Natu and Buchku each have a certain number of oranges. Natu tells Buchku that if Buchku gives Natu 10 of his oranges, then Natu will have twice as many oranges as Buchku has left. Buchku replies that if Natu gives Buchku 10 oranges, then Buchku will have the same number of oranges as Natu has left. What is the number of oranges with Natu and Buchku, respectively?

Introduction / Context:
This arithmetic reasoning puzzle involves forming and solving linear equations from a word problem. Two people, Natu and Buchku, have some oranges. They describe how their numbers of oranges would change if they gave 10 oranges to each other under two different scenarios. You must translate these statements into equations and solve for the two unknown quantities, representing the oranges each person has initially.

Given Data / Assumptions:
- Let N be the number of oranges that Natu has initially. - Let B be the number of oranges that Buchku has initially. - Scenario 1: Buchku gives 10 oranges to Natu. Then Natu has N + 10, and Buchku has B - 10. - According to Natu, in scenario 1, N + 10 equals twice the oranges left with Buchku, so N + 10 = 2 * (B - 10). - Scenario 2: Natu gives 10 oranges to Buchku. Then Natu has N - 10, and Buchku has B + 10. - According to Buchku, in scenario 2, B + 10 equals N - 10, so B + 10 = N - 10.

Concept / Approach:
The approach is to express the verbal conditions as algebraic equations in N and B, then solve the simultaneous equations. The first equation comes from the condition in which Natu has twice as many oranges as Buchku after receiving 10. The second equation comes from the condition in which Buchku and Natu have equal numbers after Buchku receives 10. Once the equations are set up correctly, we solve them by substitution or elimination to find N and B.

Step-by-Step Solution:
Step 1: From scenario 1, write the equation: N + 10 = 2 * (B - 10). Step 2: Simplify this equation: N + 10 = 2B - 20, so N = 2B - 30. Step 3: From scenario 2, write the equation: B + 10 = N - 10. Step 4: Rearrange scenario 2 equation to express N: N = B + 20. Step 5: Now we have two expressions for N: N = 2B - 30 and N = B + 20. Set them equal: 2B - 30 = B + 20. Step 6: Solve for B: 2B - B = 20 + 30, so B = 50. Step 7: Substitute B = 50 into N = B + 20 to get N = 50 + 20 = 70.

Verification / Alternative check:
Verify the numbers in both scenarios. Initially, Natu has 70 oranges and Buchku has 50 oranges. In scenario 1, Buchku gives 10 oranges to Natu, so Natu has 80 and Buchku has 40. Twice 40 is 80, which matches Natu's count, so the first condition holds. In scenario 2, Natu gives 10 oranges to Buchku, so Natu has 60 and Buchku has 60. They have equal numbers, so the second condition holds as well. This confirms that N = 70 and B = 50 satisfy both conditions exactly.

Why Other Options Are Wrong:
- For 50, 20 or 20, 50, the equations N + 10 = 2 * (B - 10) and B + 10 = N - 10 do not hold simultaneously. - For 50, 70, the person with fewer oranges would incorrectly end up with more after exchange in one of the scenarios. - The option 40, 10 also fails to satisfy both equations when tested. Only the pair 70, 50 satisfies both conditions at the same time.

Common Pitfalls:
A common mistake is to reverse the roles of giver and receiver and write the equations incorrectly. Some candidates forget that the new quantities after giving or receiving 10 must be used in the equations, not the original values. Others attempt to solve by trial and error without properly translating the word problem into algebra, which can become confusing. Systematically defining variables and writing equations from each statement is the most reliable approach.

Final Answer:
The number of oranges with Natu and Buchku, respectively, is 70 and 50.