Mixture and Motion Problems: Charts Strategy

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Why are these two very different things grouped together?

That's because they both use the same style charts as one of the most effective strategies. 

MIXTURE PROBLEMS:

You have to create a chart that has THREE columns and rows.

Top Row: Prices/Percentages

Middle Row: Volumes: (x, y and the total number)

Bottom row: Multiply everything from the top and down, and put the result in the appropriate box.

It sounds very confusing, which is why I will provide a drawing for the example problem right now.

Example Problem:

Apple Jack has wants to have a 34 gallon apple cider batch. Blend 1 has 3% pure apple juice, and Blend 2 has 58% pure apple juice. She wants her final batch to have 36% pure apple juice. How much should she use?

Chart:


IMPORTANT:

  • "Equation 1" is just: x + y = 34
  • "Equation 2" is just: 0.03x + 0.58y = 12.24 

 

 

 Now solve for x and y using system of linear equations:

0.03x + 0.58y = 12.24 

x + y = 34

...Let's use substitution for this one.

y = 34 - x

0.03x + 0.58(34 - x) = 12.24

0.03x + 19.72 - 0.58x = 12.24

-0.55x = -7.48 

x = 13.6 

y = 34 - 13.6

y = 20.4 

Apple Jack will need 13.6 gallons of Blend 1 and 20.4 gallons for Blend 2. 

As long as you know how to do Algebra and how to organize the chart, this should be very simple. TRUST ME! 

RATE PROBLEMS:

Use a chart with TWO rows and THREE columns.

WHAT SHOULD BE IN THE CHART:

First column: Rates (mph, km/h, any number PER time period)

SAME DIRECTION ==> Speed of A + Speed of B (except when it's different objects like trains)

Environment helps you move. 

OPPOSITE DIRECTION ==> Speed of A - Speed of B 

Environment prevents you from moving but you still move. 

Second column: Time (hours, minutes, seconds, any time measurement)

Third Column: Distance

(It could already be given, or it is the product of the first two columns) 

Example Problem:

Fluttershy is really tired. She is flying with the wind towards Cloudsdale for 2 hours. She comes back flying against the wind from Cloudsdale, just to bring Pinkie Pie cupcakes for 3 hours. In total, she travels 192 miles. (She went down the same path twice in this problem!) What is the speed of Fluttershy and the speed of the wind?

 

 

DEFINE YOUR VARIABLES!

f = Speed of Fluttershy

w = Speed of Wind  

Also, 192/2 = 96, each path was equal, remember? 

Total round trip = distance/2 

Solve for f and w using system of linear equations:

2(f + w) = 96 ==> f + w = 48

3(f - w) = 96 ==> f - w = 32

2w = 16

w = 8

f = 40

Fluttershy's speed is 40 mph and the wind speed is 8 mph.

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