Problem 8 - Elementary Algebra

Question

A student works 12 hours a week at a job that pays \(\$ 7.25\) per hour. Since she needs to make \(\$ 240\) per week to pay her bills, she is looking for another job. If she can find a job that pays \(\$ 10\) per hour, how many additional hours will she need to work? Round up to the nearest whole number.

Step-by-Step Solution

Answer

16 hours

Step 1: Calculate weekly earnings from current job

Multiply the number of hours worked per week by the hourly wage. Let the current job's weekly earnings be: \[ 12 \text{ hours/week} \times 7.25 \text{ dollars/hour} = 87 \text{ dollars/week} \]

Step 2: Determine the additional money needed

Subtract the earnings from the current job from the total amount needed per week. The additional amount needed: \[ 240 \text{ dollars/week} - 87 \text{ dollars/week} = 153 \text{ dollars/week} \]

Step 3: Calculate additional hours needed at new job

Divide the additional amount needed by the hourly wage of the new job. Then, round up to the nearest whole number. Let the additional hours needed be: \[ \frac{153 \text{ dollars}}{10 \text{ dollars/hour}} = 15.3 \text{ hours} \] Rounding up, the additional hours needed are: 16 hours

Key Concepts

hourly wage

An hourly wage is the amount of money paid for each hour of work. In our exercise, the student has two different hourly wages. The first job pays \(7.25 \text{ dollars/hour}\), and she is considering a new job that pays \(10 \text{ dollars/hour}\). Understanding your hourly wage helps you calculate your income more accurately. To find out how much you would earn in a week, you can multiply your hourly wage by the number of hours you work each week.

weekly earnings

Weekly earnings are the total amount of money you make in one week. For our student, this involves calculating her current weekly earnings and figuring out how much more she needs. To calculate, multiply the number of hours she works by the hourly wage at her current job: \[ 12 \text{ hours/week} \times 7.25 \text{ dollars/hour} = 87 \text{ dollars/week} \]. This tells us she makes \(87 \text{ dollars/week}\). To find out the additional earnings needed, we subtract the current weekly earnings from the total amount needed: \[ 240 \text{ dollars/week} - 87 \text{ dollars/week} = 153 \text{ dollars/week} \]. This is the amount she needs to make up.

rounding numbers

Rounding numbers means adjusting them to the nearest whole number for simplicity. In this exercise, we're figuring out how many additional hours the student needs to work. After performing our calculations, we find the additional hours required are \(15.3\). Since you can’t work a fraction of an hour in most jobs, we round up to ensure she makes enough money. So, \(15.3\) hours becomes \(16\) hours. Always remember to round up when any fraction is present in such scenarios.

algebraic problem solving

Algebraic problem solving involves setting up equations and solving for unknowns. Here, we need to solve how many additional hours our student needs to work. The equation used is: \[ \frac{153 \text{ dollars}}{10 \text{ dollars/hour}} = 15.3 \text{ hours} \]. We derived \(153\) dollars from subtracting the current earnings from what she needs. Dividing by \(10\), the hourly rate of the new job, gives us the hours needed. This equation demonstrates a simple use of algebra to solve real-world problems.