9.1 Use a Problem Solving Strategy

Problem Solving Strategy

Solving word problems in math comes down to having a reliable process you can follow every time. The strategy covered here gives you a step-by-step framework for turning word problems into math, solving them, and making sure your answer actually makes sense.

Systematic Approach for Word Problems

Every word problem can be tackled using four main phases. Here's how each one works:

1. Understand the problem

Read the problem carefully, more than once if needed. Your goal is to identify three things:

• Given information — the values, quantities, and conditions stated in the problem
• The unknown — what you're being asked to find (this becomes your target variable)
• Constraints — any limitations or restrictions mentioned (e.g., "no more than 20 students")

2. Devise a plan

Before you start calculating, think about how you'll approach the problem.

• Break the problem into smaller, manageable steps
• Identify which math concepts or formulas apply (equations, proportions, area formulas, etc.)
• Consider whether an algebraic, graphical, or arithmetic approach fits best

3. Execute the plan

Now carry out your approach step by step.

• Perform calculations in an organized way
• Simplify expressions and solve equations as needed
• Keep your work neat so you can trace back through it if something goes wrong

4. Review and check the solution

This step catches more mistakes than you'd expect.

• Does the answer make sense in context? (A person's age shouldn't be negative, a distance shouldn't be zero if someone traveled, etc.)
• Double-check your arithmetic for errors
• Confirm that your solution satisfies all the conditions given in the problem, not just some of them

Identifying and Setting Up the Problem

Before jumping into math, spend time figuring out what kind of problem you're dealing with and pulling out the right information.

• Recognize the problem type. Look for patterns or similarities to problems you've solved before. Is it a percentage problem? A ratio? A geometry question? Classifying it helps you pick the right tools.
• Extract relevant information. List out the given values, quantities, and measurements. Note any special conditions or relationships between quantities.
• Represent the problem mathematically. Assign variables to the unknowns, then write equations or expressions that capture the relationships described in the problem.
• Solve and interpret. After solving, translate your mathematical answer back into the context of the problem. Express it with the right units and check that it's reasonable.

Translating Words into Math

One of the trickiest parts of word problems is converting English sentences into mathematical expressions. Here's a reliable process:

1. Spot the key words. Certain words map directly to math operations:

2. Assign variables to unknowns. Use a letter to stand for what you don't know. Choosing a meaningful letter helps: $d$ for distance, $p$ for price, $n$ for a number.

3. Build the expression. Convert phrases piece by piece. For example, "five more than twice a number" becomes $2n + 5$.

4. Form an equation. Use the translated expressions to set up an equation that reflects the relationship described. For instance, "five more than twice a number is 17" becomes $2n + 5 = 17$.

5. Solve the equation. Use inverse operations to isolate the variable.

• $2n + 5 = 17$
• $2n = 12$ (subtract 5 from both sides)
• $n = 6$ (divide both sides by 2)

Then check: twice 6 is 12, plus 5 is 17. It works.

Additional Problem-Solving Techniques

These extra strategies are worth keeping in your back pocket:

• Estimation — Before solving, estimate a rough answer. If your final result is wildly different from your estimate, something probably went wrong.
• Logical reasoning — Think about the structure of the problem. What has to be true? What can you rule out?
• Visualization — Sketch a diagram, draw a number line, or make a table. Seeing the problem laid out visually often makes the path to a solution much clearer.