If A's income is 10% more than B's, how much percentage is B's income less than A's?

### Concept & Formula This is a classic **Relative Percentage Change** problem. When one value is $R\%$ more than another, the percentage by which the second value is less than the first is not the same $R\%$. The base of comparison changes. To find how much percent less a quantity is when compared to a larger quantity, we use the formula: $$\text{Percentage Less} = \left( \frac{R}{100 + R} \right) \times 100$$ where $R$ is the percentage by which the first quantity is greater. ### Step-by-step Solution * **Given:** A's income is $10\%$ more than B's income. Here, $R = 10$. * **Calculation (Using Formula):** $$\text{Percentage Less} = \left( \frac{10}{100 + 10} \right) \times 100$$ $$\text{Percentage Less} = \left( \frac{10}{110} \right) \times 100$$ $$\text{Percentage Less} = \frac{100}{11}\%$$ * Converting this improper fraction to a mixed number: $$100 \div 11 = 9 \text{ with a remainder of } 1$$ $$\text{Percentage Less} = 9\frac{1}{11}\%$$ ### Exam Strategy & Shortcut **The 100 Base Method:** Always assume the base value (the one being compared *against* initially) is 100. Let B = 100. Since A is 10% more, A = 110. Now, the question asks: how much is B less than A? Difference = 10. New base (compared to A) = 110. Required Percentage = $(10 / 110) \times 100 = 100 / 11 = 9\frac{1}{11}\%$. This mental model prevents you from ever needing to memorize the $R/(100+R)$ formula. ### Common Pitfall The most frequent error is assuming percentage relationships are symmetric. Students often think "If A is 10% more than B, then B must be 10% less than A." This leads directly to choosing option (d). Always remember that the *base* of the percentage changes, altering the final percentage value. ### Final Answer **Therefore, the correct answer is $9\frac{1}{11}\%$.**