Five farmers have 7, 9, 11, 13, and 14 apple trees respectively in their orchards. Each farmer finds that every tree in his own orchard yields exactly the same number of apples. It is further observed that if the 3rd farmer gives one apple to the 1st farmer, and the 5th farmer gives 3 apples each to the 2nd and the 4th farmers, then all five farmers end up with exactly the same total number of apples. What were the yields per tree in the orchards of the 3rd and 4th farmers?

Introduction / Context:
This arithmetic reasoning question involves equal yields per tree and redistribution of apples so that all farmers finally have the same number of apples. Each farmer has a different number of apple trees, but within each orchard every tree yields the same number of apples. After certain transfers of apples among farmers, all of them end up with equal totals. The aim is to determine the per tree yields of the 3rd and 4th farmers.

Given Data / Assumptions:

• Farmer 1 has 7 trees, farmer 2 has 9 trees, farmer 3 has 11 trees, farmer 4 has 13 trees, and farmer 5 has 14 trees.
• Within each orchard, every tree yields the same number of apples, so totals are 7y1, 9y2, 11y3, 13y4, and 14y5.
• The 3rd farmer gives 1 apple to the 1st farmer.
• The 5th farmer gives 3 apples to the 2nd farmer and 3 apples to the 4th farmer, a total of 6 apples.
• After these transfers, all farmers have the same total number of apples.
• We must find the yields per tree of the 3rd farmer (y3) and the 4th farmer (y4).

Concept / Approach:
Let A be the final common number of apples each farmer holds after all transfers. Express the final total for each farmer in terms of the original totals and the number of apples given or received. Equate these expressions to A to form linear equations involving y1, y2, y3, y4, and y5. The crucial relation needed to answer the question involves y3 and y4, which we can obtain by focusing on the equations for the 3rd and 4th farmers.

Step-by-Step Solution:
Step 1: Initial totals: farmer 1 has 7y1 apples, farmer 2 has 9y2, farmer 3 has 11y3, farmer 4 has 13y4, and farmer 5 has 14y5. Step 2: After transfers: farmer 1 receives 1 apple from farmer 3, so total becomes 7y1 + 1. Step 3: Farmer 2 receives 3 apples from farmer 5, so total becomes 9y2 + 3. Step 4: Farmer 3 gives 1 apple, so total becomes 11y3 - 1. Step 5: Farmer 4 receives 3 apples from farmer 5, so total becomes 13y4 + 3. Step 6: Farmer 5 gives away 6 apples in total, so total becomes 14y5 - 6. Step 7: All final totals are equal to A: 7y1 + 1 = 9y2 + 3 = 11y3 - 1 = 13y4 + 3 = 14y5 - 6 = A. Step 8: Equate the expressions for farmers 3 and 4: 11y3 - 1 = 13y4 + 3. Step 9: Rearranging gives 13y4 - 11y3 = -4. Step 10: Check the given option pairs for (y3, y4). For (11, 9), compute 13*9 - 11*11 = 117 - 121 = -4, which satisfies the equation.

Verification / Alternative check:
The other option pairs do not satisfy 13y4 - 11y3 = -4. For example, for (17, 9): 13*9 - 11*17 = 117 - 187 = -70; for (9, 11): 13*11 - 11*9 = 143 - 99 = 44; for (9, 9): 117 - 99 = 18. Only (11, 9) yields -4. Because the condition of equal final totals constrains y3 and y4 through this equation, the only consistent pair is y3 = 11 and y4 = 9.

Why Other Options Are Wrong:
Each alternative pair fails to satisfy the key relationship 13y4 - 11y3 = -4 that comes from equating the final totals of the 3rd and 4th farmers. As a result, those pairs would lead to unequal final totals and cannot describe a situation where all farmers end with the same number of apples.

Common Pitfalls:
This type of problem can be confusing because there are many variables and totals to track. A common mistake is to try to solve for all yields explicitly rather than focusing on the essential equations that involve the unknowns asked in the question. Another pitfall is miscounting the apples transferred, especially from the 5th farmer, who gives to two different farmers. Carefully constructing equations and then checking only the necessary pairs simplifies the reasoning.

Final Answer:
The yields per tree for the 3rd and 4th farmers are 11 and 9 apples respectively.