# 3.5 Solve uniform motion applications  (Page 2/7)

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Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

Wayne 21 mph, Dennis 28 mph

Jeromy can drive from his house in Cleveland to his college in Chicago in 4.5 hours. It takes his mother 6 hours to make the same drive. Jeromy drives 20 miles per hour faster than his mother. Find Jeromy’s speed and his mother’s speed.

Jeromy 80 mph, mother 60 mph

In [link] , the last example, we had two trains traveling the same distance. The diagram and the chart helped us write the equation we solved. Let’s see how this works in another case.

Christopher and his parents live 115 miles apart. They met at a restaurant between their homes to celebrate his mother’s birthday. Christopher drove 1.5 hours while his parents drove 1 hour to get to the restaurant. Christopher’s average speed was 10 miles per hour faster than his parents’ average speed. What were the average speeds of Christopher and of his parents as they drove to the restaurant?

## Solution

Step 1. Read the problem. Make sure all the words and ideas are understood.

• Draw a diagram to illustrate what it happening. Below shows a sketch of what is happening in the example.

• Create a table to organize the information.
• Label the columns rate, time, distance.
• List the two scenarios.
• Write in the information you know.

Step 2. Identify what we are looking for.

• We are asked to find the average speeds of Christopher and his parents.

Step 3. Name what we are looking for. Choose a variable to represent that quantity.

• Complete the chart.
• Use variable expressions to represent that quantity in each row.
• We are looking for their average speeds. Let’s let r represent the average speed of the parents. Since the Christopher’s speed is 10 mph faster, we represent that as $r+10.$

Fill in the speeds into the chart.

Multiply the rate times the time to get the distance.

Step 4. Translate into an equation.

• Restate the problem in one sentence with all the important information.
• Then, translate the sentence into an equation.
• Again, we need to identify a relationship between the distances in order to write an equation. Look at the diagram we created above and notice the relationship between the distance Christopher traveled and the distance his parents traveled.

The distance Christopher travelled plus the distance his parents travel must add up to 115 miles. So we write:

Step 5. Solve the equation using good algebra techniques.

$\begin{array}{cccc}\text{Now solve this equation.}\hfill & & & \hfill \begin{array}{ccc}\hfill 1.5\left(r+10\right)+r& =\hfill & 115\hfill \\ \hfill 1.5r+15+r& =\hfill & 115\hfill \\ \hfill 2.5r+15& =\hfill & 115\hfill \\ \hfill 2.5r& =\hfill & 100\hfill \\ \hfill r& =\hfill & 40\hfill \end{array}\hfill \\ & & & \text{So the parents’ speed was 40 mph.}\hfill \\ \text{Christopher’s speed is}\phantom{\rule{0.2em}{0ex}}r+10.\hfill & & & \hfill \begin{array}{c}\hfill r+10\hfill \\ \hfill 40+10\hfill \\ \hfill 50\hfill \end{array}\hfill \\ & & & \text{Christopher’s speed was 50 mph.}\hfill \end{array}$

Step 6. Check the answer in the problem and make sure it makes sense.

$\begin{array}{cccccc}\phantom{\rule{2.7em}{0ex}}\text{Christopher drove}\hfill & & & \text{50 mph (1.5 hours)}\hfill & =\hfill & \text{75 miles}\hfill \\ \phantom{\rule{2.7em}{0ex}}\text{His parents drove}\hfill & & & \text{40 mph (1 hours)}\hfill & =\hfill & \underset{\text{_______}}{\text{40 miles}}\hfill \\ & & & & & \text{115 miles}\hfill \end{array}$

$\begin{array}{cccc}\begin{array}{c}\mathbf{\text{Step 7. Answer}}\phantom{\rule{0.2em}{0ex}}\text{the question with a complete sentence.}\hfill \\ \\ \end{array}\hfill & & & \begin{array}{c}\text{Christopher’s speed was 50 mph.}\hfill \\ \text{His parents’ speed was 40 mph.}\hfill \end{array}\hfill \end{array}$

Aziza is solving this equation-2(1+x)=4x+10
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