membership to a music club costs $75.members pay $10 per music lesson and nonmembers pay $15 per music lesson. how many music lessons would you have to take for them the cost to be the same for members and nonmembers

2 $15 nonmember music lessons is equal to 3 $10 member music lesson.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4