Problem 74 - Intermediate Algebra

Question

Use a proportion to solve each problem. Shopping. A recipe for guacamole dip calls for 5 avocados. If they are advertised at 3 for \(\$ 1.98\), what will 5 avocados cost?

Step-by-Step Solution

Answer

The cost for 5 avocados is \(\$3.30\).

Step 1: Calculate the Cost of One Avocado

First, determine the cost for one avocado. Since the price for 3 avocados is \(\\( 1.98\), divide \(1.98\) by 3. \[ \text{Cost per avocado} = \frac{1.98}{3} = 0.66 \] This means each avocado costs \(\\) 0.66\).

Step 2: Set Up the Proportion

Now, set up a proportion to find the cost of 5 avocados. Let \( x \) be the cost for 5 avocados.\[ \frac{1 \, \text{avocado}}{0.66} = \frac{5 \, \text{avocados}}{x} \] Cross multiply to solve for \( x \).

Step 3: Calculate the Total Cost

Use the cross-multiplication formula from the proportion in Step 2.\[ 1 \times x = 5 \times 0.66 \] Simplify this to get the total cost.\[ x = 3.30 \] Hence, 5 avocados will cost \(\$ 3.30\).

Key Concepts

Cost Calculation

Calculating the cost of items is a fundamental skill, especially when budgeting for groceries or following a recipe. To determine the cost of a specific number of items, such as avocados, it's important to start by understanding the price per item. This process begins with determining how much one item costs when the price for a group of items is provided.
For example, if 3 avocados cost $1.98, you calculate the price per avocado by dividing the total cost by the number of avocados.

• Calculate the price of one avocado: \[\text{Price per avocado} = \frac{1.98}{3} = 0.66\]
• Thus, each avocado costs 0.66 dollars.

Understanding the cost of a single item allows you to scale this cost up or down depending on your needs, which is essential for making accurate purchases.

Cross Multiplication

Cross multiplication is a powerful mathematical method used to solve proportions easily. It involves multiplying across the proportion, which helps in finding an unknown variable. When you have a ratio set up like \[\frac{1 \text{ avocado}}{0.66} = \frac{5 \text{ avocados}}{x}\]You cross multiply to find the value of \( x \). This means you multiply 1 by \( x \) and 5 by 0.66, then equate the two results.

• Form your equation: \[1 \times x = 5 \times 0.66\]
• Solve for \( x \) by calculating the multiplication on the right side.
• Finally, equate and solve for \( x \).\[x = 3.30\]

Following these steps, you solve the equation, efficiently finding the amount or cost that the proportion represents. This method provides clarity and an understandable way to solve proportion problems.

Proportion Setup

Setting up the proportion correctly is a crucial step when solving problems involving ratios and proportions. A proportion represents two equal ratios and is typically set up when you want to find an unknown variable across two related quantities.
To set up a proportion, you need to:

• Identify the known ratios and what you need to find.
• Decide what your unknown variable will be, which is often the missing part of the comparison.
• Write the proportion as two fractions set equal to each other, with the unknown variable in one of the fractions.

In our avocado example, we knew the cost of one and needed the cost of five. We set our proportion by comparing the number of avocados to their total cost:\[\frac{1 \text{ avocado}}{0.66} = \frac{5 \text{ avocados}}{x}\]Correctly setting up this proportion allowed us to easily use cross multiplication to solve for the unknown, ensuring an accurate cost estimation.