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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 \frac{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 ?

Explanation	Calculations	Help
$We define variables for the lengths of painted and unpainted sections.$	$Let x = length of each painted section (in inches)$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate the total length description into an equation: painted sections plus unpainted sections equals 203.$	$n x + (n - 1) \times \frac{1}{2} x = 203$	Theory & Tactics Method Card TRANS1-B Click to view full details Theory & Tactics Method Card TRANS1-C Click to view full details Theory & Tactics Method Card TRANS2-B Click to view full details
$We combine like terms on the left and isolate the product.$	$n x + (n - 1) \frac{1}{2} x = \frac{( 3 n - 1 ) x}{2} \frac{( 3 n - 1 ) x}{2} = 203 (3 n - 1) x = 406$
$We introduce a new variable to represent the unpainted section length and express x in terms of this new variable.$	$k = \frac{x}{2}$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We substitute x = 2 k into the simplified equation and divide both sides by 2.$	$(3 n - 1) (2 k) = 406 (3 n - 1) k = 203$
$We factor 203 to find possible integer values for k .$	$203 = 7 \times 29$	Theory & Tactics Method Card DIV4-C Click to view full details
$We test factor pairs of 203, remembering k = x /2 must be an integer greater than 2, and solve for n .$	$For k = 7 : 3 n - 1 = \frac{203}{7} = 29 3 n = 30 n = 10$

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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 \frac{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 ?

Explanation	Calculations	Help
$We define variables for the lengths of painted and unpainted sections.$	$Let x = length of each painted section (in inches)$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate the total length description into an equation: painted sections plus unpainted sections equals 203.$	$n x + (n - 1) \times \frac{1}{2} x = 203$	Theory & Tactics Method Card TRANS1-B Click to view full details Theory & Tactics Method Card TRANS1-C Click to view full details Theory & Tactics Method Card TRANS2-B Click to view full details
$We combine like terms on the left and isolate the product.$	$n x + (n - 1) \frac{1}{2} x = \frac{( 3 n - 1 ) x}{2} \frac{( 3 n - 1 ) x}{2} = 203 (3 n - 1) x = 406$
$We introduce a new variable to represent the unpainted section length and express x in terms of this new variable.$	$k = \frac{x}{2}$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We substitute x = 2 k into the simplified equation and divide both sides by 2.$	$(3 n - 1) (2 k) = 406 (3 n - 1) k = 203$
$We factor 203 to find possible integer values for k .$	$203 = 7 \times 29$	Theory & Tactics Method Card DIV4-C Click to view full details
$We test factor pairs of 203, remembering k = x /2 must be an integer greater than 2, and solve for n .$	$For k = 7 : 3 n - 1 = \frac{203}{7} = 29 3 n = 30 n = 10$

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10