Problem 27 - Algebra for College Students

Question

The number of days needed to assemble some machines varies directly as the number of machines and inversely as the number of people working. If it takes 4 people 32 days to assemble 16 machines, how many days will it take 8 people to assemble 24 machines? 24 days

Step-by-Step Solution

Answer

It will take 24 days.

Step 1: Understand the relationship

The problem states that the number of days needed to assemble machines varies directly as the number of machines and inversely as the number of people working. This can be expressed using the equation: \( d \propto \frac{m}{p} \), where \( d \) is the number of days, \( m \) is the number of machines, and \( p \) is the number of people. This can be rewritten using a constant k as \( d = k \cdot \frac{m}{p} \).

Step 2: Find the constant of variation

Using the given information, substitute \( d = 32 \), \( m = 16 \), and \( p = 4 \) into the equation \( 32 = k \cdot \frac{16}{4} \). Simplifying, \( 32 = k \cdot 4 \), leading to \( k = 8 \).

Step 3: Set up the equation with the new scenario

With the constant \( k = 8 \) found, use the formula for the new scenario: \( d = 8 \cdot \frac{24}{8} \).

Step 4: Calculate the number of days

Substitute the values into the equation: \( d = 8 \cdot \frac{24}{8} = 8 \cdot 3 = 24 \). So, it will take 24 days for 8 people to assemble 24 machines.

Key Concepts

Proportional Relationships

Proportional relationships are a fundamental concept in mathematics, especially in algebra and real-world problem-solving. When we say that quantities are directly or inversely proportional, it describes how one quantity changes in relation to another.

In the context of this exercise, the number of days (\(d\)) required to assemble machines varies directly as the number of machines (\(m\)). This means as you increase the number of machines, the days needed generally increase if other factors remain constant. At the same time, this relationship is inversely proportional to the number of people (\(p\)) working. This suggests that more people working on the task will decrease the number of days required to finish the job.

• **Direct Proportionality:** Quantities increase or decrease together. If one doubles, the other also doubles.
• **Inverse Proportionality:** One quantity increases while the other decreases. If one doubles, the other halves.

Variation Constant

The variation constant (\(k\)) is a specific value that defines the exact nature of how the quantities are proportional. In problems involving direct and inverse variation, this constant helps formulate the relationship between the quantities.

From the exercise, we see that the equation can be written as \(d = k \cdot \frac{m}{p}\). Here, \(k\) represents the variation constant that ties together the number of days, machines, and people. It remains constant for all scenarios involving these specific relationships.

• **Why a Constant?:** The constant reflects consistent conditions across scenarios, allowing calculations to predict outcomes when one variable changes.
• **Finding the Constant:** Substitute known values and solve for \(k\). In this case, using \(d = 32\), \(m = 16\), and \(p = 4\), we derive \(k = 8\).

Problem Solving in Algebra

Algebra is an incredibly powerful language in mathematics that allows us to solve problems involving variables and unknowns efficiently. This exercise showcases how direct and inverse variations can be tackled using algebraic methods.

By setting up equations based on proportional relationships, we can solve for unknowns like the number of days needed in this scenario. The process involves:

• **Translating the Problem:** Write down the direct and inverse relationships as equations.
• **Identifying Variables:** Recognize all the given information and what needs to be found. Here \(d\), \(m\), and \(p\) are the variables.
• **Solving the Equation:** Use algebraic manipulation to find the unknowns, such as the variation constant and further use it predictively for new scenarios.

Through these steps, algebra helps systematically organize and solve complex problems.