To solve this problem, we will follow a systematic approach:

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Step 1: Define Variables

Let the initial number of apples be $5x$ and the initial number of oranges be $8x$, where $x$ is the common multiplier.

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Step 2: Represent the Changes in the Basket

1. 33 oranges are removed: The number of oranges left becomes $8x - 33$.
2. 27 apples are added: The number of apples becomes $5x + 27$.

At this point, the number of apples and oranges are equal: $5x + 27 = 8x - 33.$

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Step 3: Solve for $x$

Simplify the equation: $5x + 27 = 8x - 33.$

Rearrange terms: $27 + 33 = 8x - 5x.$

$60 = 3x.$

Solve for $x$: $x = 20.$

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Step 4: Calculate the Initial Numbers of Apples and Oranges

Substitute $x = 20$ into the initial expressions:

1. Initial number of apples: $5x = 5 \times 20 = 100$.
2. Initial number of oranges: $8x = 8 \times 20 = 160$.

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Step 5: Calculate the Final Number of Oranges

After removing 33 oranges, the final number of oranges is: $160 - 33 = 127.$

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Final Answer:

At the end, there were 127 oranges in the basket.

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Do you want further details or clarifications? Here are some related questions for practice:

1. If 50 apples were added instead of 27, what would the final number of oranges be?
2. What was the total number of fruits in the basket initially?
3. If the ratio of apples to oranges at the end was $1:1$, how many apples were there?
4. What if 15 oranges were removed instead of 33? What would be the new ratio?
5. Can you determine how many total fruits were removed from the basket?

Tip: Always double-check your equations when dealing with ratios and changes!