Final answer:
The expression 1/3a²/3b, interpreted as a²/9b, is shown to be closer to b than to a by demonstrating that a²/9b is less than b and greater than a, given the condition that 0 < a < b.
Step-by-step explanation:
The student is asking to show that 1/3a²/3b is closer to b than to a when given that 0 < a < b for two rational numbers. To solve this, let's interpret 1/3a² as (1/3)a², which simplifies to a²/3. When we divide by 3b, it becomes (a²/3)/(3b) or a²/9b. Since a < b, we know that a² < b². Comparing a²/9b to a and b:
- a²/9b will be less than b, because (a² < b²) implies that a²/9b < b²/9b, which simplifies to b/9, a fraction of b.
- The expression a²/9b will be greater than a, because a²/9b is a fraction of b, and since a < b, a fraction of b is still larger than a.
Thus, a²/9b must be closer to b than to a, satisfying the condition set by the student's question.