Rich chooses a 4-digit positive integer. He erases one of the digits of this integer. The
remaining digits, in their original order, form a 3-digit positive integer. When Rich
adds this 3-digit integer to the original 4-digit integer, the result is 6031. What is the
sum of the digits of the original 4-digit integer?
To solve this problem, we need to break it down into smaller steps:
Step 1: Let's assume the original 4-digit positive integer is ABCD, where A, B, C, and D represent the individual digits.
Step 2: We know that when one of the digits is removed, the remaining digits form a 3-digit positive integer. Let's say this 3-digit integer is XYZ, where X, Y, and Z represent the remaining digits in their original order.
Step 3: According to the problem, when we add the 3-digit integer (XYZ) to the original 4-digit integer (ABCD), we get the resulting number 6031.
Step 4: Now, let's write the equation for this problem:
ABCD + XYZ = 6031
Step 5: Since we don't know the specific values of A, B, C, D, X, Y, and Z, let's represent them as variables.
Step 6: From step 2, we know that XYZ is a 3-digit number, so it must be between 100 and 999.
Step 7: Let's rewrite the equation using the variable representation:
1000A + 100B + 10C + D + 100X + 10Y + Z = 6031
Step 8: We need to simplify this equation by combining like terms:
1000A + 100B + 10C + D + 100X + 10Y + Z = 6031
1000A + 100B + 10C + 100X + 10Y + (D + Z) = 6031
Step 9: Simplifying further, we find:
1000A + 100B + 10C + 100X + 10Y + (D + Z) = 6031
1000A + 100B + 10C + 100X + 10Y + T = 6031
1000A + 100B + 10C + 100X + 10Y = 6031 - T
Step 10: Since we are looking for the sum of the digits in the original 4-digit integer (ABCD), we know that A, B, C, and D are all between 0 and 9. Hence, the sum of the digits will be between 0 and 36.
Step 11: Based on the equation from step 9, the sum of the digits (A + B + C + D) can be represented as (6031 - T - 100X - 10Y). Since we know that the sum of digits lies between 0 and 36, we can say:
0 ≤ 6031 - T - 100X - 10Y ≤ 36
Step 12: Now, we need to consider the possible values for T, X, and Y that satisfy the inequality in step 11. We can do this by checking all possible combinations of T, X, and Y and calculating 6031 - T - 100X - 10Y for each combination.
Step 13: Find the combination that satisfies the inequality and corresponds to a valid sum of digits.
Step 14: Finally, sum the individual digits of the original 4-digit integer to find the answer.
Please note that step 12 involves trying out different combinations, which may require manual calculation or the use of a computer program.