Which values of a, b, and c correctly complete the division?

One-fourth divided by StartFraction 5 Over 6 EndFraction = StartFraction 1 Over a EndFraction times StartFraction b Over c EndFraction
a = 4, b = 5, c = 6
a = 1, b = 6, c = 5
a = 1, b = 5, c = 6
a = 4, b = 6, c = 5

1 answer

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 \(c\):

  1. Option a: \(a = 4, b = 5, c = 6\)

    \[ 3(4)(6) = 72, \quad 10(5) = 50 \quad \text{(not equal)} \]

  2. Option b: \(a = 1, b = 6, c = 5\)

    \[ 3(1)(5) = 15, \quad 10(6) = 60 \quad \text{(not equal)} \]

  3. Option c: \(a = 1, b = 5, c = 6\)

    \[ 3(1)(6) = 18, \quad 10(5) = 50 \quad \text{(not equal)} \]

  4. 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:

\[ \boxed{4, 6, 5} \]