Question

Two riders are in a bike race. The second rider starts half an hour late but rides 5 miles per hour faster than the first rider. If they both finish 2 hours after the race starts, how fast are they each riding?

Answer 1

A. 5 mph and 10 mph
B. 12.5 mph and 17.5 mph
C. 15 mph and 20 mph
D. 10 mph and 15 mph

Answer 2

The both rode the same distance, d miles.

Can you try to fill out the following table with the given information.

Distance Time Speed
First Rider
Second Rider

Answer 3

All I know is time. The first did it in 2 hours and the second did it in an hour and a half.

Answer 4

The distance is the same for both.
The two times are known.
The speed of the first rider is s. What is the speed of the second rider?

Distance Time Speed
First Rider d
Second Rider d

Answer 5

s + 5?

Answer 6

Great.

Distance Time Speed
First Rider d 2 s
Second Rider d 1.5 s + 5

Answer 7

The above is what we know.
Remember, the distance for both riders is equal.
There is one more important piece of information.

speed = distance/time

Solving for distance, we get

distance = speed * time

Ok so far?

Answer 8

Yes, I believe so.

Answer 9

Great.
Now using the info of the first rider in the table, what is the distance equal to?

Answer 10

Remember, distance = speed * time
Distance is d, replace speed and time with the table values for the first rider.

Answer 11

2s?

Answer 12

The whole equation is
d = 2s.
Good.

Now do the same using the information of the second rider.
What is distance equal to?

Answer 13

1.5s+5?

Answer 14

I think you meant this:

d = 1.5(s+5)

which is correct.

Answer 15

Yes, sorry. Thank you.

Answer 16

1.5 doesn't just multiply s. 1.5 multiplies the quantity s + 5. That's why you need parentheses.

Answer 17

Ahhh. Alright, I see that now.

Answer 18

Ok.
Now we know that d = 2s and d = 1.5(s + 5).
d is the same for both equations, so we can write
2s = 1.5(s + 5)

Now we have a simple equation with only one variable, s.
We can solve for s to find the speed of the first rider.

Answer 19

2s = 1.5(s + 5)

First, we distribute the 1.5 on the right side to get rid of the parentheses.

Answer 20

So would you dived 2s by 1.5?

Answer 21

You can do that, but I think it's easier to distribute the 1.5 on the right side like this:

2s = 1.5s + 1.5 * 5
2s = 1.5s + 7.5

Answer 22

Now subtract 1.5s from both sides.

Answer 23

So 0.5s=7.5?

Answer 24

@mathstudent55

Answer 25

Great.
Now, divide both sides by 0.5.
What do you get for s?

Answer 26

s=15? So C?

Answer 27

Correct.
s = 15
That means the speed of the first rider is 15 mph.
We were told that the second rider's speed is 5 mph higher, so the second rider's speed is 20 mph.

Answer 28

Awesome! Do you think you can help me with a couple more?

Answer 29

I can do 1 more, but then I need to go.
Please start anew post.
I'll look for it.