# Distance rate time problems

Distance rate time problems involves object moving at a constant rate. This is called uniform motion

The formula d = r × t is the formula to use to solve problems related to distance, rate, and time.

I will show you how to solve 3 types of uniform motion.

Case #1: Same direction travel

Let's say you and your wife are going to a place, but you each prefer to drive your own car.

It may be obvious that for this type of problem, the distance is the same since you left the same place (your house) and are going to the same location.

Example #1:

You leave your house at 7 AM and your drive 60 miles per hour. Your wife was taking her time. So she leaves the house at 9 AM. However, she is now speeding at 90 miles per hour.

How long will it take your wife to catch up with you assuming that the police does not stop her?

When your wife finally catches up with you, what distance have you both traveled?

It is always a good idea to make a table when solving distance rate time problems, especially if you are not familiar with distance word problems.

let t be the time it takes you to get to your destination.

t-2 is then the time it will take your wife. The reason we subtract 2 is because compared to you, your wife is late by 2 hours

 Car Rate Time Distance traveled You 60 t 60 × t Your wife 90 t-2 90 × (t - 2)

Since the distance you travel is the same as the distance your wife travels, we get:

60 × t = 90 × (t - 2)

60t = 90t - 180

60t + 180 = 90t

180 = 90t - 60t

180 = 30t

Since 30 times 6 = 180, t = 6

This means that your wife will catch up with you in 6 hours

d = 60 × 6 = 360

d = 90 × (6-2) = 90 × 4 = 360

Therefore, your wife will catch up with you 360 miles away. Of course this is assuming that you keep crusing at 60 miles per hour and she keeps cruising at 90 miles per hour.

Case #2: Round-trip travel

Let's say that you are going back to your house. When you get to your house, this is a round-trip

Suppose that you went to no other places, the distance then to go back to your house is again the same.

Example #2:

However, now you decide to speed things up a little bit. Instead of driving 60 miles per hour, you do 90 miles per hour

If the round-trip lasts 12 hours, how long does it take you to get back home?

let t be the time it takes you to get to your destination.

12 - t is then the time it will take you to get back home

 Car Rate Time Distance traveled to your location 60 t 60 × t back to your house 90 12 - t 90 × (12 - t)

Since the distance to go to your location and to go back home is the same, we get:

60 × t = 90 × (12 - t)

60t = 90 × 12 - 90t

60t = 1080 - 90t

60t + 90t = 1080 - 90t + 90t

150t = 1080

Since 150 times 7.2 = 1080, t = 7.2

12 - 7.2 = 4.8

Therefore, it took you 4.8 hours to get back home.

Case #3: Opposite direction travel

Suppose that you are driving south and your wife is going north on the same highway.

After a certain amount of time driving the distance you travel will most likely be different than your wife's distance since it is not likely that your rate will be the same.

Therefore, when solving distance rate time problems involving opposite direction travel, you can add the distances to get the total distance

Example #3:

Going south you drive 45 miles per hour. Going north your wife drives 30 miles per hour. After how long will you be 225 miles apart?

let t be the time it takes you to be 225 miles apart from your wife.

It will take your wife the same time to be 225 miles apart from you. I hope this makes sense!

 Car Rate Time Distance traveled You 45 t 45 × t Your wife 30 t 30 × t

The total distance you and your wife travel is 45t + 30t = 75t and this must be equal to 225

75 × t = 225

Since 75 times 3 = 225, t = 3

Therefore, you will be 225 miles apart after 3 hours.

The above are great examples of distance rate time problems. Try to do some now in your math textbook.

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