$(x - 968) \div 79 \times 4 = 512$

Concept & Logic This is a linear equation problem involving the reversal of arithmetic operations. To find the unknown variable $x$, isolate it by systematically moving constants to the right side of the equation using their inverse operations. Step-by-Step Solution * Given equation: $$ (x - 968) \div 79 \times 4 = 512 $$ * Step 1: Divide both sides by $4$ to isolate the division term. $$ (x - 968) \div 79 = \frac{512}{4} $$ $$ (x - 968) \div 79 = 128 $$ * Step 2: Multiply both sides by $79$ to eliminate the division. $$ x - 968 = 128 \times 79 $$ * Step 3: Calculate the product. To do this quickly, use $(80 - 1)$ instead of $79$. $$ 128 \times (80 - 1) = (128 \times 80) - 128 $$ $$ 10240 - 128 = 10112 $$ So, $x - 968 = 10112$. * Step 4: Add $968$ to both sides to solve for $x$. $$ x = 10112 + 968 $$ $$ x = 11080 $$ Exam Strategy & Shortcut **Unit Digit Method:** Once you reach $x - 968 = 128 \times 79$, look at the unit digits. The unit digit of the product $128 \times 79$ is the last digit of $8 \times 9$, which is $2$. So, $x - \text{(something ending in 8)} = \text{(something ending in 2)}$. Therefore, the unit digit of $x$ must be $8 + 2 = 10 \implies 0$. Looking at the options, only $10190$ and $11080$ end in $0$. A quick estimate of $128 \times 80 \approx 10240$, plus $968$, puts the answer comfortably over $11000$, making $11080$ the only logical choice. Common Pitfall A common error is applying the order of operations incorrectly when unwinding the equation, such as trying to divide $968$ by $79$ first before clearing the brackets. Always treat the entire bracketed expression as a single variable until it is completely isolated. Final Answer **Therefore, the correct answer is 11080.**