Problem 26 - Algebra for College Students

Question

Suppose the number of days it takes to complete a construction job varies inversely as the number of people assigned to the job. If it takes 7 people 8 days to do the job, how long will it take 10 people to complete the job? \(5 \frac{3}{5}\) days

Step-by-Step Solution

Answer

It will take 10 people \( 5 \frac{3}{5} \) days to complete the job.

Step 1: Understand Inverse Variation

Inverse variation means that as one quantity increases, the other quantity decreases in such a way that the product of the two quantities remains constant. Mathematically, if two quantities, \( x \) and \( y \), vary inversely, then \( x \times y = k \), where \( k \) is a constant.

Step 2: Define Variables and Equation

Let's define the variables: \( p \) is the number of people, and \( d \) is the number of days. According to the problem, \( p \times d = k \). Initially, 7 people take 8 days, so \( 7 \times 8 = k \). We need to find the number of days, \( d \), when 10 people are assigned.

Step 3: Calculate the Constant of Variation

Multiply the initial number of people by the number of days: \( k = 7 \times 8 = 56 \). This is the constant of variation for the construction job.

Step 4: Solve the Equation for 10 People

Using the constant \( k = 56 \), set up the equation for 10 people: \( 10 \times d = 56 \). Solve for \( d \) by dividing both sides by 10: \( d = \frac{56}{10} = 5.6 \).

Step 5: Convert Decimal to Mixed Number

Convert the decimal \( 5.6 \) days into a mixed number: \( 5.6 = 5 \frac{6}{10} = 5 \frac{3}{5} \).

Key Concepts

Constant of Variation

When we talk about inverse variation, the constant of variation plays a crucial role. Imagine it as the glue that holds the two varying quantities together. With inverse variation, as one increases, the other decreases, and they balance each other out by multiplying to give a constant value. If we have quantities like the number of people working and the days required to complete a job, these two variations work hand in hand by maintaining a steady balance.
In mathematical terms, if we represent the number of people as \( p \) and the number of days as \( d \), they have a relation: \( p \times d = k \). Here, \( k \) is a fixed number known as the constant of variation. Let’s delve deeper: when 7 people can finish a task in 8 days, \( k = 7 \times 8 = 56 \). This magic number, 56, lets us easily determine how different configurations of people and days will fit together, following the inverse relation.
This concept simplifies our task by providing the needed uniformity and calculation ease in similar problems. The constant of variation acts like a flexible glue, adapting yet staying resilient through different scenarios.

Mixed Numbers

Mixed numbers are a blend between whole numbers and fractions, often used to express more precise values. When you see a number like \( 5 \frac{3}{5} \), you're looking at a mixed number, which essentially means 5 whole units plus a fraction of a unit. It's a practical way to communicate numbers that aren't quite whole and add clarity to calculations.
In the context of our problem, suppose we computed the number of days required and arrive at a decimal like 5.6. It might be clearer to present this as a mixed number: \( 5 \frac{6}{10} \). Simplifying \( \frac{6}{10} \), you get \( \frac{3}{5} \). Thus, 5.6 can be better expressed as \( 5 \frac{3}{5} \), offering a tidier and straightforward representation.

• Whole numbers are easily understandable, like counting five apples.
• Fractions indicate you have more than the whole number but less than the next one.
• Mixed numbers provide a complete picture, clarifying the result especially in practical tasks like measurement.

Breaking down numbers into mixed numbers provides clarity, especially in real-world approximations, and helps in applying the most accurate values needed for precise outcomes.

Problem-Solving in Algebra

Algebra might seem daunting, but it's simply a structured way to solve problems using symbols and rules. The beauty of algebra comes in how it turns puzzling questions into simple, step-by-step solutions. Let’s take a walk through this process using the inverse variation example.
Firstly, identify what you know and what you need to find out. In our construction task, we knew the number of people (7) and the number of days (8) they needed. From there, establish the relationship: inverse variation tells us that \( p \times d = k \).
Secondly, calculate the constant of variation. Doing so involves simple multiplication of the given values. For our problem, we found \( k = 56 \). Once you have \( k \), you can find unknown variables by plugging other known values into the same relationship. This general framework enables solutions without getting lost in numbers.

• Define your variables clearly.
• Identify relationships and patterns, like inverse variation.
• Follow through with logical steps to arrive at a neat conclusion.

Finally, always cross-check your answer to ensure consistency and validity. Problem-solving isn't just about finding any answer but understanding the path to a correct one. As seen, algebra provides powerful tools for reaching these insights efficiently and effectively.