Have you ever noticed that for any two people with different ages, you can always figure out a moment in time when the younger person was exactly half the age of the older person?

Let’s say that you are $$18$$ and your friend is $$23$$, so there is a $$5$$ year difference in your ages. Then, when you were $$5$$, your friend was $$10$$, so that was the moment when the ratio of your ages was exactly half.

Does this work for all age differences?

Let the initial age of the younger person be $$0$$, and the age of the older person be $$a$$. Then, at any time $$t$$ after this, the age of the younger person is $$0 + t$$ and the age of the older person is $$a + t$$.

Hence the ratio of the ages of the two people at any time $$t$$ is

$\frac{0 + t}{a + t}.$

Let’s look at what happens when we substitute different values for $$t$$. We will also let $$a = 5$$ to say that the second person is $$5$$ years older than the first. If we let $t=0$, then

$\frac{0+0}{5+0}=0$.

Note: If you are viewing this in Chrome and you can’t see the fraction, you need to enable MathML by pasting chrome://flags/#enable-experimental-web-platform-features into the address bar, enabling Experimental Web Platform features, and restarting Chrome.

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The graph above plots the ratio of the two ages over time. As you move the slider, you can see that as time approaches infinity, the ratio of the two ages approaches 1.

(NOTE: The graph is very glitchy if you move the slider quickly, and it doesn’t undo the plot when you slide back to the left. However, I’ve procrastinated publishing this post for three years because I haven’t been motivated to create a proper minimal plotting library for it, so I’m just publishing it as-is…)

We can also prove this mathematically using the concept of a limit:

$limt→∞ta+t=limt→∞t/ta/t+t/t=limt→∞1a/t+1=limt→∞10+1=1$