Two baskets together contain 640 oranges. If one fifth of the oranges in the first basket are transferred to the second basket and then the numbers of oranges in the two baskets become equal, how many oranges were there in the first basket originally?

Introduction / Context:
This is a word problem involving simple algebra and ratios. It describes a transfer of oranges between two baskets and states that after moving a fraction of the first basket's contents, both baskets end up with the same number of oranges. The total number of oranges is fixed. You must set up and solve equations to find the original number in the first basket.

Given Data / Assumptions:
- Total number of oranges in both baskets together = 640.
- Let the number of oranges in the first basket initially be x.
- Then the second basket initially has 640 - x oranges.
- One fifth of the oranges in the first basket are transferred to the second basket.
- After the transfer, both baskets contain the same number of oranges.

Concept / Approach:
The key idea is conservation of total oranges and equality of the final counts. When one fifth of x is moved from the first basket to the second, the first basket is left with four fifths of x, and the second basket gains that one fifth of x. Setting these final quantities equal produces an equation that can be solved for x. Once x is known, it directly gives the original number in the first basket.

Step-by-Step Solution:
Step 1: Let the initial number of oranges in the first basket be x.Step 2: Then the initial number in the second basket is 640 - x.Step 3: One fifth of the first basket's oranges is x / 5.Step 4: After transferring x / 5 from the first to the second basket, the first basket has x - x / 5 = (4x / 5) oranges.Step 5: The second basket now has (640 - x) + x / 5 oranges.Step 6: After the transfer, the two baskets have equal numbers of oranges, so set 4x / 5 = (640 - x) + x / 5.Step 7: Multiply every term by 5 to clear the denominator: 4x = 5(640 - x) + x.Step 8: Expand: 4x = 3,200 - 5x + x = 3,200 - 4x.Step 9: Add 4x to both sides: 8x = 3,200.Step 10: Solve for x: x = 3,200 / 8 = 400.

Verification / Alternative check:
If the first basket originally had 400 oranges, the second had 640 - 400 = 240 oranges. One fifth of the first basket is 400 / 5 = 80. After moving 80 oranges, the first basket has 400 - 80 = 320, and the second basket has 240 + 80 = 320. Both baskets now have 320 oranges, which confirms that x = 400 is consistent with the conditions.

Why Other Options Are Wrong:
- 800 and 600 exceed or misfit the total of 640 or fail the equality condition when checked with the transfer rule.
- 300 does not satisfy both the total 640 and the final equality condition; testing it will show unequal final counts in the baskets.

Common Pitfalls:
Some learners forget that the second basket gains exactly the amount the first basket loses, leading to incorrect equations. Others may incorrectly assume that one fifth is taken from the total instead of from the first basket only. Always define variables clearly and translate the transfer step precisely into algebraic expressions before solving.

Final Answer:
The first basket originally contained 400 oranges.