Simplify : 1/(1 +​ (2/3) / 1 + (2/3) + ((8/9)/(1-(2/3)))

Question

Simplify: \(\frac{1}{1+\frac{\frac23}{1+\frac23+\frac{\frac89}{1-\frac23}}}\)

(a) \(\frac{11}{13}\) 

(b) \(\frac{13}{15}\)

(c) \(\frac{13}{11}\)

(d) \(\frac{15}{13}\)

Answer 1

(b) \(\frac{13}{15}\)

  \(\frac{1}{1+\frac{\frac23}{1+\frac23+\frac{\frac89}{1-\frac23}}}\) = \(\frac{1}{1+\frac{\frac23}{1+\frac23+\frac{\frac89}{\frac13}}}\)

= \(\frac{1}{1+\frac{\frac23}{1+\frac23+\frac83}}\) = \(\frac{1}{1+\frac{\frac23}{\frac{13}{3}}}\)

= \(\frac{1}{1+\frac{2}{13}} = \frac{1}{\frac{15}{13}}\) = \(\frac{13}{15}.\)