2_5classnotes

2.5 PROBLEM SOLVING
HW: ODDS 1-7, 11, 15, 17, 25, and 33

Basic Facts To Know:
Enter % in equation as a decimal number AND a % is always multiplied by some factor.

Consecutive Integers: Integer and integer immediately following.   x, x+1

Consecutive Odd (or Even) Integers: Odd integer and odd integer immediately following. x, x+2

Perimeter: The sum of all sides.

Angles of a Triangle: The sum of the interior angles of a triangle is 180 degrees.

Complementary Angles: 2 angles whose sum is 90 degrees.

Supplementary Angles: 2 angles whose sum is 180 degrees.

Problem Solving Steps:
1. Read problem carefully & reread problem!
2. List the unknown and choose a variable to represent the unknown. Be specific!
              Ex. x = measure of angle 1 in degrees
3. Draw a picture if possible.
4. Translate the problem to an equation using the variable names that you have assigned to the unknowns.
5. Solve the equation.
6. State the answer clearly, using appropriate units.

Example:
Evelyn paid $50.40 including a 5% tax for her textbook. How much did the book itself cost?

Reading the problem we note that the $50.40 includes the cost of the book plus the tax that was charged.

        x=cost of the textbook in dollars
       .05x=tax charged on the book in dollars

The equation will be the sum of the cost of the book and the tax charged on the book is equal to $50.40.

     x + .05x = 50.40

Notice the tax charged is equal to the tax rate (written as a decimal) times the cost of the book.

     1.05x = 50.40       Combined like terms

          x = 48             Divided by 1.05

The cost of the book was $48.

Notice that we state the answer clearly using the units, which for this problem was $.

Example:
The sum of two consecutive even integers is 50. Find the numbers.

Reading the problem we note that we are adding two consecutive even integers and the sum is 50.

        x=even integer
      x+2=consecutive even integer

The equation will be the sum of the two numbers is equal to 50.

       x + x+2 = 50

Solve the equation.
2x + 2 = 50      Combined like terms

    2x = 48    Subtracted 2 from each side

    x = 24     Divided by 2

Find the other number,   x+2=24+2

                                      x+2=26

The numbers are 24 and 26.

Don’t forget to state the answer clearly. This problem has no units since the unknowns are numbers.

Example:
The second angle of a triangle is 3 times as large as the first. The third angle is 10 degrees more than the sum of the other two angles. Find the measure of the third angle.

Reading the problem we note that we will use the fact that the sum of the angles of a triangle is 180 degrees to solve this problem. We have 3 unknowns in the problem, which are the measures of the 3 angles. However, we have only to state the measure of the third angle in the final answer.

       x= measure of first angle in degrees
     3x= measure of second angle in degrees
   4x+10= measure of third angle in degrees

The equation will be the sum of the 3 angles is equal to 180.

     x + 3x + 4x+10 = 180

Solve the equation.

        8x + 10 = 180    Combined like terms

              8x = 170     Subtracted 10

              x = 21.25    Divided by 8

Find the measure of the third angle. 4x+10=4(21.25)+10
4x+10=85+10                                                   4x+10=95

The measure of the third angle is 95 degrees.

Notice that we state the answer clearly using the units, which for this problem was degrees.

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