Last night three of us could not solve a story question in our son’s 7th grade math book.
Person A leaves home at 9am and he is walking 4 mph.
Person B leaves home a half hour later in the same direction running 8.5 mph.
How long will it take for B to catch A?
What would be the algebraic expression to figure this out?
(We cheated and went to the back of the book and got the answer, which was 9 something, but we could not figure out how they came up with this answer. )
Our son went to school with this unanswered, so I/we are not violating the TOS stuff.
You want to set up 2 equations for the positions of the walker/runner.
Starting with t=0 when the runner left the house I set up function 1 for the position of the walker wrt time. And then set up function 2 for the position of the runner wrt time.
Since when the runner catches up to the walker they will be at the same position you can set function 1 = function 2 and solve for t.
Assuming they’re leaving from the same house I got a time around 26 minutes.
Here’s a way to think about it. At 9:30, when person B has just left, person A has already traveled 2 miles. The difference in speed between the two people is 4.5 mph. So how long does it take for person B to go 2 miles at 4.5 mph?
He hasn’t stopped walking - but the relative velocity is 4.5 between the two - so if you reduce the runner’s speed to 4.5, then the walker has relatively stopped.
which gives 26.66 minutes (A is 30 minutes ahead of B). ShirleyUjest is using this equation:
4t = 8.5(t-30),
which gives 56.66 minutes (B is 30 minutes behind A).
They give different answers because they’re answering different questions. Using the first equation, you’re answering the question: once B leaves*, ***how long will it take to catch A? Well, B leaves 30 minutes later, and it will take 26.66 minutes once he does, so 26.66 + 30 = 56.66.
At 9:30, when B sets out, A has walked 2 miles.
It will take 0.23529 hours for B to reach that point.
However, in that time, A has walked another 0.94196 miles.
To run that extra distance will take B another 0.11082 hours.
But in that time, A will have walked another 0.22163 miles.
And so on. The sequence never converges, so A and B never meet.
The problem statement says “How long will it take for B to catch A?”. That’s why I set t=0 for when B left the house. 26.7 minutes after B leaves and 56.7 minutes after A leaves are indeed the same time; but 26.7 minutes is the direct answer to the problem statement as I read it.
I’d mark both correct, so long as the answer statement specified which.
I suppose if showing our work were important, we should have converted time to hours first. We then would have gotten 0.944 hours, which is 56 minutes.