$8888 + 848 + 88 - x = 7337 + 737$

### Concept & Formula This problem tests the ability to manipulate large algebraic equations by transposing terms. The most efficient method is to simplify both sides of the equation independently before isolating the variable. $$ \text{If } \Sigma L - x = \Sigma R, \text{ then } x = \Sigma L - \Sigma R $$ ### Step-by-Step Solution * **Given:** $8888 + 848 + 88 - x = 7337 + 737$. * **Step 1:** Calculate the sum on the Left-Hand Side (LHS). $8888 + 848 + 88$ First add the smaller numbers: $848 + 88 = 936$. Then add to the main number: $8888 + 936 = 9824$. * **Step 2:** Calculate the sum on the Right-Hand Side (RHS). $7337 + 737 = 8074$ * **Step 3:** Substitute the sums back into the original equation. $9824 - x = 8074$ * **Step 4:** Transpose $x$ and the RHS sum to isolate $x$. $x = 9824 - 8074$ * **Step 5:** Perform the final subtraction. $x = 1750$ ### Exam Strategy & Shortcut Instead of full addition, check the **last two digits**. LHS last two digits: $88 + 48 + 88 = 224$ (so it ends in $24$). RHS last two digits: $37 + 37 = 74$ (ends in $74$). Equation for last two digits: $24 - x = 74$. To subtract, borrow $100$: $124 - x = 74 \rightarrow x = 124 - 74 = 50$. The answer must end in $50$. Unfortunately, all options (a, b, c, d) end in $50$. Let's step it up and check the **hundreds digit** without summing everything. LHS hundreds sum: $800 + 800 + 0 = 1600$. Plus the carry of $2$ from the tens ($224$), so $1800$. RHS hundreds sum: $300 + 700 = 1000$. Equation for hundreds: $18 - x = 10 \rightarrow x = 8$ (or $7$ depending on borrowing in the exact calculation). Let's just do the last three digits for safety: LHS: $888 + 848 + 088 = 1824$. RHS: $337 + 737 = 1074$. $1824 - 1074 = 750$. The answer must end in $750$. Only option (d) $1750$ matches. ### Common Pitfall A common error is improper alignment of place values when performing manual addition, specifically adding $88$ to the hundreds column instead of the tens column. Always align numbers right-to-left before adding. ### Final Answer Therefore, the correct answer is **1750**.