The arithmetic mean of $15$ numbers is $41.4$. Then the sum of these numbers is

### Concept & Formula The foundational formula for average bridges three variables: Average, Sum, and Count. If you know the average and the total count, you can instantly find the total sum by rearranging the standard formula. $$ \text{Sum} = \text{Average} \times \text{Total Count} $$ ### Step-by-Step Solution * **Given:** The arithmetic mean ($\text{Average}$) is $41.4$. The total number of values ($\text{Count}$) is $15$. * **Calculation:** Apply the rearranged formula directly: $\text{Sum} = 41.4 \times 15$ To calculate easily, break it down: $\text{Sum} = 41.4 \times (10 + 5)$ $\text{Sum} = 414 + 207$ $\text{Sum} = 621$ ### Exam Strategy & Shortcut **Unit Digit & Decimal Elimination:** You need to multiply $41.4 \times 15$. You know $4 \times 5 = 20$, which means the decimal part will result in a whole number (specifically, $.4 \times 15 = 6$). There will be no decimals in the final answer. Furthermore, $41 \times 15$ is roughly $40 \times 15 = 600$. The answer must be slightly above $600$. Between options (c) $620$ and (d) $621$, exact unit digit tracking ($4 \times 5 = 20 \rightarrow \text{adds } 0$ to unit, but $1 \times 5 = 5$, plus the $6$ from the decimal gives $11$, so it ends in $1$). The answer must be $621$. ### Common Pitfall Students sometimes get confused by the decimal and write $41.4 \times 15 = 6210$, placing the decimal point incorrectly. Breaking the multiplier $15$ into $(10 + 5)$ is the safest way to avoid decimal placement errors. ### Final Answer **Therefore, the correct answer is 621.**