To solve the equation given:
\[ \frac{1}{4} \div \frac{5}{6} = \frac{1}{a} \times \frac{b}{c} \]
First, we need to perform the left side of the equation, which can be done by multiplying by the reciprocal:
\[ \frac{1}{4} \div \frac{5}{6} = \frac{1}{4} \times \frac{6}{5} = \frac{1 \cdot 6}{4 \cdot 5} = \frac{6}{20} = \frac{3}{10} \]
Now we know that:
\[ \frac{3}{10} = \frac{1}{a} \times \frac{b}{c} \]
Next, let's analyze the options given for \(a\), \(b\), and \(c\):
Option: a = 4, b = 5, c = 6
\[ \frac{1}{4} \times \frac{5}{6} = \frac{5}{24} \quad \text{(not equal to } \frac{3}{10}\text{)} \]
Option: a = 1, b = 6, c = 5
\[ \frac{1}{1} \times \frac{6}{5} = \frac{6}{5} \quad \text{(not equal to } \frac{3}{10}\text{)} \]
Option: a = 1, b = 5, c = 6
\[ \frac{1}{1} \times \frac{5}{6} = \frac{5}{6} \quad \text{(not equal to } \frac{3}{10}\text{)} \]
Option: a = 4, b = 6, c = 5
\[ \frac{1}{4} \times \frac{6}{5} = \frac{6}{20} = \frac{3}{10} \quad \text{(this is equal to } \frac{3}{10}\text{)} \]
Thus, the correct values to complete the division are:
a = 4, b = 6, c = 5.