A number, when divided by the sum of 555 and 445, gives two times their difference as the quotient and 30 as the remainder. What is the number?

Introduction / Context:
This question combines the concepts of sum and difference of two numbers with the standard relation between dividend, divisor, quotient, and remainder. It is a typical example of using basic arithmetic expressions in a word problem.

Given Data / Assumptions:
- Two numbers are 555 and 445.
- The divisor is the sum of these numbers: 555 + 445.
- The quotient is two times their difference: 2 * (555 − 445).
- The remainder is 30.
- We must find the original dividend (the number being divided).

Concept / Approach:
When a number N is divided by a divisor d and gives quotient q and remainder r, the relationship is N = d * q + r. We can compute d and q from the given sum and difference conditions and then substitute them into this formula, along with r = 30, to find N.

Step-by-Step Solution:
Compute the sum: 555 + 445 = 1000. So the divisor d = 1000. Compute the difference: 555 − 445 = 110. The quotient is two times this difference: q = 2 * 110 = 220. The remainder r is given as 30. Use N = d * q + r = 1000 * 220 + 30 = 220000 + 30 = 220030.

Verification / Alternative Check:
We can check by performing the division: 220030 ÷ 1000 gives quotient 220 and remainder 30, because 1000 × 220 = 220000 and 220030 − 220000 = 30. This exactly matches the conditions in the problem. The divisor does equal the sum 555 + 445 = 1000, and the quotient 220 is indeed double the difference 110, confirming the result.

Why Other Options Are Wrong:
22030 would give a much smaller quotient when divided by 1000, namely 22 with remainder 30, which does not match two times the difference.
1220 or 1250 are far too small and cannot produce quotient 220 when divided by 1000.
220003 divided by 1000 would not produce a quotient of exactly 220 with remainder 30.

Common Pitfalls:
A common mistake is to miscalculate the sum and difference of 555 and 445 or confuse which of them should be doubled. Others may incorrectly apply the formula for N in terms of divisor, quotient, and remainder, for example by adding instead of multiplying. Remember that N must be divisor times quotient plus remainder to satisfy the division algorithm.

Final Answer:
The required number that satisfies all the given conditions is 220030.