The numbers of oranges in three baskets are in the ratio 3 : 4 : 5. By what ratio must the numbers of oranges in the first two baskets be increased (keeping the third basket unchanged) so that the new ratio of the three baskets becomes 5 : 4 : 3?

Introduction / Context:
This question deals with adjusting quantities in two of three groups so that the overall ratio changes to a new specified ratio while one group remains unchanged. It tests the ability to set up algebraic equations from ratio conditions and to think carefully about which quantities are changed.

Given Data / Assumptions:

• Original ratio of oranges in three baskets (first : second : third) = 3 : 4 : 5.
• The number of oranges in the third basket remains unchanged.
• After increasing the numbers in the first two baskets, the new ratio becomes 5 : 4 : 3.
• We must find the ratio in which the numbers of oranges in the first and second baskets are increased.

Concept / Approach:
Let the original numbers be 3k, 4k and 5k. After changes, let the new numbers be 5t, 4t and 3t according to the new ratio. Since the third basket is unchanged, 5k must equal 3t, which links k and t. The increments in the first and second baskets can then be expressed in terms of k, and their ratio gives the required answer.

Step-by-Step Solution:
Step 1: Let original numbers be 3k, 4k and 5k. Step 2: New ratio is 5 : 4 : 3, so new numbers can be written as 5t, 4t and 3t. Step 3: The third basket is unchanged, so 5k = 3t. Step 4: From 5k = 3t, we get t = 5k / 3. Step 5: New number in first basket = 5t = 5 * (5k / 3) = 25k / 3. Step 6: New number in second basket = 4t = 4 * (5k / 3) = 20k / 3. Step 7: Increase in first basket = (25k / 3) - 3k = (25k - 9k) / 3 = 16k / 3. Step 8: Increase in second basket = (20k / 3) - 4k = (20k - 12k) / 3 = 8k / 3. Step 9: Ratio of increases in first and second baskets = (16k / 3) : (8k / 3) = 16 : 8 = 2 : 1.

Verification / Alternative check:
Pick a convenient value for k, for example k = 3. Then original numbers are 9, 12 and 15. The new numbers, according to ratio 5 : 4 : 3, must be in the form 5t, 4t and 3t with 3t = 15, so t = 5. Thus new numbers are 25, 20 and 15. Increases in first and second baskets are 25 - 9 = 16 and 20 - 12 = 8, which are in the ratio 16 : 8 = 2 : 1, matching our earlier result.

Why Other Options Are Wrong:
Ratios such as 1 : 3, 3 : 4 or 2 : 3 do not satisfy the constraints when tested with actual numbers. For these, the adjustments to the first and second baskets would not lead to the new overall ratio 5 : 4 : 3 while keeping the third basket fixed at 5k.

Common Pitfalls:
A typical mistake is to change all three baskets, which violates the condition that the third basket remains unchanged. Another error is to assume that the new ratio must share the same base multiple k instead of introducing a new variable t. Misinterpreting ratios as absolute numbers rather than relative quantities can also cause confusion.

Final Answer:
The numbers of oranges in the first and second baskets must be increased in the ratio 2 : 1.