In a colored picture, the overall ratio of blue to yellow color used is 4:3. If in the upper half of the picture the ratio of blue to yellow is 2:3, then what must be the ratio of blue to yellow in the lower half of the picture?

Introduction:
This is a ratio-combination question where an overall ratio is formed by combining two equal halves (upper and lower) that have different internal ratios. The key idea is that the picture has two halves of equal size, so the total amount of color used in the upper half equals the total amount used in the lower half. Using the given overall blue:yellow ratio and the upper-half ratio, we can compute the actual blue and yellow amounts in each half and then determine the required ratio in the lower half.

Given Data / Assumptions:

• Overall ratio (blue:yellow) = 4:3• Upper half ratio (blue:yellow) = 2:3• Upper half and lower half are equal in size, so total color used in each half is equal

Concept / Approach:
Let total color used in the whole picture be a convenient multiple of (4+3)=7 parts. Then each half uses half of the total color. Use the upper-half ratio to split its half into blue and yellow. Subtract from total blue and yellow to get lower-half blue and yellow, then form the lower-half ratio.

Step-by-Step Solution:
Step 1: Assume total color used in the picture = 7k parts.Overall blue = 4k, overall yellow = 3kStep 2: Since upper and lower halves are equal, each half uses (7k)/2 = 3.5k total color.Step 3: Upper half ratio is 2:3 (total 5 parts).Upper blue = (2/5)*3.5k = 1.4kUpper yellow = (3/5)*3.5k = 2.1kStep 4: Compute lower half amounts by subtraction.Lower blue = 4k - 1.4k = 2.6kLower yellow = 3k - 2.1k = 0.9kStep 5: Form the ratio in the lower half.Lower blue : Lower yellow = 2.6k : 0.9k = 26 : 9

Verification / Alternative check:
If lower half ratio is 26:9, then lower-half blue fraction is 26/35 and yellow fraction is 9/35. Combining with upper half (2:3) while keeping equal total amounts in each half will reproduce the overall 4:3 ratio. This supports that the computed ratio is logically consistent with the given totals.

Why Other Options Are Wrong:
1 : 1 and 2 : 1 make the lower half too balanced or too blue-heavy; then the overall would not remain 4:3.9 : 26 reverses the result, making the lower half yellow-heavy, contradicting the subtraction outcome.13 : 5 is close but not exact; it would not match the exact overall ratio when combined with the upper half.

Common Pitfalls:
• Ignoring that halves are equal and assuming unknown unequal quantities.• Averaging ratios directly instead of converting to actual amounts first.• Making mistakes while subtracting upper-half amounts from the overall totals.

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