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Chapter 2: Problem 5

APPLICATION The equation \(c=1.25 f\) shows the direct variation relationship between the length of fabric and its cost. The variable \(f\) represents the length of the fabric in yards, and \(c\) represents the cost in dollars. Use the equation to answer these questions. a. How much does \(2 \frac{1}{2}\) yards of fabric cost? b. How much fabric can you buy for \(\$ 5\) ? c. What is the cost of each additional yard of fabric?

Short Answer

Expert verified
2.5 yards of fabric costs $3.125, you can buy 4 yards for $5, and each additional yard costs $1.25.

Step by step solution

01

Convert Mixed Number to Fraction

First, we need to convert the mixed number \( 2 \frac{1}{2} \) into an improper fraction.\[2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}\]
02

Calculate the Cost

Using the equation \( c = 1.25f \), substitute \( f = \frac{5}{2} \) to find the cost:\[c = 1.25 \times \frac{5}{2} = \frac{1.25 \times 5}{2} \]Calculate \( 1.25 \times 5 = 6.25 \). So,\[c = \frac{6.25}{2} = 3.125\]The cost of \( 2 \frac{1}{2} \) yards of fabric is \( \$3.125 \).
03

Set Up the Equation for Cost

To determine how much fabric you can buy for \( \$5 \), set \( c = 5 \) in the equation and solve for \( f \):\[5 = 1.25f\]
04

Solve for Length of Fabric

Solve the equation:\[5 = 1.25f\]Divide both sides by \( 1.25 \):\[ f = \frac{5}{1.25} = 4 \]You can buy 4 yards of fabric for \( \$ 5 \).
05

Determine the Cost of Additional Yard

Each additional yard of fabric corresponds to an increase of \( f \) by 1. The equation is:\[c = 1.25 \times 1 = 1.25\]Thus, each additional yard of fabric costs \( \$1.25 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
A linear equation is essentially a mathematical statement where two expressions are set equal, typically involving one or more variables raised only to the first power.
Linear equations can be graphed as straight lines on a coordinate plane. For example, the given direct variation equation, \(c = 1.25f\), is a linear equation. It expresses that the cost \(c\) varies directly with the length \(f\) of fabric in yards.
This specific equation implies a proportional relationship between the two variables.
Key Characteristics of Linear Equations:
  • The graph of a linear equation is always a straight line.
  • No variables in a linear equation are raised to a power higher than one.
  • It often has an "equal" sign, where one side of the equation is an expression equating another expression.
  • Linear equations can often be rearranged to the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, \(m = 1.25\) and \(b = 0\).
Understanding linear equations helps in solving algebraic problems and analyzing the relationships between different quantities.
Solving Algebraic Equations
Solving algebraic equations involves finding the value of the variable that makes the equation true.
For the equation \(c = 1.25f\), you are often required to solve for either \(f\) or \(c\), depending on the known values.
In this example, solving for \(f\) when \(c = 5\) is an important task.
Steps to Solve Algebraic Equations:
  • Identify the variable you need to solve for. In our example, you might solve for \(f\) to determine how many yards you can buy with \$5.
  • Perform operations to isolate the variable. This means dividing, multiplying, adding, or subtracting to get the variable on one side of the equation. In our case, dividing \(5 = 1.25f\) by \(1.25\) gives \(f = 4\).
  • Check your solution by plugging it back into the original equation to ensure both sides are equal.
  • Rearrange and simplify the equation step-by-step if necessary, ensuring that no mistakes are made during calculation.
Knowing these steps helps to tackle various algebraic equations with confidence.
Unit Rate
The concept of a unit rate is pivotal in understanding direct variation scenarios like the equation \(c = 1.25f\).
A unit rate tells you the cost for one unit of an item, which in this case, is one yard of fabric.
The given unit rate is \(1.25\) dollars per yard.
Why Understanding Unit Rate Matters:
  • A unit rate allows easy comparison between different items. You can quickly determine which supplier offers a better deal by comparing unit rates.
  • Unit rates can aid in budgeting and planning purchases.
  • They simplify complex problems by breaking them down to their most basic form 鈥� cost, quantity, distance, time, etc., per single unit.
Calculating the unit rate involves understanding the cost of one unit and is a fundamental skill in real-world applications, such as shopping, cooking, and other daily tasks. By mastering unit rates, students can make informed decisions and enhance their problem-solving skills.

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