The Relationship Between Assembly Time and Manpower

When considering assembly processes in manufacturing or construction, the time required to assemble a certain number of items is influenced by both the number of items to assemble and the number of workers available. This relationship can be described using the concepts of direct and inverse proportions. This article explores how to solve a practical problem involving these concepts, providing insights into how to calculate the required number of workers based on assembly time and the number of items to be assembled.

Understanding Direct and Inverse Proportions

In the context of assembly processes, there are two key relationships to consider:

Direct Proportion: If the number of items to assemble increases, the time taken to assemble them also increases, provided the number of workers remains constant. Inverse Proportion: If the number of workers increases, the time required to assemble the same number of items decreases, provided the number of items remains constant.

The overall relationship can be described by the formula: [ h k frac{x}{m} ] where ( h ) is the number of hours, ( x ) is the number of machines (or items) to assemble, ( m ) is the number of men (or workers), and ( k ) is a constant of proportionality.

Solving the Problem

The problem presents a scenario where 8 men can assemble 24 machines in 8 hours. We are asked to determine the number of men needed to assemble 72 machines in 16 hours.

Step 1: Determine the Constant ( k )

Using the given scenario, we can find the constant ( k ).

[begin{aligned} 8 k frac{24}{8} 8 k cdot 3 k frac{8}{3} end{aligned}]

Step 2: Set Up the Equation for the New Scenario

We now need to find the number of men ( m ) required to assemble 72 machines in 16 hours. Substituting ( h 16 ), ( x 72 ), and ( k frac{8}{3} ) into the equation:

[ 16 frac{frac{8}{3}}{m} cdot 72 ]

Step 3: Solve for ( m )

To solve for ( m ), we multiply both sides by ( m ) to eliminate the fraction:

[ 16m frac{8}{3} cdot 72 ]

Calculating the right side:

[frac{8 cdot 72}{3} 192]

This simplifies to:

[ 16m 192 ]

Dividing both sides by 16:

[ m frac{192}{16} 12 ]

Conclusion

Therefore, to assemble 72 machines in 16 hours, 12 men are needed.

Alternative Methods

There are alternative methods to solve this problem, which yield the same result. Let's explore three different approaches:

Direct and Inverse Proportions: First, calculate the man-hours required for the initial scenario: 8 times 8 64 man-hours for 24 machines. Calculate the total man-hours required for the new scenario: 72 times frac{64}{24} 192 man-hours. To complete this in 16 hours, frac{192}{16} 12 men are needed. Simple Comparison: 1 man can assemble (frac{24}{8} 3) machines in 8 hours. In 16 hours, 1 man can assemble 6 machines. To assemble 72 machines in 16 hours, we need (frac{72}{6} 12) men. Man-Hours Calculation: 8 men can assemble 24 machines in 64 man-hours. For 72 machines, the required man-hours are frac{72}{24} times 64 192 man-hours. To complete this in 16 hours, (frac{192}{16} 12) men are needed.

Both the original approach and the alternative methods confirm that 12 men are required to assemble 72 machines in 16 hours.