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Algebra / Linear equations in one variable Difficulty: Hard

Each side of a $30$ -sided polygon has one of three lengths. The number of sides with length $8 centimeters left parenthesis cm right parenthesis$ is $5$ times the number of sides $n$ with length $3 cm$ . There are $6$ sides with length $4 cm$ . Which equation must be true for the value of $n$ ?

$5 n + 6 = 30$

$6 n + 6 = 30$

$8 n + 3 n + 4 n = 30$

$8 (5 n) + 3 n + 4 (6) = 30$

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Explanation

Choice B is correct. It’s given that each side of a $30$ -sided polygon has one of three lengths. It's also given that the number of sides with length $8$ $centimeters (cm)$ is $5$ times the number of sides $n$ with length $3 cm$ . Therefore, there are $5 times n$ , or $5 n$ , sides with length $8 cm$ . It’s also given that there are $6$ sides with length $4 cm$ . Therefore, the number of $3 cm$ , $4 cm$ , and $8 cm$ sides are $n$ , $6$ , and $5 n$ , respectively. Since there are a total of $30$ sides, the equation $n + 6 + 5 n = 30$ represents this situation. Combining like terms on the left-hand side of this equation yields $6 n + 6 = 30$ . Therefore, the equation that must be true for the value of $n$ is $6 n + 6 = 30$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.