The average of $X_1$, $X_2$ and $X_3$ is 14. Twice the sum of $X_2$ and $X_3$ is 30. What is the value of $X_1$?

### Concept & Logic The relationship between average and total sum is the key to solving this. If you know the average of a set, you can find the total sum. By finding the total sum of all three variables and subtracting the sum of the last two, you isolate the first variable. $$Total\ Sum = Average \times Number\ of\ Items$$ ### Step-by-Step Solution * **Given:** The average of $X_1$, $X_2$, and $X_3$ is 14. * Twice the sum of $X_2$ and $X_3$ is 30: $2(X_2 + X_3) = 30$. * **Calculation:** First, find the total sum of all three variables. * $X_1 + X_2 + X_3 = 14 \times 3 = 42$ * Next, solve the second equation to find the exact sum of $X_2$ and $X_3$. * $X_2 + X_3 = \frac{30}{2} = 15$ * Finally, substitute this value back into the total sum equation. * $X_1 + 15 = 42$ * $X_1 = 42 - 15 = 27$ ### Exam Strategy & Shortcut Think of this in blocks: The total weight of all 3 items is $14 \times 3 = 42$. The combined weight of the last two items is half of 30, which is 15. The first item must make up the difference: $42 - 15 = 27$. This can be done entirely mentally in under 10 seconds. ### Common Pitfall A very common mistake is subtracting 30 directly from 42, forgetting that the 30 represents *twice* the sum of $X_2$ and $X_3$. Always simplify sub-equations completely before substituting them into the main equation. ### Final Answer **Therefore, the correct answer is 27.**