finding two different average speeds knowing only the total distance, total time, and both times?

Question

okay, heres the word problem:

on a recent 6 hour bike trip of 70 miles, olivia spent 2 hours on the first part of the trip biking at an average speed which was half of her average speed for the second part of the trip. find her speeds for the first and second parts of her trip

Answer 1

For this one, we have two different speeds, one slow, and one fast. Great! Let's start with that.

We know that the Liv was going the slower speed for 2 hours, and that the total trip was 6 hours, so we know that she must have been going the faster speed for 4 hours.

So we can write out an equation to represent this problem. We'll call the slow speed 's' and the faster speed 'f'

2 * s + 4 * f = 70

You can read this as she rode 2 hours at a slow speed, and 4 hours at a fast speed, for a total of 70 miles.

So we almost got it, but we can't really solve a problem with two variables neatly, so we need to find a way to make this equation use just one variable throughout. Luckily, we know that there is a relationship between the two speeds. Specifically that the slow speed, 's' was half of the fast speed, 'f'

in other words, 's'; is the same as '1/2f' and 'f' is the same as '2s'. We can use them interchangably. From here, see if you can replace either s or f in the above equation with its counterpart, swapping 's'; for '1/2f';. or swapping 'f' for '2s'. From there you should have one of your two speeds, which you can then use to find the other.

If you get this one, I suggest trying to find a similar problem, and giving that one a try on your own to see if you can get it :)

Answer 2

Use a variable to represent speed.

Let x = the average speed for the first two hours

So 2x = the average speed for the remaining four hours.

She went 70 miles by going 2 hours at x and 4 hours at 2x .

Equation would be 70 = 2*x + 4*2x.

Make sense? See if you can solve it from there.