Word problems with equations

Question

Find bobs average speed if he ride his bike 5mi/hr uphill and 20mi/hr downhill

Answer 1

distance=rate*time

total distance=rate*total time

You are traveling the same distance up the hill as down the hill so the distance that you travel does not matter.

Pick an arbitrary distance such as 20 miles up the hill and 20 miles down the hill. I chose 20 because it is a multiple of 20 and a multiple of 5.

d=r*t

20=5*t

t=4 hours

20=20*t

t=1 hour

total distance=40 miles and total time=5 hours

total distance=rate*total time

40=rate*5

rate=40/5

rate=8 miles per hour (average speed)

Choose another distance, say 60 miles.

60=5*t

t=12 hours

60=20*t

t=3 hours

120=15*rate

rate=120/15

rate=8 miles per hour

you can also use what is called the harmonic mean because the distances are the same

here is the simple form of the harmonic mean: (2*rate1*rate2)/(rate1+rate2)

(2*5*20)/(5+20)

200/25

8 miles per hour again

the answer will be closer to 5 mph, which it is, than to 20 mph because you are spending more time traveling at the lower rate of 5 mph

Answer 2

Hi Alyssa, Bob's average speed would depend upon how far he rode in each direction, but let's assume he rode the same distance each way

You will need to find how long it took him to ride in each direction. Let's call the time it took to ride uphill at 5 mph t1 and the time it took to ride downhill at 20 mph t2. The total time for Bob's ride is t1 + t2.

Since it is assumed that the distance is the same in each direction, we can call the distance in each direction d, and that makes the total distance twice that, or 2d.

To find the time it takes to ride a certain distance, the distance is divided by the speed. For example, if Bob rode 10 miles uphill at 5 mph, then divide 10 miles by 5 mph and we get two hours going uphill. That would be t1 If the downhill ride was also 10 miles, then going 20 mph would have taken 10 miles divided by 20 mph, or one half hour, which would be t2. Add those together and Bob's total time for the ride was 2 1/2 hours.

The total distance for this ride was 20 miles, so divide that by the total time to get his average speed. That would be 20 miles / 2 1/2 hours = 8 miles per hour.

The distances do not have to be the same. It was just simple to use that as an illustration. For any given set of distances and speeds for Bob's ride this can be put into a formula:

Average speed = (d1 + d2) / ((d1/s1) + (d2/s2)), where d1 is the distance of the first leg and d2 is the distance of the second leg, and the speed of each leg is given by s1 and s2 respectively. That formula puts total distance in the numerator and total time in the denominator to give average speed for the entire trip.

Hope this helps