# 1.7: Problem-Solving Plan

**At Grade**Created by: CK-12

## Learning Objectives

- Read and understand given problem situations.
- Make a plan to solve the problem.
- Solve the problem and check the results.
- Compare alternative approaches to solving the problem.
- Solve real-world problems using a plan.

## Introduction

We always think of mathematics as the subject in school where we solve lots of problems. Throughout your experience with mathematics you have solved many problems and you will certainly encounter many more. Problem solving is necessary in all aspects of life. Buying a house, renting a car, figuring out which is the better sale are just a few examples where people use problem solving techniques. In this book, you will use a systematic plan to solve real-world problem and learn different strategies and approaches to solving problems. In this section, we will introduce a problem-solving plan that will be useful throughout this book.

## Read and Understand a Given Problem Situation

The first step to solving a word problem is to **read and understand** the problem. Here are a few questions that you should be asking yourself.

What am I trying to find out?

What information have I been given?

Have I ever solved a similar problem?

This is also a good time to define any variables. When you identify your **knowns** and **unknowns**, it is often useful to assign them a letter to make notation and calculations easier.

## Make a Plan to Solve the Problem

The next step in the problem-solving plan is to **make a plan** or **develop a strategy.** How can the information you know assist you in figuring out the unknowns?

Here are some common strategies that you will learn.

- Drawing a diagram.
- Making a table.
- Looking for a pattern.
- Using guess and check.
- Working backwards.
- Using a formula.
- Reading and making graphs.
- Writing equations.
- Using linear models.
- Using dimensional analysis.
- Using the right type of function for the situation.

In most problems, you will use a combination of strategies. For example, drawing a diagram and looking for patterns are good strategies for most problems. Also, making a table and drawing a graph are often used together. The “writing an equation” strategy is the one you will work with the most in your study of algebra.

## Solve the Problem and Check the Results

Once you develop a plan, you can implement it and **solve the problem.** That means using tables, graph and carrying out all operations to arrive at the answer you are seeking.

The last step in solving any problem should always be to **check and interpret** the answer. Here are some questions to help you to do that.

Does the answer make sense?

If you plug the solution back into the problem do all the numbers work out?

Can you use another method to arrive at the same answer?

## Compare Alternative Approaches to Solving the Problem

Sometimes a certain problem is best solved by using a specific method. Most of the time, however, it can be solved by using several different strategies. When you are familiar with all of the problem-solving strategies, it is up to you to choose the methods that you are most comfortable with and that make sense to you. In this book, we will often use more than one method to solve a problem. This way we can demonstrate the strengths and weakness of different strategies when applied to different types of problems.

Regardless of the strategy you are using, you should always implement the problem-solving plan when you are solving word problems. Here is a summary of the problem-solving plan.

**Step 1**

Understand the problem

Read the problem carefully. Once the problem is read, list all the components and data that are involved. This is where you will be assigning your variables.

**Step 2**

Devise a plan – Translate

Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart or construct a table as a start to solve your problem solving plan.

**Step 3**

Carry out the plan – Solve

This is where you solve the equation you developed in Step 2.

**Step 4**

Look – Check and Interpret

Check to see if you used all your information. Then look to see if the answer makes sense.

## Solve Real-World Problems Using a Plan

Let’s now apply this problem solving plan to a problem.

**Example 1**

*A coffee maker is on sale at 50% off the regular ticket price. On the “Sunday Super Sale” the same coffee maker is on sale at an* *additional**40% off. If the final price is $21, what was the original price of the coffee maker?*

**Solution:**

**Step 1**

**Understand**

We know: A coffee maker is discounted 50% and then 40%

The final price is $21.

We want: The original price of the coffee maker.

**Step 2**

**Strategy**

Let’s look at the given information and try to find the relationship between the information we know and the information we are trying to find.

50% off the original price means that the sale price is half of the original or \begin{align*}0.5 \ \times\end{align*} original price

So, the first sale price \begin{align*}= 0.5 \ \times\end{align*} original price

A savings of 40% off the new price means you pay 60% of the new price \begin{align*}0.6 \ \times\end{align*} new price \begin{align*} = 0.6 \ \times (0.5 \ \times\end{align*} original price\begin{align*}) = 0.3 \ \times\end{align*} original price

So, the price after the second sale \begin{align*}= 0.3 \ \times\end{align*} original price

We know that after two sales, the final price is $21

\begin{align*}0.3 \ \times\end{align*} original price \begin{align*}= \$21\end{align*}

**Step 3**

**Solve**

Since \begin{align*}0.3 \ \times\end{align*} original price \begin{align*}= \$21\end{align*}

We can find the original price by dividing $21 by 0.3.

Original price \begin{align*} = \$21 \div 0.3 = \$70\end{align*}

**Answer** The original price of the coffee maker was $70.

**Step 4**

**Check**

We found that the original price of the coffee maker is $70.

To check that this is correct let’s apply the discounts.

50% of \begin{align*}\$70 = .5 \times \$70 = \$35\end{align*} savings.

So, after the first sale you pay: original price – savings \begin{align*}= \$70 - \$35 = \$35\end{align*}.

40% of \begin{align*}\$35 = .4 \times \$35 = \$14\end{align*} savings.

So, after the second sale you pay: \begin{align*}\$35 - \$14 = \$21\end{align*}.

**The answer checks out.**

## Review Questions

- A sweatshirt costs $35. Find the total cost if the sales tax is 7.75%.
- This year you got a 5% raise. If your new salary is $45,000, what was your salary before the raise?
- It costs $250 to carpet a room that is \begin{align*}14 \ ft \times 18 \ ft\end{align*}. How much does it cost to carpet a room that is \begin{align*}9 \ ft \times 10 \ ft\end{align*}?
- A department store has a 15% discount for employees. Suppose an employee has a coupon worth $10 off any item and she wants to buy a $65 purse. What is the final cost of the purse if the employee discount is applied before the coupon is subtracted?
- To host a dance at a hotel you must pay $250 plus $20 per guest. How much money would you have to pay for 25 guests?
- It costs $12 to get into the San Diego County Fair and $1.50 per ride. If Rena spent $24 in total, how many rides did she go on?
- An ice cream shop sells a small cone for $2.95, a medium cone for $3.50 and a large cone for $4.25. Last Saturday, the shop sold 22 small cones, 26 medium cones and 15 large cones. How much money did the store earn?
- The sum of angles in a triangle is 180 degrees. If the second angle is twice the size of the first angle and the third angle is three times the size of the first angle, what are the measures of the angles in the triangle?

## Review Answers

- $37.71
- $42857
- $89.29
- $45.25
- $750
- 8 rides
- $219.65
- \begin{align*}30^\circ, 60^\circ, 90^\circ\end{align*}