 4242, which is not within the range of 4000.
This case is not valid.
3. Consider the pair {8, 4}:
Thousands digit = 4
Hundreds digit = 2 * 4 = 8
Tens digit = 4
Units digit = sum of digits - (thousands digit + hundreds digit + tens digit)
= 12 - (4 + 8 + 4)
= -4
Therefore, this case is also not valid.
Since none of the pairs of digits satisfy all the given conditions, there is no number that fits the given criteria.