Solve $2\frac{1.5}{5} + 2\frac{1}{6} - 1\frac{3.5}{15} = \left(\frac{x^{1/3}}{4}\right) + 1\frac{7}{30}$

Concept & Strategy This problem combines mixed fractions, decimals within fractions, and fractional exponents. The strategy is to first clear the decimals from the fractions by converting them into standard proper fractions, then isolate the unknown variable $x$. Step-by-Step Solution * Simplify the fractional terms containing decimals on the left-hand side (LHS): $$2\frac{1.5}{5} = 2 + \frac{1.5}{5} = 2 + 0.3 = 2\frac{3}{10} = \frac{23}{10}$$ $$1\frac{3.5}{15} = 1 + \frac{3.5}{15}$$ Multiply numerator and denominator of the fraction part by 2 to remove the decimal: $$1 + \frac{7}{30} = \frac{37}{30}$$ * Convert the remaining mixed fractions to improper fractions: $$2\frac{1}{6} = \frac{13}{6}$$ $$1\frac{7}{30} = \frac{37}{30}$$ * Rewrite the entire equation with proper fractions: $$\frac{23}{10} + \frac{13}{6} - \frac{37}{30} = \frac{x^{1/3}}{4} + \frac{37}{30}$$ * Solve the LHS by finding the LCM of 10, 6, and 30, which is 30: $$\frac{23 \times 3}{30} + \frac{13 \times 5}{30} - \frac{37}{30}$$ $$\frac{69 + 65 - 37}{30} = \frac{97}{30}$$ * Substitute the LHS sum back into the main equation and isolate $x$: $$\frac{97}{30} = \frac{x^{1/3}}{4} + \frac{37}{30}$$ * Subtract $\frac{37}{30}$ from both sides: $$\frac{97 - 37}{30} = \frac{x^{1/3}}{4}$$ $$\frac{60}{30} = \frac{x^{1/3}}{4}$$ $$2 = \frac{x^{1/3}}{4}$$ * Multiply by 4: $$x^{1/3} = 8$$ * Cube both sides to find $x$: $$x = 8^3 = 512$$ Exam Strategy & Shortcut Notice the $\frac{37}{30}$ that appears on both sides once you convert $1\frac{3.5}{15}$. If you keep it in mixed form and move $1\frac{7}{30}$ to the left side, you get $(2.3 + 2.166... - 1.233... - 1.233...)$. Spotting that $1\frac{3.5}{15}$ is mathematically identical to $1\frac{7}{30}$ saves massive amounts of calculation time, as you can just combine the integer terms and evaluate the remaining simple fractions. Common Pitfall Students often get intimidated by a decimal inside a fraction (like $\frac{1.5}{5}$) and attempt to multiply the entire mixed number by 10 unnecessarily. Simply evaluating $\frac{1.5}{5}$ as $0.3$ and converting it to $\frac{3}{10}$ is much cleaner and less error-prone. Final Answer Therefore, the correct answer is **512**.