A 4 kg sample of an alloy contains 1/5 copper and the rest zinc. Another 5 kg sample of a different alloy contains 1/6 copper and the rest zinc. If the two alloys are melted together to form a single mixture, what is the ratio of copper to zinc in the resulting alloy?

Introduction:
This alloy problem involves combining two different metal mixtures, each containing copper and zinc in different proportions. After mixing, we must determine the overall ratio of copper to zinc in the final alloy. This is a direct application of weighted averages and ratio calculations based on given fractions and masses.

Given Data / Assumptions:

• First alloy has a total mass of 4 kg.
• In the first alloy, copper is 1/5 of the mass and the rest is zinc.
• Second alloy has a total mass of 5 kg.
• In the second alloy, copper is 1/6 of the mass and the rest is zinc.
• The two alloys are completely melted and mixed to form one homogeneous alloy.
• We must find the ratio of copper to zinc in this final mixture.

Concept / Approach:
We first compute the absolute masses of copper and zinc in each alloy using the given fractions. Then we add the copper masses to get total copper and add the zinc masses to get total zinc. Finally, we express these as a simplified ratio copper : zinc. Fractions are helpful to keep the calculations exact.

Step-by-Step Solution:
Step 1: In the first alloy of 4 kg, copper mass = 1/5 of 4 kg = 4/5 kg. Step 2: Zinc mass in the first alloy = 4 - 4/5 = 20/5 - 4/5 = 16/5 kg. Step 3: In the second alloy of 5 kg, copper mass = 1/6 of 5 kg = 5/6 kg. Step 4: Zinc mass in the second alloy = 5 - 5/6 = 30/6 - 5/6 = 25/6 kg. Step 5: Total copper in the final mixture = 4/5 + 5/6. Step 6: Find a common denominator: 4/5 = 24/30, 5/6 = 25/30, so total copper = (24 + 25) / 30 = 49/30 kg. Step 7: Total zinc in the final mixture = 16/5 + 25/6. Step 8: Convert: 16/5 = 96/30, 25/6 = 125/30, so total zinc = (96 + 125) / 30 = 221/30 kg. Step 9: Ratio of copper to zinc = (49/30) : (221/30). Step 10: The denominators cancel, giving copper : zinc = 49 : 221.

Verification / Alternative check:
We can check that the total mass is preserved. Total initial mass = 4 kg + 5 kg = 9 kg. Final copper + final zinc = 49/30 + 221/30 = 270/30 = 9 kg, which matches the initial total. This confirms that no arithmetic errors have been made and that the ratio 49 : 221 is correct.

Why Other Options Are Wrong:
39 : 231, 94 : 181, and 1 : 5 do not result from the exact calculations of copper and zinc masses. When you convert these ratios back to fractions of a 9 kg mixture, they do not match the computed values of 49/30 kg and 221/30 kg.
None of these is incorrect because we have a valid matching option, 49 : 221.

Common Pitfalls:
Some learners mistakenly average the given fractions 1/5 and 1/6 directly without weighting by the masses 4 kg and 5 kg. Others subtract copper percentage incorrectly, leading to wrong zinc amounts. It is also easy to make mistakes when converting fractions to a common denominator, so care is required in these arithmetic steps.

Final Answer:
The ratio of copper to zinc in the resulting alloy is 49 : 221.