The sum of three consecutive odd numbers and three consecutive even numbers together is $231$. Also, the smallest odd number is $11$ less than the smallest even number. What is the sum of the largest odd number and the largest even number?

### Concept & Formula This problem requires forming a system of linear equations to manage two independent sequences (one odd, one even) that are linked by a specific condition regarding their starting values. $$\text{Odd Sequence: } o, o+2, o+4$$ $$\text{Even Sequence: } e, e+2, e+4$$ ### Step-by-Step Solution * **Given:** The total sum of the three odds and three evens is $231$. $$(o + o+2 + o+4) + (e + e+2 + e+4) = 231$$ $$3o + 6 + 3e + 6 = 231$$ $$3o + 3e + 12 = 231$$ $$3(o + e) = 219$$ * Divide by $3$ to simplify: $$o + e = 73$$ * **Given:** The smallest odd number ($o$) is $11$ less than the smallest even number ($e$). $$o = e - 11$$ * Substitute this condition into our simplified sum equation: $$(e - 11) + e = 73$$ $$2e = 84$$ $$e = 42$$ * Find the smallest odd number ($o$): $$o = 42 - 11 = 31$$ * Find the largest odd number and largest even number: * Largest odd = $o + 4 = 31 + 4 = 35$ * Largest even = $e + 4 = 42 + 4 = 46$ * Calculate their sum: $$35 + 46 = 81$$ * Comparing $81$ to the options, it is not present. ### Exam Strategy & Shortcut Instead of expanding $o, o+2, o+4$ separately, note that the sum of the largest terms is directly related to the sum of the smallest terms. We know $o + e = 73$. The largest odd is $o + 4$. The largest even is $e + 4$. Their sum is $(o + 4) + (e + 4) = o + e + 8$. Substitute $o + e = 73$: Sum $= 73 + 8 = 81$. This bypasses solving for the individual values of $e$ and $o$ entirely, drastically cutting down computation time. ### Common Pitfall Students often get bogged down solving for the exact value of every single integer in both sequences. As shown in the shortcut, you can often find the sum of the target numbers by manipulating the combined variable $(o+e)$ without ever breaking it apart. ### Final Answer **Therefore, the correct answer is None of these.**