In a proportion the product of 1st and 4th terms is 40 and that of 2nd and 3rd terms is $2.5x$. Then the value of $x$ is

### Concept & Proportion Rule In any mathematical proportion consisting of four terms ($a : b : : c : d$), the fundamental rule states that the product of the extreme terms equals the product of the mean terms. $$ \text{Product of Extremes} = \text{Product of Means} $$ ### Step-by-Step Solution * **Given:** Product of 1st and 4th terms (Extremes) = $40$. Product of 2nd and 3rd terms (Means) = $2.5x$. * **Equate the products:** Based on the proportion rule, set them equal to each other. $40 = 2.5x$ * **Solve for $x$:** Divide both sides by $2.5$. $x = \frac{40}{2.5}$ * **Simplify:** Multiply the numerator and denominator by $10$ to remove the decimal. $x = \frac{400}{25}$ * **Calculate:** Divide $400$ by $25$. Since $100 / 25 = 4$, then $400 / 25 = 4 \times 4 = 16$. $x = 16$ ### Exam Strategy & Shortcut Think of $2.5$ as the fraction $\frac{5}{2}$. The equation becomes $40 = \frac{5}{2}x$. Multiply both sides by $2$ to get $80 = 5x$, and then divide by $5$ to quickly find $x = 16$. Using fractions is often mentally faster than long division with decimals. ### Common Pitfall Dividing in the wrong order, such as calculating $x = 2.5 / 40$, which results in a fraction rather than isolating the variable correctly. ### Final Answer Therefore, the correct answer is **$16$**.