If 35% of the income of A is equal to 25% of the income of B, then what is the ratio of the income of A to the income of B?

Introduction / Context:
This problem focuses on converting percentage relations between two quantities into a simple ratio. Such questions appear frequently in aptitude tests and competitive examinations to check whether students can handle percentages and ratios together. Here, we know that a fixed percentage of the income of A is equal to another percentage of the income of B, and from this relation we must derive the ratio between their incomes.

Given Data / Assumptions:

• 35% of A's income is equal to 25% of B's income.
• We denote the income of A by A and the income of B by B.
• We are required to find the ratio A : B.
• All incomes are assumed to be positive real numbers.

Concept / Approach:
We translate the percentage statement into an equation. If 35% of A equals 25% of B, we write (35 / 100) * A = (25 / 100) * B. Then, we simplify this equation to isolate the ratio A / B. Once we get A / B in fractional form, we express it in the simplest whole number ratio format, which is the standard way of representing ratios in such questions.

Step-by-Step Solution:
Step 1: Write the percentage condition as an equation: (35 / 100) * A = (25 / 100) * B. Step 2: Multiply both sides of the equation by 100 to remove the denominators. This gives 35A = 25B. Step 3: Rearrange the equation to express A / B. So, A / B = 25 / 35. Step 4: Simplify the fraction 25 / 35 by dividing numerator and denominator by their greatest common divisor, which is 5. Step 5: 25 / 35 simplifies to 5 / 7. Therefore A / B = 5 / 7. Step 6: Thus, the ratio of the income of A to the income of B is 5 : 7.

Verification / Alternative check:
To verify, assume A = 5 units and B = 7 units. Then 35% of A is (35 / 100) * 5 = 1.75 units. Similarly, 25% of B is (25 / 100) * 7 = 1.75 units. Both values are equal, which confirms that the ratio A : B = 5 : 7 satisfies the condition in the question. Therefore our derivation is correct.

Why Other Options Are Wrong:
Option 7:5 would correspond to A / B = 7 / 5, which would imply 35% of A being larger than 25% of B. Option 4:7 and 4:3 do not maintain the equality of 35% of A and 25% of B when tested with actual numbers. Only the ratio 5:7 ensures that these two percentages match exactly.

Common Pitfalls:
A common error is inverting the ratio. Another frequent mistake is incorrectly handling percentages, for example treating 35% as 35 instead of 35 / 100. Sometimes students attempt to plug in random values instead of solving the equation systematically, which can lead to confusion. Writing out the algebraic steps carefully and simplifying the fraction with care helps avoid these issues.

Final Answer:
The ratio of A's income to B's income is 5 : 7.