
fractals
Green, the designing engineer from the M&P' toy manufacturing company is designing a 'ballinaballinaball · · ·' toy where, for child safety reasons, the largest ball has a diameter of 12 cm and the smallest ball must be at least 5 cm in diameter. Each ball splits in half to expose the next ball. The ball inside another ball is 10 percent smaller than the ball it came from. Also, 85 percent of each ball's volume is air space in order to provide space for the nextsmaller ball. What is the largest number of balls in this toy? How much plastic is required to build this toy?
No idea.

Did you mean "fractions" for you title? This problem has nothing to do with "fractals".
Assuming that by "10% smaller", you mean each ball's diameter is 10% smaller than the next larger ball, or 90% of the nest larger ball's diameter, then you have a geometric progression each ball's diameter is 1/.9= 1.11 times as large as the previous ball. The first ball has diameter 5 cm so the second has diameter 5(1.11)= 5.55 cm, the third has diameter $\displaystyle 5(1.11)^2$, etc. If there are n balls, then the largest has diameter $\displaystyle 5(1.11)^{n1}$. Find integer n so that is just less than 12 cm.