The solution to \(\( \frac{1}{2} \div 15 \times \left( 2 \frac{1}{16} - 1 \frac{1}{2} \right) \) is \(\frac{3}{160}\)\).
To solve the expression \(\( \frac{1}{2} \div 15 \times \left( 2 \frac{1}{16} - 1 \frac{1}{2} \right) \)\), let's break it down step by step:
1. Convert the mixed numbers to improper fractions:
- \(\( 2 \frac{1}{16} = \frac{2 \times 16 + 1}{16} = \frac{33}{16} \)\)
- \(\( 1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2} \)\)
2. Subtract the improper fractions:
- Find a common denominator for \(\(\frac{33}{16}\)\) and \(\(\frac{3}{2}\): \[\)
\(\frac{3}{2} = \frac{3 \times 8}{2 \times 8} = \frac{24}{16} \]\)
- Now, subtract the fractions:
\(\[ \frac{33}{16} - \frac{24}{16} = \frac{33 - 24}{16} = \frac{9}{16} \]\)
3. Multiply the fractions:
- First, note that \(\frac{1}{2} \div 15\) is the same as \(\(\frac{1}{2} \times \frac{1}{15}\)\):
\(\[ \frac{1}{2} \times \frac{1}{15} = \frac{1}{30} \]\)
- Now, multiply \(\(\frac{1}{30}\) by \(\frac{9}{16}\): \[\)
\(\frac{1}{30} \times \frac{9}{16} = \frac{1 \times 9}{30 \times 16} = \frac{9}{480} \]\)
4. Simplify the fraction:
- Divide the numerator and the denominator by their greatest common divisor (GCD), which is 3:
\(\[ \frac{9 \div 3}{480 \div 3} = \frac{3}{160} \]\)
So, the solution to \(\( \frac{1}{2} \div 15 \times \left( 2 \frac{1}{16} - 1 \frac{1}{2} \right) \) is \(\frac{3}{160}\)\).