One side of a parking stall is defined by a straight stripe that consists of n...

Question

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 ?

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10 We define variables for the lengths of painted and unpainted sections. Let x = length of each painted section (in inches) We translate the total length description into an equation: painted sections plus unpainted sections equals 203. n x + (n - 1) 1/2x = 203 We combine like terms on the left and isolate the product. n x + (n - 1)1/2x = (3n - 1)x/2 (3n - 1)x/2 = 203 (3n - 1)x = 406 We introduce a new variable to represent the unpainted section length and express x in terms of this new variable. k = x/2 We substitute x = 2k into the simplified equation and divide both sides by 2. (3n - 1)(2k) = 406 (3n - 1)k = 203 We factor 203 to find possible integer values for k. 203 = 7 29 We test factor pairs of 203, remembering k = x/2 must be an integer greater than 2, and solve for n. For k = 7: 3n - 1 = 203/7 = 29 3n = 30 n = 10

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

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