Problem 30 - Prealgebra

Question

Determine which is less expensive per can, a 6 -pack of soda for \(\$ 2.20\) or a 12 -pack of soda for \(\$ 4.25 .\) Explain.

Step-by-Step Solution

Answer

A 12-pack is less expensive per can than a 6-pack.

Step 1: Calculate the Cost per Can for 6-Pack

To find the cost per can, divide the total cost of the 6-pack by the number of cans. Let total cost be \$2.20 and the number of cans be 6. So, \( \text{Cost per can} = \frac{2.20}{6} \approx 0.3667 \).

Step 2: Calculate the Cost per Can for 12-Pack

Similarly, divide the total cost of the 12-pack by the number of cans. Let the total cost be \$4.25 and number of cans be 12. So, \( \text{Cost per can} = \frac{4.25}{12} \approx 0.3542 \).

Step 3: Compare the Costs per Can

Now compare the calculated cost per can of both packages. For the 6-pack, it is approximately \\(0.3667, and for the 12-pack, it is approximately \\)0.3542. Since \(0.3542 < 0.3667\), the cost per can is less in a 12-pack.

Key Concepts

Understanding Division in Math

Division in mathematics is a fundamental concept where a number, known as the dividend, is split into equal parts by another number, known as the divisor. The result of this operation is called the quotient. In the context of the soda pack problem, division helps us determine how much each can costs by distributing the overall pack cost evenly across the number of cans.
For example, when dividing the total cost of a 6-pack, which is \(2.20\) dollars, by the number of cans, which is 6, we perform the following operation:

• Dividend: 2.20 (total cost of the pack)
• Divisor: 6 (number of cans in the pack)

The cost per can, or quotient, is calculated as \(\frac{2.20}{6} \approx 0.3667\). It's this division that gives us valuable insights into how much each unit (or can) costs, allowing us to make cost-effective decisions.

Insights on Cost Analysis

Cost analysis is an essential practice for making informed purchasing decisions. It involves comparing unit prices to determine which option offers more value for money. In our soda scenario, this analysis involves breaking down the cost per can in two different pack sizes: the 6-pack and the 12-pack. By calculating the unit price, you can determine which package is less expensive per can.
The equation was:

• 6-pack: \(\frac{2.20}{6} = 0.3667\) dollars per can
• 12-pack: \(\frac{4.25}{12} = 0.3542\) dollars per can

After identifying that \(0.3542\) is less than \(0.3667\), it becomes clear that the 12-pack is more economical per can. Through cost analysis, consumers can maximize savings and make smarter purchasing decisions.

Working with Fractions in Math

Fractions are an integral part of mathematics and are commonly used to represent parts of a whole. In cost analysis, fractions are vital when expressing unit prices. When dividing the cost of a soda pack by the number of cans, the outcome is a fraction of the dollar per can. Understanding fractions helps in visualizing and interpreting such calculations.
In the soda pack example, the division resulted in:

• 6-pack: \(\frac{2.20}{6} = 0.3667\), meaning each can costs about 37 cents
• 12-pack: \(\frac{4.25}{12} = 0.3542\), or roughly 35 cents per can

When dealing with these figures, it's important to understand that \(0.3667\) and \(0.3542\) can be rewritten as fractions of a dollar. This representation helps in understanding the concept of unit prices more clearly and assists in making better economic choices. With fractions, you can precisely analyze and compare costs to ensure you're getting the best deal.