An alternative way to work on this problem, or any rate problem, is to focus on your units of measurement. With rate problems, students are often left wondering do I divide by a number, or multiply by a number. Viewing the rate problems as a ratio, and then focusing on the units of measurement makes it nearly impossible to screw up.
The final answer asks you for how many seconds. So start by writing down "sec=?".
Then write down what you know:
*Empty Space - 1/4 (Vol Cyl) = 1/4(10)pi*r^2=2.5*pi*r^2 SPACE
* r/10 SPACE/MARBLE
* 12 MARBLE/MINUTE
Since the problem does not give you a specific units of measurement for distance (for instance, inches, cm), you must create a consistent one. Since the only time I work with distance is with volume, I simply substitute the word SPACE.
Now line up your units of measurements.
1) Since your final answer is "sec" you must have time in the top. The means you must flip your "12 MAR/MIN" to "1/12 MIN/MAR".
2) MIN is not in your final answer, so convert it by inserting "60SEC/MIN"
3) MAR is not in your final answer, so you must multiply by MAR. However to cancel, MAR must be in the top, so flip that to "10/r MAR/SPACE"
4) SPACE is not in your final answer, so you must multiply by 2.5 pi* r^s SPACE. This is already set up to cancel.
You now have:
1/12 MIN/MAR * 60SEC/MIN * 10/r MAR/SPACE * 2.5 pi*r^2 SPACE
Notice that you now have SEC as your only unit of measurement left, so you know you are good. Now multiply. Notice that many things cancel. They do that on purpose to help you. First, you see that you have "pi*r" remaining which is in all of your final answers. Then the "60/12" which becomes "5" so you have "5*10*2.5" --> 125 pi * r
Focus on your units and your life will be easy peasy.