Let's reduce: $\frac{1}{\frac{1}{15}+\frac{1}{10}+\frac{1}{30}}$

ⓘ To add fractions, put them over the same denominator, then add the numerators. We choose $30$, the least common multiple of $15$ and $10$. Multiplying the numerator and the denominator by the same number does not change the fraction: $\frac{1}{15}+\frac{1}{10}$ = $\frac{1\times 2}{15\times 2}+\frac{1\times 3}{10\times 3}$ = $\frac{1\times 2+1\times 3}{15\times 2}$ = $\frac{2+3}{15\times 2}$■To add fractions, put them over the same denominator, then add the numerators: $\frac{2+3}{15\times 2}+\frac{1}{30}$ = $\frac{(2+3)\times 30+15\times 2}{15\times 2\times 30}$

$\frac{1}{\frac{(2+3)\times 30+15\times 2}{15\times 2\times 30}}$

ⓘ Parentheses are computed first: $2+3$ = $5$■Multiplications are done before additions and subtractions: $15\times 2$ = $(10+5)\times 2$ = $10\times 2+5\times 2$ = $20+10$ = $30$

ⅰ $15\times 2$ = $(10+5)\times 2$ = $10\times 2+5\times 2$ = $20+10$ = $30$

$\frac{1}{\frac{5\times 30+30}{30\times 30}}$

ⓘ Multiplications are done before additions and subtractions: $5\times 30$ = $150$

ⅰ $30\times 30$ = $900$

$\frac{1}{\frac{150+30}{900}}$

ⓘ The numerator and the denominator are computed before the division: $150+30$ = $180$

$\frac{1}{\frac{180}{900}}$

We simplify the fraction: $\frac{900}{180}$ by the common factor: $180$ to get the new expression: $5$

▶Simplified form:   $5$