Question

Suppose you have a job in which you earn \(\$ 7.35\) for each 30 minutes that you work. (a) Express this information in the form of an equivalence between dollars earned and minutes worked. (b) Use the equivalence defined in (a) to calculate the number of dollars earned in \(1 \mathrm{hr} 45 \mathrm{~min}\) (c) Use the equivalence defined in (a) to calculate the number of minutes you would have to work to earn \(\$ 333.50\).

Short answer

\)7.35/30 \text{ minutes} = \(0.245/\text{minute}, \)25.73 earned in 1 hour 45 minutes, 1360 minutes to earn $333.50.

Step-by-step solution

Create an equivalence between dollars earned and minutes worked

Given the job pays \(7.35 for each 30 minutes worked, the equivalence can be stated as \)7.35 per 30 minutes. This can be expressed as a ratio: \(7.35/30 \text{ minutes} = \)0.245/\text{minute}.

Convert 1 hour 45 minutes into minutes

1 hour is 60 minutes, and 45 minutes added to this equals 105 minutes. We will use this value to calculate dollars earned in 1 hour 45 minutes.

Calculate the dollars earned in 1 hour 45 minutes

Using the ratio from Step 1, multiply the earnings per minute by the total minutes worked: 105 minutes * \(0.245/minute = \)25.73.

Find the number of minutes to earn \(333.50

Using the ratio from Step 1, divide the total dollars by the dollars per minute to find the minutes worked: \)333.50 / (\(7.35/30 \text{ minutes}) = 1360 minutes.

Key concepts

Equivalence Ratios

Understanding equivalence ratios is essential in various fields, including chemistry and finance. These ratios create a foundational relationship between two quantities, allowing for conversion and comparison. In the given exercise, we have an example of this with earnings and time. The initial step to solving this kind of problem is to establish the equivalence ratio between the amounts of dollars earned for each time unit worked. This is expressed as \( \(7.35 per 30 \text{ minutes} \).

It is vital to express each quantity in the same terms when setting up equivalence ratios. In this case, dollars per minute would be a consistent measure. To convert the given ratio into dollars per minute, we divide the amount earned by the time worked in minutes, resulting in \( \)0.245 per \text{minute} \). This useful ratio is a tool for further calculations, enabling us to convert between time worked and dollars earned with ease.

Unit Conversion

A fundamental skill in various scientific and practical applications is the ability to perform unit conversions. In our exercise context, the conversion needed is from hours and minutes to minutes only. This process involves understanding the equivalent values of different time units and then performing the appropriate multiplication or division.

For example, to convert time from hours to minutes, we multiply by 60, as there are 60 minutes in an hour. This simple multiplication increases accuracy and simplifies calculations when applying the established equivalence ratio. The exercise demonstrated this by converting 1 hour 45 minutes into a single unit (minutes) before calculating the total earnings. Remember, consistency in units across all elements of a calculation is crucial to obtaining the correct solution.

Chemical Calculation Methods

While the provided exercise is not a chemistry problem, the methods of calculation are analogous. In chemistry, chemical calculation methods involve stoichiometry, which uses equivalence ratios known as molar ratios from balanced chemical equations. These ratios are equivalent to the exercise's dollars to minutes ratio and are used to quantify reactants and products in chemical reactions.

For a chemistry student, learning to accurately use molar ratios is comparable to learning to convert work hours into earnings. Just as we used the established ratio and unit conversions in our exercise problem, a chemist would use molar ratios and molecular weights (unit conversion) to calculate the mass of chemicals needed or produced. Thus, understanding and mastering these foundational concepts in any context, such as the financial scenario given, builds a skillset that is transferable to scientific disciplines and everyday problem-solving.