Hillary and her brother thomas were canoeing on a river. they headed west at 8: 30 a.m., traveling about 18 mph. at 11 a.m., their father followed their path west in a speed boat, traveling about 48 mph. how long will their father travel before he catches up to hillary and thomas? how long will hillary and thomas have been traveling?

The easiest way to solve this problem is to work out how far Hillary and Thomas (for the ease of writing I shall call them H&T) travel before their father sets off. As we should know already, speed tells us the distance travelled by an object over a period of time. Expressing this mathematically will give us the equation:
speed = distance/time
Now that we have this we can rearrange the equation to give us the distance travelled by an object going at a particular speed, which is:
distance = speed x time
In this question H&T are travelling at a speed of 18mph, and they set off at 8:30, which is 2 and a half hours before their father sets off too. Therefore to work out their distance travelled we do our equation for distance which is:
distance = speed x time
distance = 18 x 2.5 = 45
(Remember also that while it is 2 hours and 30 minutes, it is a half hour so the decimal following the 2 mathematically must be a 5)
So now that we know how far H&T have travelled before their father finally gets out of bed and sets off too, we can create an equation to tell us how long until he reaches them. At the point where he reaches them, the distance that he has travelled will be exactly equal to the distance that they have travelled. This means that we can create an equation to express this where the distance H&T have traveled is expressed on one side and the distance their father has travelled is on the other. Now we already know the kids have travelled 45 miles before their father sets off and we also know from our above equations that distance = speed x time and so our expression for their side of the equation, which is how far they travel before their father reaches them will be:
45 + 18a
The variable a in this case represents the time it takes for their father to reach them. The expression for their father then, given that he has no initial distance travelled and he is doing 48mph will be:
48a
Now we know these we can set them to be equal and solve for a and therefore the time it takes for the father to reach H&T:
45 + 18a = 48a
45 = 30a
a = 45/30 = 1.5
Therefore the father will travel for 1.5 hours (1 hour 30 minutes before he reaches H&T) which answers the first part of your question. The second part of your question asks how long H&T will have been travelling for, which is simply the time it takes for their father to reach them (1.5 hours) plus the time they were travelling for beforehand (2.5 hours), which we can (easily I hope) sum to be 4 hours total. Hope this helped :))

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