The sum of the three consecutive even numbers is 44 more than the average of these numbers. Which of the following is the third largest of these numbers?

### Concept & Formula To solve problems involving consecutive even numbers, we can represent them algebraically as $x$, $x+2$, and $x+4$. The problem requires translating the relationship between their sum and their average into a linear equation. ### Step-by-Step Solution * **Given:** Let the three consecutive even numbers be $x$, $x+2$, and $x+4$. * **Sum:** Their sum is $(x) + (x+2) + (x+4) = 3x + 6$. * **Average:** Their average is the middle number, which is $x+2$. (Alternatively, $\frac{3x+6}{3} = x+2$). * **Equation:** According to the question, the sum is 44 more than the average: $$3x + 6 = (x + 2) + 44$$ * **Calculation:** $$3x + 6 = x + 46$$ $$2x = 40$$ $$x = 20$$ * **Result:** The numbers are 20, 22, and 24. * The "third largest" of three numbers is simply the smallest number, which is 20. Since 20 is not among the given options (16, 18, 24), the answer is "None of these". ### Exam Strategy & Shortcut Instead of writing out the full equation, you can use the property of averages for consecutive numbers in an arithmetic progression: the average is always the middle number. Let the middle number be $M$. The numbers are $M-2$, $M$, and $M+2$. Sum = $3M$. Average = $M$. The question states: $3M = M + 44 \Rightarrow 2M = 44 \Rightarrow M = 22$. The smallest number (third largest) is $M-2 = 20$. This mental math takes less than 10 seconds. ### Common Pitfall The most common mistake is misinterpreting "third largest". In a set of three numbers, the 1st largest is the maximum, the 2nd largest is the middle, and the 3rd largest is the minimum. Students often get confused and look for a non-existent number, or forget to check the "None of these" option when their calculated answer (20) doesn't match the numerical options. ### Final Answer Therefore, the correct answer is None of these.