The sum of seven consecutive numbers is $175$. What is the difference between twice the largest number and thrice the smallest number?

### Concept & Strategy The sum of an odd number of consecutive terms (which form an Arithmetic Progression) yields the middle term when divided by the number of terms. This is because the average of an AP is its perfectly centered value. $$\text{Middle Term} = \frac{\text{Sum of Terms}}{\text{Number of Terms}}$$ ### Step-by-Step Solution * **Given:** The sum of $7$ consecutive numbers is $175$. * Calculate the middle term directly by finding the average: $$\text{Middle Term} = \frac{175}{7} = 25$$ * Since $25$ is the middle term of $7$ consecutive integers, there are exactly $3$ terms before it and $3$ terms after it. * The sequence is: $22, 23, 24, \textbf{25}, 26, 27, 28$. * Identify the smallest and largest numbers: * Smallest Number = $22$ * Largest Number = $28$ * The question asks for the difference between twice the largest and thrice the smallest: * Twice the largest = $2 \times 28 = 56$ * Thrice the smallest = $3 \times 22 = 66$ * Calculate the absolute difference: $$|56 - 66| = 10$$ ### Exam Strategy & Shortcut Never use the standard algebraic setup $x + (x+1) + ... + (x+6) = 175$, which simplifies to $7x + 21 = 175$. While it works, it is mathematically heavier. Identifying the middle term via division ($175 / 7 = 25$) allows you to immediately step back by $3$ to find the smallest ($22$) and step forward by $3$ to find the largest ($28$). ### Common Pitfall The phrasing "difference between A and B" in aptitude exams implies the absolute difference (larger value minus smaller value) unless a specific order of subtraction is stated. Calculating $56 - 66 = -10$ and assuming the answer isn't present (leading to "None of these") is a frequent trap. ### Final Answer **Therefore, the correct answer is 10.**