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Re: One side of a parking stall is defined by a straight stripe [#permalink]
1
Thanks Brent, although I have a question. What if x=29 (indeed >2) and 3n-1=2*7=14? This is feasible (because 14 is 1 less than some multiple of 3) and yields n=5.
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Re: One side of a parking stall is defined by a straight stripe [#permalink]
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fraise wrote:
Thanks Brent, although I have a question. What if x=29 (indeed >2) and 3n-1=2*7=14? This is feasible (because 14 is 1 less than some multiple of 3) and yields n=5.


Great question!!

If x = 29, then each painted stripe has length 29.
This also means the length of each unpainted section (between the stripes) must have length 14.5 (since we're told that each unpainted section is 1/2 as long as each stripe).
This creates a problem since we're told that each unpainted section is an integer greater than 2. So, the length of an unpainted section cannot be 14.5
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Re: One side of a parking stall is defined by a straight stripe [#permalink]
GreenlightTestPrep wrote:
Carcass 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.
Image
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 = nx
Likewise, 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 3
When 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 3
If 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



I tried backsolving and tried A and it works.
if n= 5
x(3n-1)
x(3x5-1)=406
x= 29
since x is an integer .. it works

and also 10 works. I think there is an error in the problem
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Re: One side of a parking stall is defined by a straight stripe [#permalink]
Asmakan wrote:
GreenlightTestPrep wrote:
Carcass 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.
Image
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 = nx
Likewise, 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 3
When 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 3
If 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



I tried backsolving and tried A and it works.
if n= 5
x(3n-1)
x(3x5-1)=406
x= 29
since x is an integer .. it works

and also 10 works. I think there is an error in the problem


I need an answer
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Re: One side of a parking stall is defined by a straight stripe [#permalink]
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Length of Painted section = x
Length of unpainted section = x/2

Number of Painted section = n
Number of unpainted section = n-1

\(n*x + (n-1)*\frac{x}{2} = 203 \)

\(\frac{x}{2}(3n-1) = 203\)

\(\frac{x}{2}(3n-1) = 1*203 = 7*29\)

Since x is an integer and greater than 2, reject 1*203

3n-1 = 7 or 29

If 3n-1 = 7
n= 8/3 (rejected)

If 3n-1 = 29

n= 10
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Re: One side of a parking stall is defined by a straight stripe [#permalink]
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Asmakan wrote:

I tried backsolving and tried A and it works.
if n= 5
x(3n-1)
x(3x5-1)=406
x= 29
since x is an integer .. it works

and also 10 works. I think there is an error in the problem


The problem here is that x = 29 breaks one of the conditions outlined in the question.

We're told that "...the length, in inches, of each unpainted section is an INTEGER greater than 2"
0.5x = the length of each unpainted section
So, if x = 29, then the length of each unpainted section = 14.5, which is not an integer.
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Re: One side of a parking stall is defined by a straight stripe [#permalink]
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