Problem 21 - Prealgebra

Question

Write a proportion that could be used to solve for each variable. Then solve. 5 quarts for \(\$ 6.25\) \(d\) quarts for \(\$ 8.75\)

Step-by-Step Solution

Answer

The value of \(d\) is 7 quarts.

Step 1: Set Up the Proportion

We start by setting up a proportion to compare quarts to dollars in each of the given scenarios. For 5 quarts, it's for $6.25. Thus, we have the proportion: \( \frac{5}{6.25} = \frac{d}{8.75} \).

Step 2: Cross-Multiply

To solve the proportion \( \frac{5}{6.25} = \frac{d}{8.75} \), we cross-multiply. This gives us the equation: \( 5 \times 8.75 = 6.25 \times d \).

Step 3: Calculate the Cross Products

Calculate the left-hand product: \( 5 \times 8.75 = 43.75 \). Now, the equation is \( 43.75 = 6.25 \times d \).

Step 4: Solve for \(d\)

To find \(d\), divide both sides of the equation by 6.25: \( d = \frac{43.75}{6.25} \).

Step 5: Simplify the Solution

Calculate \( d = \frac{43.75}{6.25} = 7 \). Therefore, \( d = 7 \) quarts.

Key Concepts

Cross Multiplication

Cross multiplication is a fundamental technique used to solve proportions. It involves multiplying the numerator of one ratio by the denominator of the other ratio, which makes it easier to manipulate and solve the equation. Let's break it down in a simple way.
When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication means doing the following:

• Multiply \( a \) and \( d \) (these are the 'extremes' of the proportion).
• Multiply \( b \) and \( c \) (these are the 'means' of the proportion).

This gives you the equation \( a \times d = b \times c \). This transformation is particularly helpful because it eliminates the fractions, making the equation much more straightforward to solve.
In our original exercise, this technique is utilized to convert the proportion \( \frac{5}{6.25} = \frac{d}{8.75} \) into the equation \( 5 \times 8.75 = 6.25 \times d \). As a result, you have an easy-to-handle equation to find the unknown value.

Solving for a variable

To solve for a variable, like the \(d\) in our problem, you must isolate the variable on one side of the equation. This process often involves operations such as addition, subtraction, multiplication, or division, applied in a way that undoes what has been done to the variable.
In our example, once cross-multiplication has been applied, we get the equation \( 5 \times 8.75 = 6.25 \times d \). To isolate \( d \), you need to remove the coefficient (in this case, 6.25) that is "attached" to \( d \). This is most commonly achieved by dividing both sides of the equation by that coefficient:

• Calculate \( 5 \times 8.75 \), giving \( 43.75 \).
• Divide \( 43.75 \) by \( 6.25 \) to isolate \( d \), resulting in \( d = 7 \).

With the variable isolated, you have effectively solved the equation. This key skill is transferable across different types of mathematical problems and ensures you can systematically arrive at the correct solution.

Word Problems

Word problems are common in math, helping students apply learned skills to real-world situations. To approach these problems effectively, you first need to identify and organize the relevant information.
Here are steps to break down the problem given in the exercise:

• Identify what you're solving for. In our case, it's \(d\), the number of quarts.
• Write down the known quantities, such as 5 quarts for \(\\(6.25\) and \(d\) quarts for \(\\)8.75\).
• Set up a relationship or proportion between the known values and the unknown variable: \(\frac{5}{6.25} = \frac{d}{8.75}\).

Once the mathematical expression mirrors the word problem, it's a matter of applying algebraic methods (such as cross-multiplication and solving for a variable) to find the solution.
Understanding how to translate words into numbers and operations is essential in solving word problems, providing a crucial bridge between practical scenarios and mathematical reasoning.