Question

Express: $1.\overline{32}+0.\overline{35}$ as a fraction in simplest form.

Answer 1

Let $x=1.\overline{32}$

$x=\mathrm{1.323232.......}$ (1)

$100x=\mathrm{132.3232......}$ (2)

Subtract (1) from (2),

$100x-x=131.0000$

$99x=131$

$x=\frac{131}{99}$

Let $y=0.\overline{35}$

$y=\mathrm{0.3535.......}$ (1)

$100y=\mathrm{35.3535......}$ (2)

Subtract (1) from (2),

$100y-y=35.00$

$99y=35$

$y=\frac{35}{99}$

Now,

$x+y=\frac{131}{99}+\frac{35}{99}$

$=\frac{166}{99}$