Megan's regular wage at her job is p dollars per hour for the first 8 hours of work in a...
GMAT Algebra : (Alg) Questions
Megan's regular wage at her job is \(\mathrm{p}\) dollars per hour for the first \(8\) hours of work in a day plus \(1.5\) times her regular hourly wage for work in excess of \(8\) hours that day. On a given day, Megan worked for \(10\) hours, and her total earnings for that day were \($137.50\). What is Megan's regular hourly wage?
1. TRANSLATE the problem information
- Given information:
- Regular pay: \(\mathrm{p}\) dollars/hour for first 8 hours
- Overtime pay: \(\mathrm{1.5p}\) dollars/hour for hours beyond 8
- Total hours worked: 10 hours
- Total earnings: $137.50
- What this tells us: She worked 8 regular hours + 2 overtime hours
2. INFER the pay structure breakdown
- Since Megan worked 10 hours total:
- First 8 hours at regular rate \(\mathrm{p}\)
- Remaining 2 hours at overtime rate \(\mathrm{1.5p}\)
- Total pay = (regular hours × regular rate) + (overtime hours × overtime rate)
3. TRANSLATE into mathematical equation
Set up the equation: \(\mathrm{8p + 2(1.5p) = 137.50}\)
4. SIMPLIFY the equation algebraically
- Distribute: \(\mathrm{8p + 2(1.5p) = 8p + 3p = 11p}\)
- Equation becomes: \(\mathrm{11p = 137.50}\)
- Divide both sides by 11: \(\mathrm{p = 137.50 \div 11 = 12.50}\)
Answer: B. $12.50
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER reasoning: Misunderstanding the overtime pay structure by thinking ALL 10 hours are paid at the overtime rate of \(\mathrm{1.5p}\), rather than recognizing the split between regular and overtime hours.
Students might set up: \(\mathrm{10(1.5p) = 137.50}\), leading to \(\mathrm{p = 137.50 \div 15 = 9.17}\), which doesn't match any answer choice. This leads to confusion and guessing.
Second Most Common Error:
Poor TRANSLATE execution: Calculating the average hourly wage instead of the regular hourly wage by simply dividing total earnings by total hours worked.
Students calculate: \(\mathrm{137.50 \div 10 = 13.75}\) and incorrectly think this represents the regular wage. This may lead them to select Choice D ($13.75).
The Bottom Line:
The key challenge is recognizing that overtime problems involve two different pay rates for different portions of the work hours, not a single rate applied to all hours. Students must carefully parse the pay structure before setting up their equation.