Cool question.

Bruce's interest earned between t=10 and t=11 is equal to Robbie's interest earned between t=16 and t=17, so we can write

100(1+r)^11 - 100(1+r)^10 = 50(1+r)^17 - 100(1+r)^16

We need some common terms, so rewrite as

100(1+r)^10(1+r) - 100(1+r)^10 = 50(1+r)^16(1+r) - 100(1+r)^16

Pull some terms out to get

100(1+r)^10((1+r) - 1) = 50(1+r)^16((1+r) - 1)

Hmm, delicious

2(1+r)^10((1+r) - 1) = (1+r)^16((1+r) - 1)

2(1+r)^10/((1+r)^16) = ((1+r) - 1)/((1+r) - 1)

Shaping up!

2(1+r)^10/((1+r)^16) = 1

2/((1+r)^6) = 1

2 = (1+r)^6

2^(1/6) = 1+r

1.12246 = 1+r

r = 0.12246

Now that we've got r, just plug it into one side of

100(1+r)^11 - 100(1+r)^10 = 50(1+r)^17 - 100(1+r)^16

to find X.

100(1+0.12246)^11 - 100(1+0.12246)^10 = 38.87792

ROCK AND ROLL!