Solving Multi-Step Equations with Fractions from Word Problems

🔑 Key Concepts

  • Define a variable to represent the unknown amount in the situation.
  • Model a fraction of a total by multiplying the total expression by that fraction.
  • Use the reciprocal to undo multiplication by a fractional coefficient.
  • Perform the same operation on both sides to keep the equation balanced.
  • Interpret the solution using the units and context of the original problem.
  • Check the solution by substituting it into the equation.

✏️ Worked Example — Modeling Merchandise Profit with a Fraction

🧠 Math Vocabulary

  • Variable: A symbol used to represent an unknown quantity.
  • Fractional coefficient: A fraction that multiplies a variable or expression.
  • Reciprocal: A number formed by switching the numerator and denominator of a fraction.
  • Total profit: The complete amount earned after combining all sources of profit.
  • Inverse operations: Operations that undo one another and help isolate a variable.
  • Substitution: Replacing a variable with its value to check whether an equation is true.
  • Interpret: To explain what a mathematical result means in the original situation.

💡 Main Idea

In this lesson, students model a real-world financial situation with a multi-step equation involving a fraction. The expression \(1500+h\) represents the total profit from hat and hoodie sales, and multiplying that total by \(\displaystyle \frac{2}{5}\) represents the singer’s share. Students multiply by the reciprocal to remove the fractional coefficient, isolate the variable, and interpret the solution in context.

📚 What You Should Already Know

Before this lesson, students should understand how fractions represent parts of a whole, how to multiply fractions, and how reciprocals undo multiplication by a fraction. Students should also know how to write an expression for a total and solve basic multi-step equations using inverse operations.

🚀 What Comes Next

Next, students can write and solve equations involving percentages, discounts, commissions, markups, and proportional relationships. These same modeling strategies also support work with inequalities, formulas, and more complex financial situations.

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