Benjamin starts walking along a path at a constant four miles per hour. One and a half hours after Benjamin leaves, his sister Hannah starts jogging along the same route at a constant six miles per hour. How long will it take Hannah to catch up with Benjamin?
Let’s write down what we know. Benjamin is travelling at four miles per hour. And he’s been going for one and a half hours. Hannah can travel at six miles per hour. Let’s turn this information into a function.
We’ll let 𝑦 be equal to the distance travelled. 𝑥 will be equal to the number of hours travelling. Benjamin travels at four miles per hour. His rate is then four 𝑥. In the same way, we can show that Hannah’s function is 𝑦 equals six 𝑥. We want to know when the two paths would cross.
We can’t forget, though, that Benjamin started an hour and a half before Hannah. And that means we need to add the distance that Benjamin has already travelled. He travels at four miles per hour for one and a half hours. If we multiply four by one and a half, we get six. Benjamin travels four miles per hour. And he’s already six miles ahead of Hannah.
Now that we know that, we want to solve for 𝑥: when are their distances equal; when will Hannah catch up with Benjamin.
We start by subtracting four 𝑥 from both sides to get both 𝑥s on the same side. On the left, we’re left with six miles. Six 𝑥 minus four 𝑥 equals two 𝑥. And then we’ll divide both sides of the equation by two. Two divided by two cancels out, leaving us with 𝑥. Six divided by two equals three. 𝑥 equals three.
And that means that after three hours, Hannah would catch up to Benjamin. And that is quite a lot of jogging.