If $25a + 25b = 115$, what is the average of $a$ and $b$?

### Concept & Formula The average of two numbers, $a$ and $b$, is defined by dividing their sum by $2$. We can find the sum $(a + b)$ as a single entity by factoring out the common coefficient from the given algebraic equation. $$ \text{Average} = \frac{a + b}{2} $$ ### Step-by-Step Solution * **Given:** The equation provided is $25a + 25b = 115$. * **Calculation:** First, factor out the $25$ from the left side of the equation: $25(a + b) = 115$ Divide both sides by $25$ to isolate the sum $(a + b)$: $a + b = \frac{115}{25}$ $a + b = 4.6$ Now, substitute this sum directly into the average formula: $\text{Average} = \frac{4.6}{2} = 2.3$ The calculated average is $2.3$, which does not match options (a), (b), (c), or (d). ### Exam Strategy & Shortcut Instead of calculating the exact decimal immediately, keep the fractions. The sum is $\frac{115}{25}$. The average is half of the sum, which means calculating $\frac{115}{50}$. Multiplying the numerator and the denominator by $2$ gives $\frac{230}{100}$, which instantly yields $2.3$. This fractional approach is often much faster and less prone to silly division mistakes. Since $2.3$ is not present in the options, the answer is apparent. ### Common Pitfall A very common mistake is finding the sum $(a + b) = 4.6$ and hurriedly selecting option (d) without remembering that the question asks for the **average** of $a$ and $b$, not their total sum. Always double-check what the question is explicitly asking for before marking the final option. ### Final Answer **Therefore, the correct answer is None of these.**