The owner of a hair salon charges $20 more per haircut than the assistant. Yesterday the assistant gave 12 haircuts. The owner gave 6 haircuts. The total earnings from haircuts wer... The owner of a hair salon charges $20 more per haircut than the assistant. Yesterday the assistant gave 12 haircuts. The owner gave 6 haircuts. The total earnings from haircuts were $750. How much does the owner charge for a haircut? Solve by writing and solving a system of equations.
Understand the Problem
The problem describes a scenario at a hair salon where the owner charges more per haircut than the assistant. Given information about the number of haircuts each person gave and the total earnings, the goal is to determine the price the owner charges for a haircut by setting up and solving a system of equations.
Answer
The owner charges $55 for a haircut.
Answer for screen readers
The owner charges $55 for a haircut.
Steps to Solve
- Define the variables
Let $x$ be the price the assistant charges for a haircut. Since the owner charges $20 more than the assistant, the owner charges $x + 20$ for a haircut.
- Set up the equations
The assistant gave 12 haircuts, so the assistant earned $12x$. The owner gave 6 haircuts, so the owner earned $6(x + 20)$. The total earnings were $750, so $12x + 6(x + 20) = 750$.
- Solve for $x$
Expand the equation: $12x + 6x + 120 = 750$ Combine like terms: $18x + 120 = 750$ Subtract 120 from both sides: $18x = 630$ Divide both sides by 18: $x = \frac{630}{18} = 35$
- Find the price the owner charges
The owner charges $x + 20$. Since $x = 35$, the owner charges $35 + 20 = 55$.
The owner charges $55 for a haircut.
More Information
The assistant charges $35 for a haircut, and the owner charges $55 for a haircut. To verify these solutions, we can calculate the total earnings as follows: Assistant's earnings: $12 \times 35 = 420$ Owner's earnings: $6 \times 55 = 330$ Total earnings: $420 + 330 = 750$, which matches the problem statement.
Tips
A common mistake could be to incorrectly set up the equation, for instance not distributing the 6 properly across $(x+20)$, resulting in $12x + 6x + 20 = 750$. Another common mistake is to solve for x (the assistant's price) but forget to calculate the owner's price by adding $20.
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