If $a$ and $b$ are positive integers and $\frac{(a - b)}{3.5} = \frac{4}{7}$, then

### Concept & Formula This problem tests basic algebraic manipulation and the properties of positive integers. ### Step-by-Step Solution 1. Start with the given equation: $\frac{a - b}{3.5} = \frac{4}{7}$. 2. Multiply both sides by 3.5 to isolate the term $(a - b)$: $a - b = 3.5 \times \frac{4}{7}$. 3. Notice that 3.5 is exactly half of 7, which means it can be written as the fraction $\frac{7}{2}$. 4. Substitute this into the equation: $a - b = \frac{7}{2} \times \frac{4}{7}$. 5. The 7s in the numerator and denominator cancel out, leaving: $a - b = \frac{4}{2}$. 6. Simplify the fraction: $a - b = 2$. 7. Rearrange to express $a$ in terms of $b$: $a = b + 2$. 8. The problem states that $a$ and $b$ are positive integers (1, 2, 3, ...). Since you must add a positive number (2) to $b$ to get $a$, $a$ must be strictly greater than $b$. 9. Therefore, $b < a$. ### Exam Strategy & Shortcut Observe the relationship between the denominators. Since 3.5 is exactly half of 7, the numerator on the left side $(a - b)$ must be exactly half of the numerator on the right side (4) to maintain the equality. Half of 4 is 2. So, $a - b = 2$. If their difference is a positive number, the first term ($a$) is larger. Thus, $b < a$. ### Common Pitfall Miscalculating the decimal arithmetic (like $3.5 \times 4 / 7$) or overlooking the constraint that $a$ and $b$ are positive integers, which definitively proves the strict inequality rather than just $a \ge b$. ### Final Answer **Therefore, the correct answer is $b < a$.**