The sum of four consecutive even numbers $A, B, C$ and $D$ is $180$. What is the sum of the set of next four consecutive even numbers?

### Concept & Logic Consecutive even numbers are spaced exactly $2$ units apart (e.g., $x, x+2, x+4, x+6$). When shifting to the "next set" of four consecutive even numbers, each individual number in the new set is a specific, constant distance away from its corresponding counterpart in the original set. ### Step-by-Step Solution * **Given:** * Set 1: $A, B, C, D$ (four consecutive even numbers). * $A + B + C + D = 180$. * **Calculation / Deduction:** * **Method 1: Standard Algebra** 1. Let the numbers be $x, x+2, x+4, x+6$. 2. Set up the equation: $x + (x+2) + (x+4) + (x+6) = 180$. 3. Combine terms: $4x + 12 = 180$. 4. Solve for $x$: $4x = 168 \implies x = 42$. 5. The first set is $42, 44, 46, 48$. 6. The "next four" consecutive even numbers continue the sequence: $50, 52, 54, 56$. 7. Sum the new set: $50 + 52 + 54 + 56 = 212$. ### Exam Strategy & Shortcut Use the "Shift" strategy. You do not need to find the actual numbers to solve this! Think about how the sets align: Set 1: $A$, $B$, $C$, $D$ Set 2: $A+8$, $B+8$, $C+8$, $D+8$ Why $+8$? The next even number after $D$ is $D+2$. Since there are four numbers in the set, the first number of the *new* set is $4 \times 2 = 8$ units larger than the first number of the *old* set. Therefore, every single number in the new set is exactly $8$ greater than its counterpart. Total increase $= 4 \text{ numbers} \times 8 = 32$. New Sum $= \text{Old Sum} + 32 = 180 + 32 = 212$. This takes 5 seconds. ### Common Pitfall The most common trap is wasting time executing the full algebraic calculation to find $x = 42$. While it yields the correct answer, it consumes precious time that could be saved by applying the logical shift shortcut. Another minor mistake is finding the new set of numbers but making an arithmetic error when adding $50+52+54+56$. ### Final Answer Therefore, the correct answer is 212.