1. **State the problem:** We need to find the value of $x$ in the proportion $\frac{15}{66} = \frac{x}{99}$.\n\n2. **Use the cross-multiplication formula:** For two ratios $\frac{a}{b} = \frac{c}{d}$, the cross products are equal: $a \times d = b \times c$.\n\n3. **Apply cross multiplication:** $15 \times 99 = 66 \times x$.\n\n4. **Calculate the left side:** $15 \times 99 = 1485$. So, $1485 = 66x$.\n\n5. **Solve for $x$ by dividing both sides by 66:** $$x = \frac{1485}{66}$$\n\n6. **Simplify the fraction:** Both numerator and denominator can be divided by 3.\n$$x = \frac{\cancel{3}495}{\cancel{3}22} = \frac{495}{22}$$\n\n7. **Divide 495 by 22:** $495 \div 22 = 22.5$.\n\n**Final answer:** $x = 22.5$