If $A$ varies directly proportional to $C$ and $B$ also varies directly proportional to $C$, which one of the following is not correct?

### Concept & Direct Proportionality If two variables $A$ and $B$ are directly proportional to a third variable $C$, they can be algebraically expressed using proportionality constants: $A = k_1C$ and $B = k_2C$, where $k_1$ and $k_2$ are constants. ### Step-by-Step Solution 1. Given $A \propto C$, we can write $A = k_1C$. 2. Given $B \propto C$, we can write $B = k_2C$. 3. Let's test each option mathematically to identify the incorrect statement: - **Option (a):** $(A + B) = k_1C + k_2C = (k_1 + k_2)C$. Since $(k_1 + k_2)$ is a constant, $(A + B) \propto C$. This is a true statement. - **Option (c):** $\sqrt{AB} = \sqrt{(k_1C)(k_2C)} = \sqrt{k_1k_2C^2} = C\sqrt{k_1k_2}$. Since $\sqrt{k_1k_2}$ is a constant, $\sqrt{AB} \propto C$. This is a true statement. - **Option (d):** $\frac{A}{B} = \frac{k_1C}{k_2C} = \frac{k_1}{k_2}$. Since $\frac{k_1}{k_2}$ is a constant, the ratio $\frac{A}{B}$ is constant. This is a true statement. - **Option (b):** $(A - B) = k_1C - k_2C = (k_1 - k_2)C$. This means $(A - B)$ is directly proportional to $C$, i.e., $(A - B) \propto C$. The option incorrectly states that it is inversely proportional to $C$ (proportional to $\frac{1}{C}$). ### Exam Strategy & Shortcut Since $A$ and $B$ both scale linearly with $C$, any linear addition or subtraction ($A+B$ or $A-B$) will also scale linearly with $C$. Therefore, $(A-B)$ must be directly proportional to $C$. Option (b) claims an inverse relationship ($\frac{1}{C}$), which breaks fundamental proportionality rules and stands out immediately as dimensionally incorrect. ### Common Pitfall Misreading the prompt and looking for the "correct" statement rather than the "not correct" one, which often leads to prematurely selecting option (a). ### Final Answer Therefore, the correct answer is **$(A - B) \propto \frac{1}{C}$**.