Question

Emma gets \(\$ 9\) per hour for the first 40 hours worked per week and time and a half for hours over that. If she works 48 hours one week, what fractional part of her paycheck is overtime? A. \(\frac{1}{13}\) B. \(\frac{3}{13}\) C. \(\frac{7}{13}\) D. \(\frac{10}{13}\)

Short answer

The fractional part of Emma's paycheck that is overtime is \( \frac{3}{13} \).

Step-by-step solution

- Calculate Regular Pay

Determine Emma's regular pay. She earns \(\text{\textdollar}9\) per hour for the first 40 hours. Calculate this by multiplying the hourly wage by the number of regular hours: \[ 9 \text{\textdollar} \times 40 = 360 \text{\textdollar} \]

- Calculate Overtime Pay Rate

Time and a half means 1.5 times the regular hourly wage. Calculate Emma's overtime pay rate: \[ 9 \text{\textdollar} \times 1.5 = 13.5 \text{\textdollar} \]

- Calculate Overtime Pay

Emma worked 48 hours, so she worked 8 hours over the regular 40 hours. Calculate the overtime pay by multiplying the overtime hours by the overtime pay rate: \[ 13.5 \text{\textdollar} \times 8 = 108 \text{\textdollar} \]

- Calculate Total Pay

Add the regular pay to the overtime pay to get the total pay check: \[ 360 \text{\textdollar} + 108 \text{\textdollar} = 468 \text{\textdollar} \]

- Calculate Fractional Part of Overtime Pay

Find the fractional part of the paycheck that is overtime by dividing the overtime pay by the total pay: \[ \frac{108 \text{\textdollar}}{468 \text{\textdollar}} = \frac{108}{468} \]

- Simplify the Fraction

Simplify the fraction \( \frac{108}{468} \) to its simplest form: \[ \frac{108 \text{(divide by common factor 36)}}{468 \text{(divide by common factor 36)}} = \frac{3}{13} \]

Key concepts

Overtime Pay Calculations

Overtime pay is additional compensation for hours worked beyond the standard 40-hour workweek. Emma's regular hourly wage is \(9\text{\textdollar}\). To calculate her overtime pay rate, multiply her regular hourly wage by 1.5. For instance, \(9\text{\textdollar}\times1.5 = 13.5\text{\textdollar}\). This new rate compensates Emma for every overtime hour she works. Emma worked 48 hours in a week, which is 8 hours more than the standard 40 hours. Multiply the 8 overtime hours by the overtime pay rate: \(8\times13.5\text{\textdollar} = 108\text{\textdollar}\). Add this \(108\text{\textdollar}\) to her regular weekly earnings to find her total paycheck.

Fraction Simplification

Simplifying fractions makes them easier to work with by reducing them to their smallest form. To find the fractional part of Emma's paycheck that comes from overtime, divide her overtime pay by her total pay: \(\frac{108\text{\textdollar}}{468\text{\textdollar}}\). First, find the greatest common divisor (GCD) of 108 and 468, which is 36. Divide both the numerator and the denominator of the fraction by 36: \(\frac{108}{36} = 3\) and \(\frac{468}{36} = 13\). This gives us the simplified fraction \(\frac{3}{13}\). Always check your result to confirm its accuracy.

Word Problems

Word problems require reading comprehension and critical thinking to translate words into mathematical expressions. Start by identifying important information and defining variables if needed. For Emma's scenario, note her pay rate, hours worked, and the fact that she gets overtime pay. Break the problem into smaller steps, solving for regular and overtime pay separately. Combine your results to answer the question. Practice is essential for mastering word problems; they help develop problem-solving skills that are useful beyond math class. Always re-read the problem to ensure you've addressed all parts of the question.