# Miscellaneous Counting Problems

*In a legislative body of 200 people, the number of Democrats is 50 less than 4 times the number of Republicans. If one fifth of the legislators are neither Republican nor Democrat, how many of the legislators are Republicans?*

*A. 42
B. 50
C. 71
D. 95
E. 124*

Let *D* be the number of Democrats and let *R* be the number of Republicans. "One fifth of the legislators are neither Republican nor Democrat," so there are 200/5 = 40 legislators who are neither Republican nor Democrat.

Hence, there are 200 â€“ 40 = 160 Democrats and Republicans, or *D* + *R* = 160. Translating the clause "the number of Democrats is 50 less than 4 times the number of Republicans" into an equation yields *D* = 4*R* â€“ 50. Plugging this into the equation *D* + *R* = 160 yields.

4*R* â€“ 50 + *R* = 160

5*R* â€“ 50 = 160

5*R* = 210

*R* = 42

The answer is (A).

*Speed bumps are being placed at 20 foot intervals along a road 1015 feet long. If the first speed bump is placed at one end of the road, how many speed bumps are needed?*

*A. 49
B. 50
C. 51
D. 52
E. 53*

Since the road is 1015 feet long and the speed bumps are 20 feet apart, there are 1015/20 = 50.75, or 50 full sections in the road.

If we ignore the first speed bump and associate the speed bump at the end of each section with that section, then there are 50 speed bumps (one for each of the fifty full sections). Counting the first speed bump gives a total of 51 speed bumps.

The answer is (C).