Simplify the expression: ( (1/6 + 0.1 + 1/15) : (1/6 + 0.1 - 1/15) * 2.52 ) / ( (0.5 - 1/3 + 0.25 - 1/5) : (0.25 - 1/6) * 7/13 )

Solution:

1. Numerator calculation:
   • First part: \( \frac{1}{6} + 0.1 + \frac{1}{15} = \frac{1}{6} + \frac{1}{10} + \frac{1}{15} = \frac{5+3+2}{30} = \frac{10}{30} = \frac{1}{3} \)
   • Second part: \( \frac{1}{6} + 0.1 - \frac{1}{15} = \frac{1}{6} + \frac{1}{10} - \frac{1}{15} = \frac{5+3-2}{30} = \frac{6}{30} = \frac{1}{5} \)
   • Division: \( \frac{1}{3} : \frac{1}{5} = \frac{1}{3} \times 5 = \frac{5}{3} \)
   • Multiplication: \( \frac{5}{3} \times 2.52 = \frac{5}{3} \times \frac{252}{100} = \frac{5}{3} \times \frac{63}{25} = \frac{1}{1} \times \frac{21}{5} = \frac{21}{5} \)
2. Denominator calculation:
   • First part: \( 0.5 - \frac{1}{3} + 0.25 - \frac{1}{5} = \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} = \frac{30-20+15-12}{60} = \frac{13}{60} \)
   • Second part: \( 0.25 - \frac{1}{6} = \frac{1}{4} - \frac{1}{6} = \frac{3-2}{12} = \frac{1}{12} \)
   • Division: \( \frac{13}{60} : \frac{1}{12} = \frac{13}{60} \times 12 = \frac{13}{5} \)
   • Multiplication: \( \frac{13}{5} \times \frac{7}{13} = \frac{1}{5} \times \frac{7}{1} = \frac{7}{5} \)
3. Final division: \( \frac{21}{5} : \frac{7}{5} = \frac{21}{5} \times \frac{5}{7} = \frac{21}{7} = 3 \)

Ответ: 3