Problem 53 - Elementary Algebra

Question

Margaret bought two cases of soda at the local discount store for \(\$ 23.52\). If each case contained 24 bottles, how much did she spend per bottle?

Step-by-Step Solution

Answer

The cost per bottle is $0.49.

Step 1: Identify the total cost for all bottles

Margaret spent \(\$ 23.52\) for two cases of soda. This represents the total amount for all bottles.

Step 2: Calculate the total number of bottles

Each case contains 24 bottles, and Margaret bought 2 cases. Therefore, the total number of bottles is calculated as follows: \[ 24 \text{ bottles/case} \times 2 \text{ cases} = 48 \text{ bottles} \]

Step 3: Calculate the cost per bottle

Divide the total cost by the total number of bottles to find the cost per bottle: \[ \frac{\\( 23.52}{48 \text{ bottles}} = \\) 0.49 \text{ per bottle} \]

Key Concepts

Division

Division is a fundamental mathematical operation that can be thought of as distributing a number into equal parts. In the context of unit price calculation, division helps us determine how much one item costs when you know the total cost and the number of items.

For example, in calculating the cost per bottle of soda, we know the total cost Margaret paid (\(23.52\)). By dividing this amount by the total number of bottles (48), we find the cost per single bottle. Here, \(23.52 \div 48 = 0.49\), which gives the unit price of one soda bottle.

• The divisor is the number of bottles (48).
• The dividend is the total cost (\(23.52\)).
• The quotient is the cost per bottle (\(0.49\)).

Understanding division in real-world contexts like pricing helps clarify its application, revealing how essential it is to discern the value of individual items from a bulk purchase.

Word Problem

Word problems are math exercises presented in a narrative format. They require interpretation and mathematical operations to solve a problem within a story or real-life setting.

In Margaret's soda case, the word problem asks us to determine the cost per bottle from given information: the total cost and quantity of bottles purchased. Solving these involves:

• Reading carefully to grasp the scenario: two cases of soda at total cost \(23.52\).
• Extracting relevant numerical data: 24 bottles per case, 2 cases in total.
• Using appropriate mathematical operations (multiplication and division) to find the unit price.

Word problems are valuable as they reflect real-world issues, improving problem-solving skills by encouraging students to translate situations into mathematical expressions.

Multiplication

Multiplication is a mathematical operation that assists in finding the total number of items when you know the number of groups and the size of each group. It is essential in computing the total quantity in scenarios involving repeated addition.

In this problem, multiplication calculates how many bottles Margaret has in all her soda cases. Since each case contains 24 bottles and Margaret has 2 cases, we multiply to find the total: \(24 \times 2 = 48\).

• This shows that multiplication simplifies adding 24 bottles twice.
• It quickly gives the total when dealing with identical groups.

Understanding multiplication supports figuring out larger quantities efficiently and is a cornerstone of arithmetic operations used in everyday shopping decisions.