Problem 56 - Prealgebra

Question

How many hours does a person making \(\$ 6.78\) per hour have to work in order to earn \(\$ 257.64 ?\)

Step-by-Step Solution

Answer

The person needs to work approximately 38 hours.

Step 1: Understanding the Problem

We need to calculate the number of hours a person must work at a rate of $6.78 per hour to earn a total of $257.64.

Step 2: Forming the Equation

We will use the formula for calculating total earnings, which is: \[ \text{Total Earnings} = \text{Hourly Wage} \times \text{Hours Worked} \]Here, the Total Earnings is \(257.64, and the Hourly Wage is \)6.78. We need to find the Hours Worked.

Step 3: Setting Up the Equation

Plug the given values into the formula: \[257.64 = 6.78 \times \text{Hours Worked} \]We need to solve this equation for the number of hours worked.

Step 4: Solving for Hours Worked

Rearrange the equation to solve for Hours Worked:\[ \text{Hours Worked} = \frac{257.64}{6.78} \]Calculate this division to find the number of hours.

Step 5: Calculating the Answer

Perform the division:\[ \text{Hours Worked} = \frac{257.64}{6.78} \approx 38 \]The person needs to work approximately 38 hours to earn $257.64.

Key Concepts

Rate Problems

Rate problems are mathematical situations that deal with the relationship between two quantities measured against each other. In this exercise, we are considering money earned per hour worked. A rate signifies how one quantity changes concerning another.
For example:

• The rate of $6.78 per hour indicates that for every hour of work, the person earns $6.78.
• The challenge is to find how many hours one must work to accumulate a specific total, i.e., $257.64, at this rate.

Understanding rate problems requires grasping this fundamental relationship, which often involves multiplication or division to predict total or individual quantities.

Solving Equations

Solving equations involves finding unknown variables by manipulating an equation to isolate the unknown. This is a core part of algebra.
In our exercise:

• We started with an equation from the formula \[ \text{Total Earnings} = \text{Hourly Wage} \times \text{Hours Worked} \]
• With \(257.64 set as the total earnings, and \)6.78 as the hourly wage, we wanted to solve for 'Hours Worked'.
• Rearranging the equation into \[ 6.78 \times \text{Hours Worked} = 257.64 \] lets us solve for the hours by dividing both sides by 6.78.

This process of rearranging the equation to isolate the variable is the essence of solving algebraic equations.

Division in Mathematics

Division is a fundamental mathematical operation that allocates a number into specified equal parts. It is the opposite of multiplication.
In our context, division was used to solve for the number of hours worked:

• The formula was transformed into \[ \text{Hours Worked} = \frac{257.64}{6.78} \]
• Here, division helps in finding how many times \(6.78 can "fit" into \)257.64, which represents the number of hours needed.

Division provides us with an immediate solution by answering how many times a quantity splits another into equal measures. It's a strategic problem-solving tool when dealing with rates or allocations.