parkhydel wrote:
One side of a parking stall is defined by a straight stripe that consists of n painted sections of equal length with an unpainted section 1/2 as long between each pair of consecutive painted sections. The total length of the stripe from the beginning of the first painted section to the end of the last painted section is 203 inches. If n is an integer and the length, in inches, of each unpainted section is an integer greater than 2, what is the value of n ?
A. 5
B. 9
C. 10
D. 14
E. 29
One side of a parking stall is defined by a straight stripe that consists of n painted sections of equal length with an unpainted section 1/2 as long between each pair of consecutive painted sections. Here's an idea of what all of this looks like.
If we let
x = the length of 1 painted section, then
0.5x = the distance between painted sections.
The total length of the stripe from the beginning of the first painted section to the end of the last painted section is 203 inches.Important: If we examine the above diagram, we can see that,
IF the entire stripe consisted of
3 painted sections, then there would be
2 spaces In general: (the number of spaces) = (the number of painted sections) - 1 So, if there are
n painted sections, then there must be
(n-1) spaces
We're now ready to write an equation!
If there are
n painted sections, and each painted section has a length of
x, then the total length of the painted sections =
nxLikewise, if there are
n - 1 spaces, and each space has a length of
0.5x, then the total length of the spaces =
0.5x(n - 1)Since the total length is 203 inches, we can write:
nx +
0.5x(n - 1) = 203
Simplify to get: nx + 0.5nx - 0.5x = 203
Simplify: 1.5nx - 0.5x = 203
To get integer coefficients, we'll multiply both sides of the equation by 2 to get: 3nx - x = 406
Factor both sides to get: x(3n - 1) = (2)(7)(29)
Since we're told that n and x are both positive
integers, we know that 3n is a multiple of 3, which means 3n - 1 is
1 less than some of multiple of 3When we examine the three prime factors of 406 (2, 7, and 29), we see that 2 and 29 are both
1 less than some of multiple of 3If 3n - 1 = 2, then n = 1, and 1 is not among the answer choices (
Also, if we have just 1 painted section, then we have 0 spaces, which breaks the condition that each unpainted section is an integer greater than 2)
If 3n - 1 = 29, then n = 10. This works perfectly
Answer: C
Cheers,
Brent