16Hit1600
AlgebraHardSolve a linear equation with fractions on both sides

Algebra practice question

An archivist at the Ellington Jazz Archive is determining n, the number of reel-to-reel tapes in an unlabeled collection. The archivist's records lead to the equation (2/3)(n - 9) = (3/4)n - 8.

What value of n satisfies the equation?

Answer: 24

Multiply both sides by 12, the least common denominator: 12 * (2/3)(n - 9) = 8(n - 9) and 12 * ((3/4)n - 8) = 9n - 96. So 8n - 72 = 9n - 96, which gives 24 = n. Check: (2/3)(24 - 9) = (2/3)(15) = 10 and (3/4)(24) - 8 = 18 - 8 = 10. A student who multiplies only the first term on each side by 12 (failing to distribute to the constant -8) gets 8n - 72 = 9n - 8, or n = -64; a student who distributes 2/3 only to n and not to -9 gets (2/3)n = (3/4)n - 8, or n = 96.

Common wrong answers

If you answered 96
You distributed 2/3 to n but not to -9, dropping the -6 entirely. (2/3)(n - 9) = (2/3)n - 6, not just (2/3)n.
If you answered -64
When you multiplied by 12, you scaled (2/3)(n-9) and (3/4)n but left the -8 alone. Every term on both sides must be multiplied: 12(-8) = -96, giving 8n - 72 = 9n - 96.
If you answered -24
You solved 8n - 72 = 9n - 96 but made a sign slip at the end: subtracting 8n from both sides gives -72 = n - 96, so n = 24, not -24.

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