Solve Mixture and Uniform Motion Applications

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

• Draw a diagram to illustrate what it happening. Shown below is a sketch of what is happening in the example.
  
  
• Create a table to organize the information.
  • Label the columns “Rate,” “Time,” and “Distance.”
  • List the two scenarios.
  • Write in the information you know.
    

Step 2. Identify what you are looking for.

  You are asked to find the speed of both bikers.

  Notice that the distance formula uses the word “rate,” but it is more common to use “speed”

  when we talk about vehicles in everyday English.

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 the speed of the bikers. Let’s let r represent Wayne’s speed. Since Dennis’ speed is 7 mph faster, we represent that as $r+7$
  $\begin{array}{ccc}\hfill r+7& =\hfill & \text{Dennis’ speed}\hfill \\ \hfill r& =\hfill & \text{Wayne’s speed}\hfill \end{array}$
  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.
  The equation to model this situation will come from the relation between the distances. Look at the diagram we drew above. How is the distance travelled by Dennis related to the distance travelled by Wayne?
  Since both bikers leave from Riverside and travel to the beach, they travel the same distance. So we write:
  
  

Step 5. Solve the equation using algebra techniques.

Now solve this equation.	So Wayne’s speed is 21 mph.
Find Dennis’ speed.	Dennis’ speed 28 mph.

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

$\begin{array}{cc}\text{Dennis}\hfill & 28\text{mph}(1.5\text{hours})=42\text{miles}\hfill \\ \text{Wayne}\hfill & 21\text{mph}(2\text{hours})=42\text{miles}✓\hfill \end{array}$

Step 7. Answer the question with a complete sentence.

Wayne rode at 21 mph and Dennis rode at 28 mph.

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 Carina and her brother.

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 Carina's brother. Since Carina’s speed is 15 mph faster, we represent that as $r+15.$
  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 Carina traveled and the distance her brother traveled.
  The distance Carina traveled plus the distance her brother travel must add up to 410 miles. So we write:
  
  

Step 5. Solve the equation using algebra techniques.

Now solve this equation.	So Carina’s brother's speed was 50 mph.
Carina’s speed is $r+15.$	Carina’s speed was 65 mph.

Step 6. Check the answer in the problem and make sure it makes sense.
$\begin{array}{cccc}\hfill \text{Carina drove}& 65\text{mph}(3\text{hours})\hfill & =\hfill & 195\text{miles}\hfill \\ \hfill \text{Her brother drove}& 50\text{mph}(4\text{hours})\hfill & =\hfill & \underset{\text{—————}}{200\text{miles}}\hfill \\ & & & 395\text{miles}✓\hfill \end{array}$

Step 7. Answer the question with a complete sentence.

Carina drove 65 mph and her brother 50 mph.

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

• Draw a diagram to illustrate what it happening.
  
• 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 amount of time the trucks will travel until they are 325 miles apart.

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 the time travelled. Both trucks will travel the same amount of time.
  Let’s call the time t. Since their speeds are different, they will travel different distances.
  
  
• 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.
  We need to find a relation between the distances in order to write an equation. Looking at the diagram, what is the relationship between the distances each of the trucks will travel?
  The distance travelled by the truck going west plus the distance travelled by the truck going east must add up to 325 miles. So we write:
  
  

Step 5. Solve the equation using algebra techniques.

$\begin{array}{cccccc}\text{Now solve this equation}\hfill & & & \hfill 70t+60t& =\hfill & 325\hfill \\ & & & \hfill 130t& =\hfill & 325\hfill \\ & & & \hfill t& =\hfill & 2.5\hfill \end{array}$

So it will take the trucks $2.5$ hours to be 325 miles apart.

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

$\begin{array}{cccc}\hfill \text{Truck going West}& 70\text{mph}(2.5\text{hours})\hfill & =\hfill & 175\text{miles}\hfill \\ \hfill \text{Truck going East}& 60\text{mph}(2.5\text{hours})\hfill & =\hfill & \underset{\text{—————}}{150\text{miles}}\hfill \\ & & & 325\text{miles}✓\hfill \end{array}$

Step 7. Answer the question with a complete sentence.
It will take the trucks 2.5 hours to be 325 miles apart.

First, we draw a diagram that represents the situation to help us see what is happening.

We are asked to find her speed walking and riding her bike. Let’s call her walking speed r. Since her biking speed is three miles per hour faster, we will call that speed $r+3.$ We write the speeds in the chart.

The speed is in miles per hour, so we need to express the times in hours, too, in order for the units to be the same. Remember, 1 hour is 60 minutes. So:

We write the times in the chart.

Next, we multiply rate times time to fill in the distance column.

The equation will come from the fact that the distance from Naoko’s home to her school is the same whether she is walking or riding her bike.
So we say:

Translate to an equation.
Solve this equation.
Clear the fractions by multiplying by the LCD of all the fractions in the equation.
Simplify.
	$$6
Let’s check if this works. $\begin{array}{ccc}\text{Walk 3 mph (0.5 hour)}\hfill & \hfill =& \hfill 1.5\text{miles}\\ \text{Bike 6 mph (0.25 hour)}\hfill & \hfill =& \hfill 1.5\text{miles}\end{array}$

Yes, either way Naoko travels 1.5 miles to school.

Naoko’s walking speed is 3 mph and her speed riding her bike is 6 mph.

A diagram will help us model this trip.

Next, we create a table to organize the information. We know the total distance is 51 miles. We are looking for the rate of speed for each part of the trip. The rate while biking is 1.6 times the rate of running. If we let r = the rate running, then the rate biking is 1.6r.

The times here are given as clock times. Cruz started from home at 6:00 a.m. and started biking at 7:30 a.m. So he spent 1.5 hours running. Then he biked from 7:30 a.m until 9:45 a.m. So he spent 2.25 hours biking.

Now, we multiply the rates by the times.

By looking at the diagram, we can see that the sum of the distance running and the distance biking is 255 miles.

Translate to an equation.
Solve this equation.
Check. $\begin{array}{cccc}\text{Run}\hfill & \hfill 10\text{mph}(1.5\text{hours})& \hfill =& \hfill 15\text{mi}\\ \text{Bike}\hfill & \hfill 16\text{mph}(2.25\text{hours})& \hfill =& \hfill \underset{\text{——}}{36\text{mi}}\\ & & & \hfill 51\text{mi}\end{array}$