R S Aggarwal Solutions for Exercise 11: EXERCISE 7K

Using geometric series, express $0.\overline{3}$ as a rational number.

Using geometric series, express $0.\overline{231}$ as a rational number.

Find the value of $0.4\overline{23}$ in the form of a simple fraction.

Find the rational number whose decimal form is $0.3\overline{56}$?

Find the value of $3.5\overline{2}$ in the form of a simple fraction.

The sum of an infinite geometric series is $6$. If its first term is $2$, find its common ratio. (Write the answer in fraction.)

One side of an equilateral triangle is $24 \mathrm{cm}$. The midpoints of its sides are joined to form another triangle whose midpoints, in turn, are joined to form still another triangle. The process continues indefinitely. Find the sum of the perimeters of all the triangles. (Write the answer in $\mathrm{cm}$.)

An equilateral triangle is drawn by joining the midpoints of the sides of a given equilateral triangle. A third equilateral triangle is drawn inside the second triangle in the same manner. This process is repeated indefinitely. If each side of the first equilateral triangle is $6 \mathrm{cm}$, then the sum of the areas of all the triangles is$k\sqrt{3} {\mathrm{cm}}^2$. Hence, find the value of $k$.