18/55 : 24/25 : 15/16 × 33/40

To simplify the expression \( \frac{18}{55} : \frac{24}{25} : \frac{15}{16} \times \frac{33}{40} \), we'll handle the division and multiplication step by step. 1. **Convert the ratio into multiplication by taking the reciprocal:** \[ \frac{18}{55} : \frac{24}{25} = \frac{18}{55} \div \frac{24}{25} = \frac{18}{55} \times \frac{25}{24} = \frac{18 \times 25}{55 \times 24} \] \[ = \frac{450}{1320} \] We can simplify this fraction by finding the greatest common divisor. Both the numerator and the denominator can be divided by 30: \[ = \frac{15}{44} \] 2. Now take this result and divide it by \( \frac{15}{16} \): \[ \frac{15}{44} : \frac{15}{16} = \frac{15}{44} \div \frac{15}{16} = \frac{15}{44} \times \frac{16}{15} = \frac{16}{44} \] \[ = \frac{4}{11} \] 3. Now multiply this result by \( \frac{33}{40} \): \[ \frac{4}{11} \times \frac{33}{40} = \frac{4 \times 33}{11 \times 40} \] \[ = \frac{132}{440} \] We can simplify this by dividing the numerator and the denominator by 44: \[ = \frac{3}{10} \] Therefore, the final result of the expression \( \frac{18}{55} : \frac{24}{25} : \frac{15}{16} \times \frac{33}{40} \) is \( \frac{3}{10} \).