Math Problem Statement

Determine the prices of a winter hat and a cap based on the given equations involving their costs.

Solution

To solve the problem:

  1. Define the variables:

    • Let the price of the winter hat = HH
    • Let the price of the cap = CC
  2. From the first equation:
    H+H+C=18H + H + C = 18
    Simplify to:
    2H + C = 18 \tag{1}

  3. From the second equation:
    C = H + H \tag{2}
    Simplify to:
    C=2HC = 2H

  4. Substitute C=2HC = 2H from equation (2) into equation (1):
    2H+2H=182H + 2H = 18
    Combine terms:
    4H=184H = 18
    Solve for HH:
    H=4.5H = 4.5

  5. Use H=4.5H = 4.5 in equation (2) to find CC:
    C=2H=2(4.5)=9C = 2H = 2(4.5) = 9

Final Solution:

  • The winter hat costs $4.50.
  • The cap costs $9.00.

Would you like a breakdown of the steps or further explanation?

Here are 5 related questions to practice:

  1. If the price of the hat were $5, what would the cap's price be?
  2. How would the solution change if the total was $20 instead of $18?
  3. What happens if the cap costs $10? Can you solve for the hat's price?
  4. Could you rewrite the equations for three hats and one cap summing to $27?
  5. Can you use a graphical method to solve the problem instead?

Tip: Always double-check your substitution in equations to ensure accuracy!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations

Formulas

Substitution method in solving systems of linear equations

Theorems

Properties of equality

Suitable Grade Level

Grades 6-8