In a coloured picture, blue and yellow colours are used in the overall ratio of 4 : 3 (blue : yellow). In the upper half of the picture, the ratio of blue to yellow is 2 : 3. Assuming that the upper and lower halves are of equal area, what is the ratio of blue to yellow in the lower half of the picture?

Introduction:
This question tests your understanding of ratio, proportion, and mixing of two components in different parts of a whole. The overall ratio of blue and yellow colours is known, one half has a particular ratio, and you must deduce the ratio for the other half so that the combined picture maintains the overall ratio.

Given Data / Assumptions:

• Overall ratio of blue : yellow in the entire picture = 4 : 3.
• In the upper half, blue : yellow = 2 : 3.
• Upper half and lower half have equal area and therefore equal total amount of colour units.
• Let the ratio in the lower half be x : y (blue : yellow), which we need to find.

Concept / Approach:
We treat the amount of colour as proportional to area. If each half has the same area, then the total colour units in the upper and lower halves are equal. Let us use a common scaling factor to represent quantities in each half and then combine them to match the overall ratio of 4 : 3. This is a standard mixture and proportion problem.

Step-by-Step Solution:
Step 1: Let the total number of colour units in the upper half be 5k (since 2 + 3 = 5 parts). Step 2: Then in the upper half, blue = 2k and yellow = 3k. Step 3: Because the lower half has equal area, let its total colour units also be 5k for convenience. Let its blue and yellow be B and Y respectively such that B + Y = 5k. Step 4: Overall blue = 2k + B and overall yellow = 3k + Y, and their ratio must be 4 : 3: (2k + B) : (3k + Y) = 4 : 3. Step 5: Translate this into an equation: (2k + B) / (3k + Y) = 4/3. Step 6: Cross multiply: 3(2k + B) = 4(3k + Y). Step 7: Expand: 6k + 3B = 12k + 4Y. Step 8: Rearrange to group B and Y: 3B − 4Y = 6k. Step 9: Use the relation B + Y = 5k. Solve the two equations: From B + Y = 5k, we get B = 5k − Y. Substitute into 3B − 4Y = 6k: 3(5k − Y) − 4Y = 6k ⇒ 15k − 3Y − 4Y = 6k ⇒ 15k − 7Y = 6k. Step 10: Simplify: 15k − 6k = 7Y ⇒ 9k = 7Y ⇒ Y = 9k/7 and B = 5k − 9k/7 = (35k − 9k)/7 = 26k/7. Step 11: Therefore, the ratio in the lower half is B : Y = 26k/7 : 9k/7 = 26 : 9.

Verification / Alternative check:
Using upper half (2k, 3k) and lower half (26k/7, 9k/7), total blue = 2k + 26k/7 = (14k + 26k)/7 = 40k/7 and total yellow = 3k + 9k/7 = (21k + 9k)/7 = 30k/7. The overall ratio is 40 : 30 = 4 : 3 as required, so the result is correct.

Why Other Options Are Wrong:

• 1 : 1 or 2 : 1 would give a much higher proportion of blue and would not maintain the overall 4 : 3 ratio.
• 9 : 26 would produce more yellow than blue in the lower half and disturb the overall blue dominance.
• 3 : 2 is also inconsistent with the required combined ratio.

Common Pitfalls:
A common error is to average the ratios directly, which is incorrect since ratios must be combined using weighted sums, not simple averaging. Another mistake is ignoring that the areas of the upper and lower halves are equal, which is crucial for setting up the equations correctly.

Final Answer:
The ratio of blue to yellow in the lower half is 26 : 9.