Problem 60 - Elementary Algebra Within Reach

Question

Assembly Line Production You have a job on an assembly line for which you earn \(\$ 10\) per hour plus \(\$ 0.75\) for each unit you produce. Your earnings for an 8 -hour day are \(\$ 146\). Find the number of units you produced.

Step-by-Step Solution

Answer

The worker produced 88 units that day.

Step 1: Determine basic earnings

Begin by calculating the earnings from working hours alone. It's known that the pay is \(\$ 10\) per hour and the total work duration is 8 hours. Hence, basic earnings = hours worked * hourly rate, i.e. basic earnings = 8 * \(\$ 10\) = \(\$ 80\).

Step 2: Calculate earnings from units produced

The total day’s earnings ( \(\$ 146\)) include the earnings from the units produced as well. To discover how much money was made from producing units, subtract the basic earnings from the total earnings: earnings from units = total earnings - basic earnings, i.e. earnings from units = \(\$ 146\) - \(\$ 80\) = \(\$ 66\).

Step 3: Find the number of units produced

Since it is known that for each unit produced, \(\$ 0.75\) are earned, to get the quantity of units produced, divide the earnings from units by the earning per unit: units produced = earnings from units / earning per unit, i.e. units produced = \(\$ 66\) / \(\$ 0.75\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions representing a straight line when plotted on a graph. In the context of the assembly line production, a linear equation can express total earnings as the sum of fixed hourly earnings and the variable earnings dependent on the number of units produced.
This equation is structured as:

• Y = mX + C

where 'Y' is the total earnings, 'm' is the unit rate (amount earned per unit), 'X' is the number of units produced, and 'C' is the basic earnings.
For this problem, the equation would be:

• Y = 0.75X + 80

Here, 0.75 represents the additional earnings per unit produced, and 80 is the total fixed earnings from hourly work. Understanding this equation helps in breaking down and organizing the calculation of total earnings.

Hourly Earnings

Hourly earnings are your earnings just from the hours worked, not considering any extra pay per unit produced. In our assembly line problem, this is represented by the fixed part of the linear equation. You get \( \\( 10 \) per hour, and you work for 8 hours.
Let's make this clearer:

• Hourly rate: \( \\) 10 \) per hour
• Hours worked per day: 8 hours

That results in \( 10 \times 8 = \$ 80 \), which forms the base of your day's earnings from just the time put in. This calculation is simple and vital as it forms the strong base from which all further calculations will follow.

Unit Rate Calculation

The unit rate calculation tells us how much you earn for each individual unit produced beyond your fixed hourly wage. In the assembly line problem, this is plainly stated: \( \\( 0.75 \) per unit.
Calculating unit rate is crucial because it determines how your earnings increase with every additional unit you make.

• Each unit produced: adds an additional \( \\) 0.75 \) to earnings

Unit rate involves considering how many units one might typically produce and using this number along with the constant unit rate to predict or verify earnings, showcasing how performance on the line can affect total income.

Basic Earnings Calculation

Basic earnings calculation is an essential step to figure out total earnings. It is the process wherein you deduce the portion of earnings that are due to just the hours worked, without counting the unit-based earnings. This starts by multiplying the hourly pay rate by the total hours worked, i.e., \( 10 \times 8 = \\( 80 \).
Once basic earnings are established, you can subtract these from total day's earnings \( \\) 146 \) to determine additional income from units. This subtraction leads to \( \$ 66 \) coming from units produced.
This separates your fixed earnings from variable earnings, allowing clear identification of income from work effort (units) versus time spent (hours). Understanding basic earnings helps in concluding precisely how unit production enhances income, beyond the hourly rate.