If $(x + 1) : 8 = 3.75 : 7$, then the value of $x$ is

### Concept & Solving Proportions A proportion equates two ratios, forming a linear equation when a variable is present. It can be solved quickly via cross-multiplication. $$ \frac{a}{b} = \frac{c}{d} \implies ad = bc $$ ### Step-by-Step Solution * **Given:** $\frac{x + 1}{8} = \frac{3.75}{7}$ * **Cross-multiply:** Multiply the diagonals. $7(x + 1) = 8 \times 3.75$ * **Calculate the right side:** $8 \times 3.75 = 30$ * **Simplify the equation:** $7(x + 1) = 30$ $7x + 7 = 30$ * **Solve for $x$:** $7x = 30 - 7$ $7x = 23$ $x = \frac{23}{7}$ * **Convert to mixed fraction:** $\frac{23}{7} = 3$ with a remainder of $2$, which is $3\frac{2}{7}$. ### Exam Strategy & Shortcut To quickly multiply $8 \times 3.75$, double the $3.75$ to get $7.5$, then double again to get $15$, then double a third time to get $30$. Thus, $7(x+1) = 30$, leading directly to $x+1 = \frac{30}{7}$, so $x = \frac{30}{7} - \frac{7}{7} = \frac{23}{7} = 3\frac{2}{7}$. ### Common Pitfall Forgetting to distribute the $7$ to both the $x$ and the $1$, or forgetting to subtract the $1$ at the very end when isolating $x$. ### Final Answer Therefore, the correct answer is **$3\frac{2}{7}$**.