Question

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

Answer 1

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:

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