Is this correct?
$ \frac{1 + 0.05}{1 + x} = \frac{1 + 0.02}{1 + 0.06} $
$\frac{1.05}{1 + x} = \frac{1.02}{1.06} $
$\frac{1.05}{1 + x} = 0.9623 $
$1.05 = 0.9623(1 + x) $
$ \frac{1.05}{0.9623} = 1 + x$
$1.0911 = 1 + x$
$x = 1.0911 – 1$
$x = 0.0911 = 9.11\% $
On
Wolfram Alpha confirms Claude's answer and by extension roughly validates yours. It is perfectly alright to round answers in many circumstances, in fact in science often it is actually a rule that you do so, however you should have rounded to 0.0912, not 0.0911, because 0.09117 is closer to 0.0912 than it is to 0.0911. Thus you should round up rather than down.
I should avoid the division on your third line. At the second line, you had (and this is correct)
1.05 / (1+ x) = 1.02 / 1.06
Multiply both sides by 1.06 (1+x). You then get 1.05 * 1.06 = 1.02 * (1 + x) that is to say 1.113 = 1.02 + 1.02 * x; then x = 93 / 1020 = 31 / 340 which is the exact value for your problem. Now, make the division and you get x = 0.0911765 which is not exactly your answer even if very close.
In my opinion, you must delay the divisions as long as you can because you can loose accuracy when you start substracting result from several divisions.