Question

Calvin went cycling in San Francisco, which is extremely hilly. He can pedal up a hill at a speed of 12 mph, and down a hill at a speed of 36 mph. If he went up and down a hill (back to the same point), what is his average speed (in mph) for the entire journey?

Answer 1

i think it's 24 mph, not sure.

Answer 2

any more answers

Answer 3

You get 24 mph if you just take the average of 12 and 36, but that's not appropriate here since Calvin was riding at these different speeds for different lengths of time. He spends 3 times as long riding up the hill as he does riding down the hill, so we need to account for that to find the average speed.

In general, the average speed is just the distance/time. We don't have any particular information about either, so we'll try a particular distance for the hill and use that to find the average. For simplicity, let's say the hill is 36 miles (the exact number doesn't matter since we're going to be dividing this number out when calculating the average speed). Going up the hill, Calvin rides 12 mph, so it takes 3 hours to go up the hill. Going down the hill, Calvin rides 36 mph, so it takes 1 hour to go down the hill. His average speed is the total distance, 72 miles, over the total time, 4 hours, which works out to be 72/4=18 mph.

Answer 4

mslano has the correct answer.
a more general way to do the problem is say the distance up (or down) the hill is D

use rate*time= distance to find the time it takes to go up the hill:
\[ 12 t_{up} = D \\ t_{up} = \frac{D}{12}\]
similarly, the time to go down the hill, going 36 mph is
\[ t_{down} = \frac{D}{36}\]
the total time up and back down is T
\[ T = t_{up} + t_{down} \\ T = \frac{D}{12}+ \frac{D}{36} \\T= \frac{4D}{36}= \frac{D}{9}\]

now use
rate*time = distance
again, with total distance up and back down= 2D, and T = D/9
you get
\[ rate = \frac{2D}{\frac{D}{9}} = 2D \cdot \frac{9}{D}= 18\]

Answer 5

More simply that,
Let the distance of the hill be D m. Then, Calvin spent D÷12 hours going up the hill, and D÷36 hours going up the hill. Thus he spent a total of 4D÷36 hours cycling 2D m. Hence, his average speed is 2D÷4D÷36=18 mph.

The answer is not (12+36)÷2=24 because he did not spend the same amount of time on each leg of the trip.

Answer 6

More challanging,
Jack and Danny are bricklayers. Jack can build a certain brick wall in 8 hours whereas Danny takes 11 hours. If they work together, their combined rate decreases by 7 bricks an hour, and it only takes them 5 hours to build the wall together. Find the number of bricks in the wall.

Answer 7

if the wall contains N bricks, we can say
Jack lays N bricks/ 8 hrs
Dan lays N bricks/ 11 hrs

together they lay
\[ \frac{N}{8}+ \frac{N}{11} - \frac{7 \text{ bricks}}{1 \text{ hour}} \]
where we account for a slow down of 7 bricks per hour

at this rate this finish the wall in 5 hours, meaning that together they lay at a rate of N bricks/ 5 hours
\[ \frac{N}{8}+ \frac{N}{11} - \frac{7}{1} = \frac{N}{5} \\ N\left( \frac{1}{8}+\frac{1}{11}- \frac{1}{5}\right)=7\]
solve for N bricks

Answer 8

**More simply that,***

no, that is exactly the same thing, expressed with less detail, and harder to follow for someone who is clueless.

Answer 9

May i ask another question?