2. Problem Solving: Applications Tutorial 6g

3. 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.

4. 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.

5. 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.

6. Example 1 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? 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. 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. Review 5-Steps:

7. 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? Example 1 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. 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). Review 5-Steps: Let 2n = cost of the CD in dollars. 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?

8. Example 1 Step 3:Write an equation based on the given facts. Step 2: Let n = cost of the cassette in dollars. Let 2n = cost of the CD in dollars. Review 5-Steps: The cost of the CD is $18. 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? 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. The information in the 2 nd sentence gives us the information we need to write an equation. To write the equation, we will simply translate the sentence into math symbols 2n = 18 Solve the following equation: 2n = 18

9. Example 1 Step 3:Write an equation based on the given facts. Review 5-Steps: 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? 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. 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. n = 9 2 1 9 11

10. Example 1 Review 5-Steps: 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? 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. 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 n = 9 2 1 9 11 Step 2: Let n = cost of the cassette in dollars. Let 2n = cost of the CD in dollars. Step 5:Check the answer. Check the results with the statements of the problem: Is the cost of the CD ($18) twice the cost of the cassette? 2n = 18 2 9 = 18 18 = 18 Yes  Answer: The cassette cost $9.00 Review:

11. Example 2 Jim is 19 years old. He was 15 when his brother Bill was born. How old is Bill? Define your variable: Let b = Bill’s age. Let b + 15 = Jim’s age. Step 3:Write an equation: Relate: Jim is19 years old. Write: b + 15 =19 Step 4:Solve the equation: Solve: b + 15 = 19 Step 2:Choose and define your variable and use it with the given facts to represent the number(s) described in the problem. Jim is 19 years old. He was 15 when his brother Bill was born. How old is Bill?

12. Example 2 Jim is 19 years old. He was 15 when his brother Bill was born. How old is Bill? Let b = Bill’s age Let b + 15 = Jim’s age. Step 4:Solve the equation: Solve: b + 15 = 19 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. b = 4 -15 -15 Step 5:Check the answer. b + 15 = 19 4 + 15 = 19 19 = 19  Answer:Bill is 4 years old.