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A walkway consists of k identical rectangular slabs placed end to end, with a mortar joint between each pair of consecutive slabs. Each mortar joint has a length that is \frac{1}{3} of a slab’s length. The total length of the walkway from the beginning of the first slab to the end of the last slab is 172 inches. If k is an integer and the length, in inches, of each mortar joint is an integer greater than 1, what is the value of k ?

Explanation	Calculations	Help
$We use x to represent the length of each slab in inches.$	$x = length of each slab in inches$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We use j to represent the length of each mortar joint in inches.$	$j = length of each mortar joint in inches$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate "length that is \frac{1}{3} of a slab's length" into a mathematical equation.$	$j = \frac{1}{3} \times x$	Theory & Tactics Method Card TRANS2-B Click to view full details
$We translate "total length of the walkway from the beginning of the first slab to the end of the last slab is 172 inches" into an equation using the sum of slab lengths and joint lengths.$	$k \times x + (k - 1) \times j = 172$	Theory & Tactics Method Card TRANS1-B Click to view full details Theory & Tactics Method Card TRANS1-C Click to view full details
$We substitute the expression for j into the equation and combine like terms.$	$k \times x + (k - 1) \times \frac{1}{3} \times x = \frac{( 4 k - 1 ) \times x}{3} \frac{( 4 k - 1 ) \times x}{3} = 172 (4 k - 1) \times x = 516$
$We solve the joint relationship for x .$	$x = 3 \times j$
$We substitute the expression for x into the combined equation.$	$(4 k - 1) \times 3 \times j = 516 (4 k - 1) \times j = 172$
$We test the integer values for j greater than 1 to find k .$	$(4 k - 1) \times 4 = 172 4 k - 1 = \frac{172}{4} 4 k - 1 = 43 4 k = 44 k = 11$

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C

A walkway consists of k identical rectangular slabs placed end to end, with a mortar joint between each pair of consecutive slabs. Each mortar joint has a length that is \frac{1}{3} of a slab’s length. The total length of the walkway from the beginning of the first slab to the end of the last slab is 172 inches. If k is an integer and the length, in inches, of each mortar joint is an integer greater than 1, what is the value of k ?

Explanation	Calculations	Help
$We use x to represent the length of each slab in inches.$	$x = length of each slab in inches$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We use j to represent the length of each mortar joint in inches.$	$j = length of each mortar joint in inches$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate "length that is \frac{1}{3} of a slab's length" into a mathematical equation.$	$j = \frac{1}{3} \times x$	Theory & Tactics Method Card TRANS2-B Click to view full details
$We translate "total length of the walkway from the beginning of the first slab to the end of the last slab is 172 inches" into an equation using the sum of slab lengths and joint lengths.$	$k \times x + (k - 1) \times j = 172$	Theory & Tactics Method Card TRANS1-B Click to view full details Theory & Tactics Method Card TRANS1-C Click to view full details
$We substitute the expression for j into the equation and combine like terms.$	$k \times x + (k - 1) \times \frac{1}{3} \times x = \frac{( 4 k - 1 ) \times x}{3} \frac{( 4 k - 1 ) \times x}{3} = 172 (4 k - 1) \times x = 516$
$We solve the joint relationship for x .$	$x = 3 \times j$
$We substitute the expression for x into the combined equation.$	$(4 k - 1) \times 3 \times j = 516 (4 k - 1) \times j = 172$
$We test the integer values for j greater than 1 to find k .$	$(4 k - 1) \times 4 = 172 4 k - 1 = \frac{172}{4} 4 k - 1 = 43 4 k = 44 k = 11$

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C