Lesson 3.1: Use a Problem-Solving Strategy

New Concept

Master a powerful 7-step strategy to confidently translate any word problem into a solvable equation. You'll learn to systematically break down problems, from simple number puzzles to real-world scenarios, and find the correct solution.

What’s next

You'll now walk through interactive examples using this 7-step method, followed by practice cards to test your new problem-solving skills.

Property

Use a Problem-Solving Strategy to Solve Word Problems.

1. Read the problem. Make sure all the words and ideas are understood.
2. Identify what we are looking for.
3. Name what we are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

Examples

• A shirt is on sale for 25 dollars, which is one-third of its original price. What was the original price? Let $p$ be the original price. The equation is $25 = \frac{1}{3}p$. Solving for $p$ gives $p=75$. The original price was 75 dollars.

• A book club has 17 women. The number of women is two more than three times the number of men. How many men are in the club? Let $m$ be the number of men. The equation is $17 = 3m + 2$. Solving for $m$ gives $m=5$. There are 5 men in the club.

Property

To solve number problems, translate key words from the sentence into mathematical operations. Restate the problem in a single sentence to clarify the relationship between the numbers. Common key words include:

• Sum: Addition (+)
• Difference: Subtraction (-)
• Product/Times: Multiplication (⋅)
• Quotient: Division (÷)
• Twice a number: $2n$

Examples

• The sum of twice a number and nine is 21. Find the number. Let $n$ be the number. The equation is $2n + 9 = 21$. Solving gives $2n = 12$, so $n=6$. The number is 6.

• The difference of a number and three is 15. Find the number. Let $x$ be the number. The equation is $x - 3 = 15$. Solving gives $x=18$. The number is 18.

Property

When a problem involves two unknown quantities, name the first unknown with a variable (e.g., $n$).
Then, read the problem to find the relationship between the two unknowns and write an expression for the second unknown using the same variable (e.g., $n+5$ or $2n-10$).
Finally, form an equation that relates both unknowns and solve.

Examples

• One number is ten more than another. Their sum is 52. Find the numbers. Let the first number be $n$. The second is $n+10$. The equation is $n + (n+10) = 52$. This simplifies to $2n=42$, so $n=21$. The numbers are 21 and 31.

• The sum of two numbers is negative twenty. One number is six less than the other. Find the numbers. Let the first number be $x$. The second is $x-6$. The equation is $x + (x-6) = -20$. This simplifies to $2x = -14$, so $x=-7$. The numbers are -7 and -13.

Property

To solve problems involving consecutive integers, use a variable to represent the first integer and then write expressions for the following integers based on the pattern.

• Consecutive Integers: $n, n+1, n+2, \dots$
• Consecutive Even Integers: $n, n+2, n+4, \dots$ (where $n$ is an even integer)
• Consecutive Odd Integers: $n, n+2, n+4, \dots$ (where $n$ is an odd integer)

Examples

• The sum of two consecutive integers is 51. Find the integers. Let the integers be $n$ and $n+1$. The equation is $n+(n+1)=51$. This gives $2n=50$, so $n=25$. The integers are 25 and 26.

• Find three consecutive even integers whose sum is 96. Let them be $n, n+2, n+4$. The equation is $n+(n+2)+(n+4)=96$. This gives $3n=90$, so $n=30$. The integers are 30, 32, and 34.