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

Question

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:

$$\frac{1}{4}÷\frac{5}{6}$$

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: $a=4,b=5,c=6$

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

2. Option b: $a=1,b=6,c=5$

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

3. Option c: $a=1,b=5,c=6$

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

4. Option d: $a=4,b=6,c=5$

   $$3(4)(5)=60,10(6)=60\text{(equal)}$$

So the correct values that satisfy the equation are:

$$\boxed{{\displaystyle 4,6,5}}$$