To solve this problem, we start by simplifying the left-hand side:
\[ \frac{1}{4} \div \frac{5}{6} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{1}{4} \times \frac{6}{5} = \frac{1 \times 6}{4 \times 5} = \frac{6}{20} \]
Now we can simplify \(\frac{6}{20}\):
\[ \frac{6}{20} = \frac{3}{10} \]
Next, we set this equal to the right-hand side of the equation:
\[ \frac{3}{10} = \frac{1}{a} \times \frac{b}{c} \]
To solve for \(a\), \(b\), and \(c\), let's rewrite \(\frac{1}{a} \times \frac{b}{c}\):
\[ \frac{b}{ac} \]
So we have:
\[ \frac{3}{10} = \frac{b}{ac} \]
This implies that:
\[ 3ac = 10b \]
Now we can test the provided options for \(a\), \(b\), and \(c\):
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Option a: \(a = 4, b = 5, c = 6\)
\[ 3(4)(6) = 72, \quad 10(5) = 50 \quad \text{(not equal)} \]
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Option b: \(a = 1, b = 6, c = 5\)
\[ 3(1)(5) = 15, \quad 10(6) = 60 \quad \text{(not equal)} \]
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Option c: \(a = 1, b = 5, c = 6\)
\[ 3(1)(6) = 18, \quad 10(5) = 50 \quad \text{(not equal)} \]
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Option d: \(a = 4, b = 6, c = 5\)
\[ 3(4)(5) = 60, \quad 10(6) = 60 \quad \text{(equal)} \]
So the correct values that satisfy the equation are:
\[ \boxed{4, 6, 5} \]