$5566 - 7788 + 9988 = x + 4444$

### Concept & Formula This problem requires transposing terms across an algebraic equation to isolate the unknown variable $x$. $$ \text{If } A + B = x + C, \text{ then } x = A + B - C $$ ### Step-by-Step Solution * **Given:** The equation $5566 - 7788 + 9988 = x + 4444$. * **Step 1:** Transpose $4444$ to the left side of the equation to isolate $x$. $x = 5566 - 7788 + 9988 - 4444$ * **Step 2:** Group terms logically to simplify mental calculations. Notice that $(9988 - 7788)$ is an easy subtraction. $x = 5566 + (9988 - 7788) - 4444$ * **Step 3:** Perform the grouped subtraction. $x = 5566 + 2200 - 4444$ * **Step 4:** Add the positive terms. $x = 7766 - 4444$ * **Step 5:** Perform the final subtraction. $x = 3322$ ### Exam Strategy & Shortcut Observe the digits. Almost all numbers are repeating multiples of $1111$. You can temporarily factor out $1111$ to simplify: $1111 \times (5 - 7 + 9 - 4) = x$ Calculate the simple bracket: $5 - 7 + 9 - 4 = 14 - 11 = 3$. Multiply back: $1111 \times 3 = 3333$. Wait, let's re-verify: $5566$ is $1111 \times 5$ ? No, $5566$ is $1111 \times 5.01...$ Ah, the numbers are $5566, 7788, 9988, 4444$. They are multiples of $11$, not $1111$. Let's stick to the fast unit digit tracking: $6 - 8 + 8 - 4 = 2$. Tens: $6 - 8 + 8 - 4 = 2$. Hundreds: $5 - 7 + 9 - 4 = 3$. Thousands: $5 - 7 + 9 - 4 = 3$. Result is precisely $3322$. ### Common Pitfall Students often waste time doing sequential borrowing and carrying on large numbers instead of recognizing grouping opportunities (like $9988 - 7788 = 2200$) or checking last digits. ### Final Answer Therefore, the correct answer is **3322**.