SAT Math Practice Question #470 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #470

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

A merchant sells two types of gems: rubies and sapphires. A ruby costs $120 and a sapphire costs $85. The merchant sold a total of \( N \) gems for exactly $4,165. If the merchant sold at least one of each type of gem, what is the maximum possible value of \( N \)?
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Socratic AI Engine Step-by-Step Derivation
0ms Precomputed
Tactical Insight (Hint 1)

Check for special right triangles (\( 30^\circ-60^\circ-90^\circ \) or \( 45^\circ-45^\circ-90^\circ \)) or the complementary angle identity \( \sin(x) = \cos(90^\circ - x) \).

Elimination Framework (Hint 2)

Desmos Tactical Shortcut: Type the function directly into Desmos. Tap the peak or trough of the curve to instantly reveal the extrema coordinates \( (h, k) \).

Masterclass Solution & Distractor Trap Analysis

Let's translate this word problem into a linear Diophantine equation! Let \( r \) be the number of rubies sold, and \( s \) be the number of sapphires sold. We are told: \( 120r + 85s = 4165 \) where \( r \ge 1 \) and \( s \ge 1 \) are inte...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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