If the average of a number, its $75\%$ and its $25\%$ is $240$, then the number is

### Concept & Formula The average of multiple terms is their total sum divided by the number of terms. By expressing all terms as percentages of a single unknown variable, you can establish a simple linear equation. $$\text{Average} = \frac{\text{Sum of Terms}}{\text{Number of Terms}}$$ ### Step-by-Step Solution **Given:** * The three terms are: a number, its $75\%$, and its $25\%$. * The average of these three terms is $240$. **Calculation:** * Let the unknown base number be $x$. * The three terms expressed algebraically are $x$, $0.75x$, and $0.25x$. * Set up the mathematical average equation: $$\frac{x + 0.75x + 0.25x}{3} = 240$$ * Combine the like terms in the numerator: $$\frac{2x}{3} = 240$$ * Isolate $x$ by multiplying both sides by $3$: $$2x = 720$$ * Divide by $2$ to find the final number: $$x = 360$$ ### Exam Strategy & Shortcut Recognize percentage complements to simplify the arithmetic immediately. $75\%$ and $25\%$ perfectly add up to $100\%$, which constitutes exactly $1$ whole number. Thus, you have the original number ($1$) plus the two percentages which equal a second whole number ($1$), yielding a total sum of $2$ times the number. If the average of $3$ distinct items is $240$, their total sum must be $240 \times 3 = 720$. Since $2 \times \text{Number} = 720$, the number is simply $360$. ### Common Pitfall A standard error is forgetting to include the original base number itself in the average calculation, effectively only averaging the $75\%$ and $25\%$ terms. This leads to incorrect denominator usage and a fundamentally flawed equation. ### Final Answer **Therefore, the correct answer is 360.**