Math Problem Statement

The cost of four scarves and six hats is $52. The cost of two hats is $1 more than the cost of one scarf. What are the costs of one scarf and one hat?

Solution

The problem states the following:

  • The cost of four scarves and six hats is $52.
  • The cost of two hats is $1 more than the cost of one scarf.

Let ss be the cost of one scarf, and hh be the cost of one hat.

We can set up the following system of equations based on the given information:

  1. 4s+6h=524s + 6h = 52 (Equation for the total cost of 4 scarves and 6 hats)
  2. 2h=s+12h = s + 1 (Equation stating that the cost of two hats is $1 more than one scarf)

Solving the system:

Step 1: Solve the second equation for ss:

From 2h=s+12h = s + 1, solve for ss: s=2h1s = 2h - 1

Step 2: Substitute this expression into the first equation:

Substitute s=2h1s = 2h - 1 into 4s+6h=524s + 6h = 52: 4(2h1)+6h=524(2h - 1) + 6h = 52 Simplify: 8h4+6h=528h - 4 + 6h = 52 14h4=5214h - 4 = 52 14h=5614h = 56 h=4h = 4

Step 3: Find ss:

Substitute h=4h = 4 into s=2h1s = 2h - 1: s=2(4)1=81=7s = 2(4) - 1 = 8 - 1 = 7

Conclusion:

The cost of one scarf is $7, and the cost of one hat is $4.

Thus, the correct answer is D: $7; $4.


Would you like more details or have any questions? Here are 5 follow-up questions to expand your understanding:

  1. How can you check the solution for both equations in the system?
  2. Can you solve this problem using a different approach, such as graphing?
  3. How would the problem change if the cost of two hats were $2 more than one scarf?
  4. Can this system be solved using substitution or elimination differently?
  5. How does solving systems of equations apply to real-world problems like budgeting?

Tip: When solving word problems, always define your variables clearly to avoid confusion when setting up equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

4s + 6h = 52
2h = s + 1

Theorems

Substitution method for solving systems of equations

Suitable Grade Level

Grades 6-8