3.1 Use a Problem-Solving Strategy

Problem-Solving Strategy

Word problems follow predictable patterns once you know how to break them apart. The key is having a repeatable strategy you can apply to any problem, rather than staring at the page hoping inspiration strikes.

The Step-by-Step Strategy

Every word problem, no matter the type, can be solved using these steps:

1. Read the problem carefully. Read it at least twice. On the first pass, get the big picture. On the second pass, identify the specific numbers, relationships, and what the question is actually asking you to find.

2. Define your variables. Assign a letter (like $x$ or $y$) to each unknown quantity. Write down what each variable represents in plain English. For example: "Let $x$ = the number of adult tickets sold."

3. Identify relationships and translate to equations. Look for keywords that signal math operations:

   • "sum," "more than," "increased by" → addition
   • "difference," "less than," "decreased by" → subtraction
   • "product," "times," "of" → multiplication
   • "quotient," "per," "divided by" → division
   • "is," "was," "will be" → equals sign

4. Solve the equation. Use algebraic techniques: combine like terms, apply properties of equality, and isolate the variable using inverse operations.

5. Check your answer. Ask yourself:

   • Does it answer the actual question that was asked?
   • Does it make sense in context? (Ages shouldn't be negative. Prices shouldn't be millions of dollars for a sandwich.)
   • Does it check out when you plug it back into the original equation?
   • Is it in the right ballpark? A quick estimate can catch big errors.

Common Word Problem Types

Age Problems

These give you information about people's ages at different points in time.

• Let $x$ = Sarah's current age, so John's current age = $x + 4$
• In 5 years: Sarah is $x + 5$, John is $(x + 4) + 5 = x + 9$
• Set up the equation from "twice Sarah's age": $x + 9 = 2(x + 5)$
• Solve: $x + 9 = 2x + 10$, so $x = -1$... which doesn't make sense for an age, meaning you'd need to re-read the problem. (This is exactly why Step 5 matters.)

The trick with age problems is keeping track of when: past, present, and future ages each get their own expression.

Mixture Problems

These involve combining two or more things with different concentrations, prices, or percentages.

• Assign variables to the unknown quantities of each component (e.g., $x$ = liters of 80% juice solution).
• Set up one equation for the total amount and another for the concentration or value.
• Solve the system of equations.

Distance, Rate, and Time Problems

These all revolve around one formula:

$distance = rate \times time$

• Assign variables to whichever quantity is unknown ($d$, $r$, or $t$).
• If two objects are moving, set up a separate equation for each one.
• Use the relationship the problem describes (they meet, they're a certain distance apart, one catches up to the other) to connect the equations.

Work Problems

These ask how long it takes people or machines to complete a task together.

• If Machine A finishes a job in 4 hours, its work rate is $\frac{1}{4}$ of the job per hour.
• If Machine B finishes in 6 hours, its rate is $\frac{1}{6}$ per hour.
• Working together: $\frac{1}{4} + \frac{1}{6} = \frac{1}{t}$, where $t$ is the time to finish together.

The key insight is that you add rates, not times. Two machines that each take 4 hours don't take 8 hours together.

When You Get Stuck

• Draw a picture or diagram. Sketching the situation often reveals relationships you missed while reading.
• Re-read the problem and make sure your equation actually matches what's described. A common mistake is setting up an equation that answers a slightly different question.
• Try plugging in a simple number for the unknown to see if your equation behaves the way the problem describes. This can help you catch setup errors before you spend time solving.