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# Formulas for Problem Solving

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Use the Formula for Distance to Find Distances, Rates and Times
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“What did you do this summer for fun?” Kevin asked Laila at lunch during the first week of school.

“We went camping at Yellowstone National Park and it was the best,” Laila said taking a bite of her sandwich.

“Really, that must have taken forever,” Kevin commented.

“No, we flew into Denver and then drove from Denver to Yellowstone so that we could see some of the country.”

“How far is it from Denver to Yellowstone?”

“It is about 540 miles. I know because I tracked it on the map. After leaving Denver it only took us nine hours to get there. We just made little stops and drove straight through so we could get there in time,” Laila explained.

Kevin began figuring out the math in his head. He wondered how fast the car was traveling if it made it in just nine hours.

Figuring out this problem will require you to use formulas. Formulas are often used in math as a method for solving a particular type of problem.

### Guidance

Did you know that you can use formulas to calculate distance, rate and time?

You measure distance -as in how far you can travel by car, boat, train or foot. You measure rate or the speed that you can travel by each different mode, and finally we can measure the time or how long that it takes us.

Look at the following situation and then let’s look at how we can calculate distance, rate and time by using a formula.

The Murphy Family drove for three and a half hours from Manhattan, New York to Providence, Rhode Island at a rate of fifty-three miles per hour. Determine the distance at which the Murphy’s were traveling.

First, we need to think about what we need to solve for. In this problem, we need to figure out the distance that the Murphy family traveled. To do this, there is a simple formula that we can use. In fact, we can use this formula whenever we are calculating distance.

$Distance = rate \times time$

Now we can take the given information from the problem and substitute that information into the formula. Once we have done that, we will be able to calculate the distance.

$D=53(3.5)$

53 represents the rate or speed that the car was traveling.

3.5 represents the three and a half hours that the family traveled.

$D=185.5 \ miles$

The distance that the Murphy family traveled was 185.5 miles.

That is a great question and the answer is “sort of”. You can use the same pieces of the formula only you will need to rewrite it to help you with the math. Here are two different versions of the same formula that you can use when looking to find the rate or the time.

To find rate , divide both sides of the equation by time.

$\text{Rate} = \frac{Distance}{Time}$

To find time , divide both sides of the equation by rate.

$\text{Time} = \frac{Distance}{Rate}$

Write these three formulas down in your notebooks.

#### Example A

Use the distance formula to solve for rate.

$\text{Distance} = 285 \ miles$

$\text{Time} = 9.5 \ hours$

$\text{Rate} = x$

Solution: The answer is 30 mph.

#### Example B

Use the distance formula to solve for time.

$\text{Distance} = 550 \ miles$

$\text{Rate} = 55 \ mph$

$\text{Time} = x$

Solution: The answer is 10 hours.

#### Example C

A car traveled 45 mph for 6 hours. How many miles did it travel?

Solution: $45(6) = 270$ miles

Now let's go back to the dilemma at the beginning of the Concept.

To solve this problem you will need to use the formula:

$D=R \times T$

Now that you have identified the correct formula, let’s think about the information that you have been given.

You know the distance from Denver to Yellowstone. It is 540 miles.

You know that it took the family 9 hours to get there with little quick stops.

We are looking to find the speed the car traveled.

Let’s substitute the given information into the formula.

$540=R(9)$

Next, we can solve this problem by solving the equation. We want to figure out “ $R$ ” so we divide both sides of the equation by nine.

$\frac{540}{9} &= R\\60 \ mph &= R$

The family was traveling about 60 mph. We could say that they were even traveling a little bit slower because there were quick stops.

### Vocabulary

Distance
how far a vehicle or person travels in a certain amount of time given a specific rate.
Rate
is the speed of travel.
Time
is calculation of the length of time for travel.

### Guided Practice

Here is one for you to try on your own.

A train traveled 255 miles in 300 minutes. Determine the rate at which the train was traveling.

Solution

Step 1: When using the distance formula, it is essential that you pay attention to the units used in the problem. If the rate in the problem gives miles per hour (mph), then time must be in hours. If the time is given in minutes, divide by sixty to determine the number of hours prior to solving the equation. Therefore prior to solving the problem above, divide 300 minutes by 60.

$300 \ minutes \div 60 \ minutes = 5 \ hours$

Step 2: Since the problem is asking you to determine the rate, use the formula $\text{Rate} = \frac{\text{Distance}}{\text{Time}}$ . Plug the known information into the problem.

$\text{Rate} = \frac{Distance}{Time}$

Step 3: Solve

$\text{Rate} &= \frac{255}{5}\\\text{Rate} &= 51 \ miles \ per \ hour$

The train was traveling at a rate of 51 miles per hour.

### Practice

Directions: Find the number of miles traveled given the rate and time.

1. Four hours at a rate of 33 mph.
2. Six hours at a rate of 55 mph.
3. Eight hours at a rate of 65 mph.
4. 12 hours at a rate of 50 mph.
5. 14 hours at a rate of 60 mph.
6. 19 hours at a rate of 50 mph.
7. 11 hours at a rate of 55 mph.
8. 18 hours at a rate of 35 mph.
9. 12 hours at a rate of 70 mph.
10. 10 hours at a rate of 58 mph.
11. 15 hours at a rate of 57 mph.
12. 21 hours at a rate of 66 mph.

Directions: Use the given information to figure out each rate or time.

1. A car traveled 450 miles at a speed of 30 mph. How many hours did it take?
2. A car traveled 600 miles in 12 hours. What was the speed of the car?
3. A runner traveled 6 miles in 30 minutes. How fast was the runner going?
4. A car traveled 520 miles at a speed of 65 mph. How many hours did it take?