Difficulty: Hard
Correct Answer: 5:2
Explanation:
Introduction / Context:This is an alloy composition (mixture) problem. Each brass type has a fixed copper-to-zinc ratio, which means each type has a fixed fraction of copper and a fixed fraction of zinc. When you mix them in a given amount ratio, you must compute copper and zinc contributed by each portion and then add the totals. Finally, convert the resulting totals into the simplest copper:zinc ratio. This is similar to mixing solutions by concentration.
Given Data / Assumptions:
Concept / Approach:Convert ratios to fractions. Type 1 total parts = 11, so copper fraction = 8/11 and zinc fraction = 3/11. Type 2 total parts = 22, so copper fraction = 15/22 and zinc fraction = 7/22. Multiply each fraction by its mixed amount (5 or 2), then add copper totals and zinc totals, and simplify.
Step-by-Step Solution: Type 1 (amount 5): copper = 5*(8/11) = 40/11, zinc = 5*(3/11) = 15/11 Type 2 (amount 2): copper = 2*(15/22) = 30/22 = 15/11, zinc = 2*(7/22) = 14/22 = 7/11 Total copper = 40/11 + 15/11 = 55/11 = 5 Total zinc = 15/11 + 7/11 = 22/11 = 2 Copper:Zinc = 5:2
Verification / Alternative check:The totals became clean integers after combining, which strongly indicates correct fraction handling and common denominators. Since 5 and 2 share no common factor, the ratio is already simplified.
Why Other Options Are Wrong: 3:2 and 3:4 do not match the computed totals and imply incorrect contribution calculations. 2:3 reverses dominance and would mean zinc exceeds copper, which is impossible here because both brass types have more copper than zinc. 7:5 suggests too much zinc relative to copper compared to both input alloys.
Common Pitfalls:Using 8/3 or 15/7 directly as copper fraction, forgetting to convert to total parts (8+3 and 15+7), or forgetting to scale each type by the mixing ratio 5:2.
Final Answer:The ratio of Copper to Zinc in the new alloy is 5:2.
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