Problem 29 - Elementary Algebra Within Reach

Question

Determine which product has the lower unit price. (a) An 18-ounce jar of peanut butter for \(\$ 1.92\) (b) A 28 -ounce jar of peanut butter for \(\$ 3.18\)

Step-by-Step Solution

Answer

The 18-ounce jar of peanut butter for $1.92 (Product A) has the lower unit price.

Step 1: Calculate Unit Price for Product A

Unit price for product A is calculated by dividing the total cost of the product by its total mass. The total cost of product A is \$1.92, and it weighs 18-ounces. The unit price (in dollars per ounce) can therefore be calculated as \(\$1.92/18 = 0.1067\).

Step 2: Calculate Unit Price for Product B

The unit price for Product B is similarly calculated as the total cost of product B divided by its total mass. The total cost of product B is \$3.18, and it weighs 28-ounces. Hence, the unit price can be calculated as \(\$3.18/28 = 0.1136\).

Step 3: Compare the Unit Prices

Compare the unit prices for Product A and Product B to determine which product has the lower unit price. From Step 1 and Step 2, it can be seen that product A has a unit price of \$0.1067 per ounce and product B has a unit price of \$0.1136 per ounce. The lower of the two is \$0.1067, which corresponds to product A. Hence product A has the lower unit price.

Key Concepts

Cost Comparison

The concept of cost comparison is crucial when trying to get the best deal on purchases. It involves looking at the price of different items and determining which one offers the most value for money.

This kind of analysis is often used in everyday life, such as grocery shopping, where we compare prices of similar products to decide which one to buy. In the context of our exercise, we compared the cost per ounce of peanut butter in two differently sized jars to see which one costs less per ounce.

To conduct a cost comparison:

• Determine the total cost of each item.
• Find out the quantity or size of each item.
• Calculate the unit price by dividing the total cost by the quantity for each item.
• Compare the unit prices to see which item is more cost-effective.

Through this process, we determined that the 18-ounce jar has the lower unit price, making it the more economical choice per ounce, despite the actual price being lower than the 28-ounce jar.

Division in Algebra

When we talk about division in algebra, we're referring to the mathematical process of finding out how many times one number is contained within another. It is one of the four basic arithmetic operations and crucial for a variety of calculations.

In the provided exercise, division helps us find the unit prices by dividing the total cost by the quantity. For instance, when calculating the unit price for product A, \text{\(\frac{1.92}{18}\)} illustrates the use of the division in algebra to distribute the total cost over each ounce. Algebraically, it could be represented as \text{\(u = \frac{p}{q}\)}, where \(u\) is the unit price, \(p\) is the total price, and \(q\) is the quantity.

This operation is essential in algebra as it allows us to simplify expressions, solve equations, and understand how different values relate to one another. Knowing how to divide properly can uncover relationships between numbers that aren't obvious at first glance.

Practical Applications of Algebra

Algebra is not just an abstract branch of mathematics, but has countless practical applications in real life. One of the most direct applications is in making informed financial decisions, such as the cost comparison of products.

Algebra equips us with the tools to create formulas and solve for unknowns, which can be applied in various real-world scenarios, from budgeting to constructing buildings. In the solution provided, algebra is applied to deduce which size of peanut butter jar provides the best value. This is just one example that showcases how algebra can be used to make everyday life more economical.

Besides financial literacy, algebra is also used in fields such as engineering, science, economics, and technology. By learning algebra, students equip themselves with a powerful toolset that can be translated into smarter decisions, better problem-solving skills, and a stronger understanding of the world around them.