What misstep am I making in trying to simplify $\frac{18 - \frac 7 {3x}} {\frac 7 {18x} - 3}$?

Question

Simplify $\dfrac{18 - \dfrac 7 {3x}} {\dfrac 7 {18x} - 3}$?

I'm having a hard time simplifying this particular expression and am seeking any type of assistance in solving it.

In the expression $$\frac{18 - \dfrac 7 {3x}} {\dfrac 7 {18x} - 3}$$ I split the problems into two separate entities.

For the numerator, I get $3x$ for the LCD and then rewrite the fraction as $$54x-7\frac 7 {3x}$$ As for the denominator, I get $18x$ for the LCD and then rewrite the fraction as $$7-\frac{54x}{18x}$$

When I begin to divide, I switch the sign from division to multiplication and swap the numerator with the denominator ($7-\frac{54x}{18x}$ becomes $\frac{18x}{7-54x}$).

The product I get is $$\frac{972x^2-126x}{21x - 162x^2}$$ When I simplify I get $6-6$ which is zero. Is this answer correct?

Answer 1

Using your method, the numerator $N$ is $$N=18-\dfrac{7}{3x}=\dfrac{54x}{3x}-\dfrac{7}{3x}=\dfrac{54x-7}{3x}$$ The denominator is $$D=\dfrac{7}{18x}-3=\dfrac{7}{18x}-\dfrac{54x}{18x}=\dfrac{7-54x}{18x}$$ Thus, the given fraction is $$F=N÷D=\dfrac{54x-7}{3x}÷\dfrac{7-54x}{18x}$$ $$\implies F=\dfrac{54x-7}{3x}×\dfrac{18x}{7-54x}=\boxed{-6}$$

Hope this helps. Ask anything if not clear :)

Answer 2

It doesn't need to be this complicated. multilply the numerator and denominator by $18x$ leaving $$\frac{18 - \frac 7 {3x}} {\frac 7 {18x} - 3} =\frac{324x- 42} {7 - 54x}=-6$$

obviously you are a middle school student so well done for asking for help

Answer 3

$\require{cancel}$ As we discussed about technique in another post

$$\dfrac{18 - \dfrac 7 {3x}} {\dfrac 7 {18x} - 3} =\dfrac{\cancel{54x-7}}{3x } \cdot \dfrac{18x}{(-1)(\cancel{54x-7})} =-6$$