# An Introduction to Problem Solving

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An Introduction to Problem Solving
OBJECTIVES 1. Write algebraic expressions that can be simplified 2. Apply the steps for problem solving

Consecutive Integer Problems

Choose something to represent the first integer---let’s choose “x”.
x = first integer Then to represent a consecutive integer, that would mean the integer right after x or x+1. x +1 = second integer The sum of two consecutive integers is Find the integers. Means add together The hard part is done! x + x +1 = 137 Now solve for x to get the first integer. Just add 1 to get the second.

The sum of two consecutive integers is 65. What is the second number?
Let x = the 1st number Let x + 1 = the 2nd number Sum means??? x = 32 x + 1 = 33 x + x + 1 = 65 = 65  2x + 1 = 65 -1 -1 2x = 64

The sum of two consecutive integers is 27. What are the two integers?

What two consecutive integers have a sum of 39?

The sum of two consecutive integers is 9. What are the two integers?

Let’s consider a problem that asked for consecutive even integers
Let’s consider a problem that asked for consecutive even integers. Your first integer will still be “x”. x = first integer Then to represent a consecutive even integer, you would need to add 2 instead of 1 and get x+2. x +2 = second integer The sum of two consecutive even integers is Find the integers. Now you are ready to solve. x + x +2 = 626

Let’s consider a problem that asked for consecutive odd integers
Let’s consider a problem that asked for consecutive odd integers. Your first integer will still be “x”. x = first integer Now what would you do to x to get to the next odd integer? Most students initial reaction is “add 1” but try x = 3 (an odd integer) and see what happens when you add 1. Not an odd integer. So what would you add to 3 to get the next odd integer? x +2 = second integer So whether the problem says even integer or odd integer, the setup would look the same. If x happens to be odd then when you add 2 you will be at the next odd integer and if it happens to be even and you add 2 you will be at the next even integer.

The perimeter of a rectangle
with sides of length x and 2x - 1 The perimeter of a rectangle is the sum of the lengths of the sides x In words: sides sides 2 sides 2 sides 2x - 1 Translate: 2(x) (2x-1) Then: x x-2 6x simplify

General Strategy for Problem Solving
UNDERSTAND the problem. During this step, become comfortable with the problem. Some ways of doing this are: Read and reread the problem. Choose a variable to represent the unknown. Construct a drawing. Propose a solution and check. Pay careful attention to how you check your proposed solution. This will help when writing an equation to model the problem.

TRANSLATE the problem into an equation.
SOLVE the equation. INTERPRET the results: Check the proposed solution in the stated problem and state your conclusion.

FINDING THE ORIGINAL PRICE OF A COMPUTER
Suppose that a computer store just announced an 8% decrease in the price of a particular computer model. If this computer sells for \$2162 after the decrease, find the original price of this computer. What are the steps to solving this problem?

UNDERSTAND. Read and reread the problem. Recall that a percent decrease means a percent of the original price. Let’s guess that the original price of the computer is \$ The amount of decrease is then 8% of \$2500, or (0.08)(\$2500) = \$200. This means that the new price of the computer is the original price minus the decrease, or \$ \$200 = \$ Our guess is incorrect, but we now have an idea of how to model this problem

2. TRANSLATE: In words: Original price of computer minus
8% of original price is new price Translate: x - 0.08 x = 2160

0.92x = 2162 Divide both sides of equation
3. SOLVE the equation x – 0.08x = 2162 0.92x = Divide both sides of equation x = Solution

4. INTERPRET. Check: If the original price of the computer was \$2350, the new price is \$2350 – (0.08)(\$2350) = \$ \$188 = \$2162 State: The original price of the computer was \$2350

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