Problem%20Solving:%20Applications

1
Problem Solving Applications

• Tutorial 6g

2
Introduction

• Before building bridges, skyscrapers, or
  roller-coasters, engineers and architects make
  models. Not all models are made of wood, metal,
  or plastic.
• Mathematical models are made of equations. These
  equations can predict how cables, beams, and
  coaster cars will perform in real life.
• You can solve many math problems by writing
  equations to model a situation.

3
Introduction

• Sometimes a problem cannot be solved directly, or
  with the previous problem solving methods
  explored in tutorial 3a, instead algebraic
  equations are necessary.
• Often the hardest part of problem solving is,
  understanding what is being asked. Therefore, a
  helpful tool is to use the following five-step
  strategy for breaking down the problem.

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Five-Step Strategy

1. Read the problem carefully and decide what
   numbers are being asked for.
2. Choose and define your variable. Then use it
   with the given facts to represent the number(s)
   described in the problem. Making a drawing may
   be helpful.
3. Write an equation based on the given facts.
4. Solve the equation and find the required numbers.
   
5. Check your answer.

5
Example 1
Review 5-Steps

1. Read the problem carefully and decide what
   numbers are being asked for.
2. Choose and define your variable and use it with
   the given facts to represent the number(s)
   described in the problem. Making a drawing may
   be helpful.
3. Write an equation based on the given facts.
4. Solve the equation and find the required numbers.
   
5. Check your answer.

• Maria bought a hit CD. The cost of the CD is 18.
  Her friend bought a cassette of the same music.
  If the CD cost twice as much as the cassette,
  what was the cost of the cassette?

Step 1 Read the problem carefully and decide
what number(s) are being asked for. As you
read, look for key words or phrases such as sum,
difference, product, is less than, or twice as
much as. Sometimes it is helpful to underline
key phrases (see above).

• This problem asks for the number of dollars in
  the cost of the cassette.

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Example 1
Review 5-Steps

1. Read the problem carefully and decide what
   numbers are being asked for.
2. Choose and define your variable and use it with
   the given facts to represent the number(s)
   described in the problem. Making a drawing may
   be helpful.
3. Write an equation based on the given facts.
4. Solve the equation and find the required numbers.
   
5. Check your answer.

• Maria bought a hit CD. The cost of the CD is 18.
  Her friend bought a cassette of the same music.
  If the CD cost twice as much as the cassette,
  what was the cost of the cassette?

Maria bought a hit CD. The cost of the CD is 18.
Her friend bought a cassette of the same music.
If the CD cost twice as much as the cassette,
what was the cost of the cassette?
Step 2 Choose and define your variable and use
it with the given facts to represent the
number(s) described in the problem. Making a
drawing may be helpful.
Let n cost of the cassette in dollars. Next try
to write the other values discussed in the
problem in terms of n (see above).
Let 2n cost of the CD in dollars.
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Example 1
Review 5-Steps

1. Read the problem carefully and decide what
   numbers are being asked for.
2. Choose and define your variable and use it with
   the given facts to represent the number(s)
   described in the problem. Making a drawing may
   be helpful.
3. Write an equation based on the given facts.
4. Solve the equation and find the required numbers.
   
5. Check your answer.

• Maria bought a hit CD. The cost of the CD is 18.
  Her friend bought a cassette of the same music.
  If the CD cost twice as much as the cassette,
  what was the cost of the cassette?

Step 2 Let n cost of the cassette in
dollars. Let 2n cost of the CD in dollars.
Step 3 Write an equation based on the given
facts.
The information in the 2nd sentence gives us the
information we need to write an equation. To
write the equation, we will simply translate the
sentence into math symbols
The cost of the CD is 18.
Solve the following equation 2n 18

18
2n
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Example 1
Review 5-Steps

1. Read the problem carefully and decide what
   numbers are being asked for.
2. Choose and define your variable and use it with
   the given facts to represent the number(s)
   described in the problem. Making a drawing may
   be helpful.
3. Write an equation based on the given facts.
4. Solve the equation and find the required numbers.
   
5. Check your answer.

• Maria bought a hit CD. The cost of the CD is 18.
  Her friend bought a cassette of the same music.
  If the CD cost twice as much as the cassette,
  what was the cost of the cassette?

Step 3 Write an equation based on the given
facts.
Step 4 Solve the equation and find the required
numbers.
Solve the following equation 2n 18

• Think to yourself What is being done to the
  variable (n)?
• A 2 is being multiplied to the variable (n).
  Division undoes multiplication, therefore you
  should divide a 2 to the left side to get n alone
  on that side.
• However, whatever you do to one side of an
  equation you must also do to the other side.

9
1
2 2
1
1
n 9
Solve the following equation 2n 18
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Example 1
Review 5-Steps

1. Read the problem carefully and decide what
   numbers are being asked for.
2. Choose and define your variable and use it with
   the given facts to represent the number(s)
   described in the problem. Making a drawing may
   be helpful.
3. Write an equation based on the given facts.
4. Solve the equation and find the required numbers.
   
5. Check your answer.

• Maria bought a hit CD. The cost of the CD is 18.
  Her friend bought a cassette of the same music.
  If the CD cost twice as much as the cassette,
  what was the cost of the cassette?

Step 2 Let n cost of the cassette in
dollars. Let 2n cost of the CD in dollars.
Review
Step 5 Check the answer.
Solve the following equation 2n 18
9
1

• Check the results with the statements of the
  problem
• Is the cost of the CD (18) twice the cost of
  the cassette? 2n 18

Answer The cassette cost 9.00
2 2
1
1
n 9
2 9 18 18 18 Yes ?
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Example 2

• Jim is 19 years old. He was 15 when his brother
  Bill was born. How old is Bill?

Jim is 19 years old. He was 15 when his brother
Bill was born. How old is Bill?
Step 2 Choose and define your variable and use
it with the given facts to represent the
number(s) described in the problem.
Define your variable Let b Bills age.
Let b 15 Jims age. Step 3 Write an
equation Relate
is
19 years old.
Jim
Write

19
b 15
Step 4 Solve the equation Solve b 15 19
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Example 2
Let b Bills age Let b 15 Jims age.

• Jim is 19 years old. He was 15 when his brother
  Bill was born. How old is Bill?

Jim is 19 years old. He was 15 when his brother
Bill was born. How old is Bill?
Step 4 Solve the equation b 15 19

• Think to yourself What is being done to the
  variable (b)?
• A 15 is being added to the variable (b).
  Subtraction undoes addition therefore you should
  subtract a 15 on the left to get b alone on one
  side.
• However, whatever you do to one side of an
  equation you must also do to the other side.

-15 -15
b 4
Step 5 Check the answer.
b 15 19 4 15 19 19 19 ?
Step 4 Solve the equation Solve b 15 19
Answer Bill is 4 years old.
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The End Time to move on to the assignment or the
next lesson