Question

Find the unit rate. \(\$ 420\) for 40 hours.

Short answer

The unit rate is \$10.5 per hour.

Step-by-step solution

Identify the Values

The first step is to identify the values in the problem. In this case, the payment amount is \$420 and the total number of hours worked is 40.

Perform Division

The next step is to divide the total pay (\$420) by the number of hours (40). This operation will provide the unit rate, or the amount earned per hour.

Calculate the Result

Perform the division: \$420 divided by 40 is \$10.5. So, the unit rate is \$10.5 per hour. This means for each hour worked, \$10.5 is earned.

Key concepts

Algebraic Problem-Solving

Grasping algebraic problem-solving is a key skill in understanding many mathematical and real-world problems. Let's take the example of calculating a unit rate from the problem, \(\$420\) for 40 hours of work. The problem-solving process involves identifying the values and the relationships between them.

First, you recognize the total pay as the numerator and the total hours worked as the denominator. Recognizing that calculating the unit rate is a division problem is the next crucial step. By expressing this relationship algebraically, you can create an equation \(\$420 \div 40 = x\), where \(x\) represents the unit rate, or pay per hour.

Once the values are plugged in, you simply perform the division to solve for \(x\), yielding the unit rate. This process is fundamental in algebra and builds the foundation for more complex problem-solving scenarios.

Rational Numbers Division

Understanding how to divide rational numbers, which are numbers that can be expressed as the quotient of two integers, is essential when dealing with rates. In our problem, the division \(\$420 \div 40\) needs to be carried out. Here, both \(\$420\) and \(40\) are rational numbers, with \(\$420\) being the numerator and \(40\) the denominator.

During division, the numerator is divided by the denominator, simplifying the fraction. In division involving rational numbers like our example, if the result is not a whole number, it may be expressed as a decimal, as is the case with a unit rate of \(\$10.5\) per hour. This makes the result more interpretable in the context of the rate of pay calculation.

Rate of Pay Calculation

The rate of pay calculation is a practical application of math that impacts everyday life, especially in the workforce. To calculate the rate of pay, we use the unit rate formula, which is the total amount of money earned divided by the total number of hours worked.

In our scenario, you have earned \(\$420\) over 40 hours, so the rate of pay is found by dividing \(\$420\) by \(40\). The calculation yields a rate of \(\$10.5\) per hour, which is the unit rate. This rate helps employers and employees standardize earnings regardless of time worked and allows for a fair comparison of compensation rates across different jobs or tasks.

Remember that accuracy in this calculation directly affects one's understanding of their compensation and financial planning. Emphasizing the need for precise division and understanding of rational numbers is crucial in providing the correct unit rate calculation.