To understand how the ratio of the ages of A and B changes over time, let's consider their current ages and how they change with the passage of time.
Let's define:
- A's current age as [tex]\( A \)[/tex]
- B's current age as [tex]\( B \)[/tex]
where [tex]\( A > B \)[/tex], meaning A is older than B.
Initially, the ratio of their ages is given by:
[tex]\[ \text{Initial Ratio} = \frac{A}{B} \][/tex]
Now, let's assume some years pass. Suppose the number of years that pass is [tex]\( t \)[/tex]. After [tex]\( t \)[/tex] years:
- A's new age will be [tex]\( A + t \)[/tex]
- B's new age will be [tex]\( B + t \)[/tex]
The new ratio of their ages will then be:
[tex]\[ \text{New Ratio} = \frac{A + t}{B + t} \][/tex]
To determine how this new ratio compares to the initial ratio, let's analyze the scenario. Since A is older than B, adding the same number of years [tex]\( t \)[/tex] to both their ages affects a larger portion of B's age compared to A's age. This implies:
[tex]\[ \frac{A + t}{B + t} < \frac{A}{B} \][/tex]
This inequality shows that the ratio of their ages decreases over time. Consequently, the correct answer is:
[tex]\[ \boxed{c) \text{ratio of the ages of A and B decreases.}} \][/tex]
With the passage of time, the ratio of the ages of A and B decreases, since the same constant increment in age impacts the younger individual (B) more significantly than the older individual (A), reducing the overall ratio.
