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# Solving Problems Involving Distance, Speed/Rate, and Time

When you are solving problems for distance, speed, and time, you will find it helpful to use diagrams and/or charts to organize the information and help you solve the problem. You will also apply the formula that solves distance, speed/rate and time which is distance = speed/rate times time:

*d = rt.*

### DISTANCE, RATE, AND TIME EXAMPLE

Let’s begin with an example of how to read a word problem and plug numbers into the formula:

A train leaves Deb’s house and travels at 50 miles per hour. Two hours later, another train leaves from Deb’s house on the track beside or parallel to the first train but it travels at 100 miles per hour.

How far away from Deb’s house will the faster train pass the other train?

### SOLVING THE PROBLEM

Remember, d will represent the distance in miles from Deb’s house and t will represent the time that the slower train has been traveling.

You may wish to draw a diagram to show what is happening. Organize the information you have in a chart format if you haven’t solved these types of problems before. Remember the formula:

distance = speed x time

When identifying the parts of the word problem, remember distance is typically given in units of miles, meters, kilometers, inches, etc. Time is in units of seconds, minutes, hours, or years. Rate or speed is distance per time, so its units could be miles per hour, meters per second, inches per year, and so on.

Now you can solve the system of equations:

50*t* = 100 (*t* – 2) — multiply both values inside the parentheses by 100

50*t* = 100*t* – 200

200 = 50*t *— divide 200 by 50 to solve for t

*t* = 4

Now substitute *t* = 4 into train 1

*d* =50*t*

=50(4)

=200

Now you can write your statement. “The faster train will pass the slower train 200 miles from Deb’s house.

Now try solving similar problems:

Remember to use the formula that supports what you’re looking for – distance? speed/rate? time?

d = rt (Multiply)

r = d/t (Divide)

t = d/r (Divide)

### PRACTICE QUESTION 1

A train left Chicago and traveled towards Dallas. Five hours later another train left for Dallas traveling at 40 miles per hour with a goal or catching up with the first train bound for Dallas. The second train finally caught up with the first train after traveling for three hours. How fast was the train that left first going?

Remember to use a diagram to arrange your information. Then write the 2 equations to solve your problem.

Answer: 15 Miles per hour.

### PRACTICE QUESTION 2

One train left the station and traveled toward its destination at a speed of 65 miles per hour. Later, another train left the station traveling in the opposite direction of the first train, it was going at a speed of 75 miles per hour. After the first train had traveled for 14 hours it was 1960 miles apart from the second train. How long did the second train travel?

Answer: 14 Hours

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