How do I clear out or solve ratios with equations in the denominator?

Question

Example 7/(4q+1)=14/(7-2q). My book says answer is q=1/2.

Answer 1

7/(4q + 1) = 14/(7 - 2q)

Divide by 7

1/(4q + 1) = 2/(7 - 2q)

Invert

4q + 1 = (7 -2q)/2

Multiply by 2

8q + 2 = 7 - 2q

Add to 2q to each side

10q + 2 = 7

Subtract 2 from each side

10q = 5 and q = 1/2

 

You could have simply done a cross multiplication, but then the collection process would have been messier, not harder but messier.